Two-parameter Quantum Group of Exceptional Type G_{2} and Lusztig’s Symmetries

Naihong Hu^{⋆} and Qian Shi

Abstract. We give the defining structure of two-parameter quantum group of type G_{2} defined over a field ℚ(r, s) (with r ≠ s), and prove the Drinfel’d double structure as its upper and lower triangular parts, extending an earlier result of [BW1] in type A case and a recent result of [BGH1] in type B, C, D cases. We further discuss the Lusztig’s ℚ-isomorphisms from U_{r,s}(G_{2}) to its associated object U_{s^{−1},r^{−1}}(G_{2}), which give rise to the usual Lusztig’s symmetries defined on U_{q}(G_{2}) only when r = s^{−1} = q (this fact also reflects the distinguished difference between our newly defined two-parameter object and the standard Drinfel’d-Jimbo quantum groups). Especially, arising from this discussion, some interesting (r, s)-identities held in U_{r,s}(G_{2}) are derived.

1. Two-parameter Quantum Group U_{r,s}(G_{2})

Let 𝕂 = ℚ(r, s) be a field of rational functions with two indeterminates r, s. Let Φ be a finite root system of G_{2} with Π a base of simple roots, which is

a subset of a Euclidean space E = ℝ^{3} with an inner product ( , ). Let ϵ_{1}, ϵ_{2}, ϵ_{3} denote an orthonormal basis of E, then Π = {α_{1} = ϵ_{1} − ϵ_{2}, α_{2} = ϵ_{2} + ϵ_{3} − 2ϵ_{1}} and Φ = ±{α_{1}, α_{2}, α_{2} + α_{1}, α_{2} + 2α_{1}, α_{2} + 3α_{1}, 2α_{2} + 3α_{1}}. In this case, we set r_{1} = r^{\frac{(α_{1}, α_{1})}{2}} = r, r_{2} = r^{\frac{(α_{2}, α_{2})}{2}} = r^{3} and s_{1} = s^{\frac{(α_{1}, α_{1})}{2}} = s, s_{2} = s^{\frac{(α_{2}, α_{2})}{2}} = s^{3}.

We begin by giving the definition of two-parameter quantum group of type G_{2}, which is new.

Definition 1.1. Let U = U_{r, s}(G_{2}) be the associative algebra over ℚ(r, s) generated by symbols e_{i}, f_{i}, ω_{i}^{±1}, ω_{i}^{′±1} (1 ≤ i ≤ 2) subject to the relations

(G1) [ ω_{i}^{±1}, ω_{j}^{±1} ] = [ ω_{i}^{±1}, ω_{j}^{′±1} ] = [ ω_{i}^{′±1}, ω_{j}^{′±1} ] = 0, ω_{i}ω_{i}^{−1} = 1 = ω′_{j}ω_{j}^{′−1}. (G2) ω_{1} e_{1} ω_{1}^{−1} = (rs^{−1}) e_{1}, ω_{1} f_{1} ω_{1}^{−1} = (r^{−1}s) f_{1},

ω_{1} e_{2} ω_{1}^{−1} = s^{3} e_{2}, ω_{1} f_{2} ω_{1}^{−1} = s^{−3} f_{2},

ω_{2} e_{1} ω_{2}^{−1} = r^{−3} e_{1}, ω_{2} f_{1} ω_{2}^{−1} = r^{3} f_{1},
ω_{2} e_{2} ω_{2}^{−1} = (r^{3}s^{−3}) e_{2}, ω_{2} f_{2} ω_{2}^{−1} = (r^{−3}s^{3}) f_{2}.

1991 Mathematics Subject Classification. Primary 17B37, 81R50; Secondary 17B35. Key words and phrases. 2-parameter quantum group, Hopf (skew-)dual pairing, Drinfel’d

quantum double, Lusztig’s symmetries.

⋆N.H., Corresponding Author, supported in part by the NNSF (Grant 10431040), the TRAPOYT and the FUDP from the MOE of China, the SRSTP from the STCSM, the Shanghai Priority Academic Discipline from the SMEC.

(G3)  ω′_{1} e_{1} ω_{1}^{′−1} = (r^{−1}s) e_{1},  ω′_{1} f_{1} ω_{1}^{′−1} = (rs^{−1}) f_{1},
ω′_{1} e_{2} ω_{1}^{′−1} = r^{3} e_{2}, ω′_{1} f_{2} ω_{1}^{′−1} = r^{−3} f_{2},
ω′_{2} e_{1} ω_{2}^{′−1} = s^{−3} e_{1}, ω′_{2} f_{1} ω_{2}^{′−1} = s^{3} f_{1},
ω′_{2} e_{2} ω_{2}^{′−1} = (r^{−3}s^{3}) e_{2}, ω′_{2} f_{2} ω_{2}^{′−1} = (r^{3}s^{−3}) f_{2}.

(G4) For 1 ≤ i, j ≤ 2, we have

[ e_{i}, f_{j} ] = δ_{ij} \frac{ω_{i} − ω′_{i}}{r_{i} − s_{i}} .

(G5) We have the following (r, s)-Serre relations:

e_{2}^{2}e_{1} − (r^{−3} + s^{−3}) e_{2}e_{1}e_{2} + (rs)^{−3} e_{1}e_{2}^{2} = 0, (G5)_{1}

\begin{matrix}e_{1}^{4}e_{2} − (r+ & s)(r^{2} + s^{2}) e_{1}^{3}e_{2}e_{1} + rs(r^{2} + s^{2})(r^{2} + rs + s^{2}) e_{1}^{2}e_{2}e_{1}^{2} \\ − & (rs)^{3}(r + s)(r^{2} + s^{2}) e_{1}e_{2}e_{1}^{3} + (rs)^{6}e_{2}e_{1}^{4} = 0. \end{matrix} (G5)_{2}

(G6) We have the following (r, s)-Serre relations:

f_{1}f_{2}^{2} − (r^{−3} + s^{−3}) f_{2}f_{1}f_{2} + (rs)^{−3} f_{2}^{2}f_{1} = 0, (G6)_{1}

\begin{matrix}f_{2}f_{1}^{4} − (r+ & s)(r^{2} + s^{2}) f_{1}f_{2}f_{1}^{3} + rs(r^{2} + s^{2})(r^{2} + rs + s^{2}) f_{1}^{2}f_{2}f_{1}^{2} \\ − & (rs)^{3}(r + s)(r^{2} + s^{2}) f_{1}^{3}f_{2}f_{1} + (rs)^{6} f_{1}^{4}f_{2} = 0. \end{matrix} (G6)_{2}

It is straightforward to check

Proposition 1.2. The algebra U_{r,s}(G_{2}) is a Hopf algebra under the comultiplication, the counit and the antipode below:

Δ(ω_{i}^{±1}) = ω_{i}^{±1} ⊗ ω_{i}^{±1}, Δ(ω′_{i}^{±1}) = ω′_{i}^{±1} ⊗ ω′_{i}^{±1},
Δ(e_{i}) = e_{i} ⊗ 1 + ω_{i} ⊗ e_{i}, Δ(f_{i}) = 1 ⊗ f_{i} + f_{i} ⊗ ω′_{i},
ε(ω_{i}^{±}) = ε(ω′_{i}^{±1}) = 1, ε(e_{i}) = ε(f_{i}) = 0,
S(ω_{i}^{±1}) = ω_{i}^{∓1}, S(ω′_{i}^{±1}) = ω′_{i}^{∓1},
S(e_{i}) = −ω_{i}^{−1}e_{i}, S(f_{i}) = −f_{i} ω′_{i}^{−1}.
□

Remark 1.3. (I) When r = q = s^{−1}, the Hopf algebra U_{r, s}(G_{2}) modulo the Hopf ideal generated by elements ω′_{i} − ω_{i}^{−1}(1 ≤ i ≤ 2), is just the quantum group U_{q}(G_{2}) of Drinfel’d-Jimbo type.

(II) In any Hopf algebra H, there exist the left-adjoint and the right-adjoint action defined by its Hopf algebra structure as follows

ad_{l} a (b) = ∑_{(a)} a_{(1)} b S(a_{(2)}), ad_{r} a (b) = ∑_{(a)} S(a_{(1)}) b a_{(2)},

where Δ(a) = ∑_{(a)} a_{(1)} ⊗ a_{(2)} ∈ H ⊗ H, for any a, b ∈ H.

From the viewpoint of adjoint actions, the (r, s)-Serre relations (G5), (G6) take the simplest forms:

(ad_{l} e_{i})^{1−a_{ij}} (e_{j}) = 0, for any i ≠ j,
(ad_{r} f_{i})^{1−a_{ij}} (f_{j}) = 0, for any i ≠ j.

TWO-PARAMETER QUANTUM GROUP OF EXCEPTIONAL TYPE G_{2} 3

2. Drinfel’d Quantum Double

Definition 2.1. A (Hopf) dual pairing of two Hopf algebras A and U (see [BGH1], or [KS]) is a bilinear form ⟨ , ⟩ : U × A ⟶ 𝕂 such that

⟨f, 1_{A}⟩ = ε_{U}(f), ⟨1_{U}, a⟩ = ε_{A}(a), (1)

⟨f, a_{1}a_{2}⟩ = ⟨△_{U}(f), a_{1} ⊗ a_{2}⟩, ⟨f_{1}f_{2}, a⟩ = ⟨f_{1} ⊗ f_{2}, △_{A}(a)⟩, (2)

for all f, f_{1}, f_{2} ∈ U, and a, a_{1}, a_{2} ∈ A, where ε_{U} and ε_{A} denote the counits of U and A, respectively, and △_{U} and △_{A} are their comultiplications.

A direct consequence of the defining properties above is

⟨S_{U}(f), a⟩ = ⟨f, S_{A}(a)⟩, f ∈ U, a ∈ A,

where S_{U}, S_{A} denote the respective antipodes of U and A.

Definition 2.2. A bilinear form ⟨ , ⟩ : U × A ⟶ 𝕂 is called a skew-dual pairing of two Hopf algebras A and U (see [BGH1]) if ⟨ , ⟩ : U^{cop} × A ⟶ 𝕂 is a Hopf dual pairing of A and U^{cop}, where U^{cop} is the Hopf algebra having the opposite comultiplication to U, and S_{U^{cop}} = S_{U}^{−1} is invertible.

Let B = B(G_{2}) (resp. B′ = B′(G_{2})) denote the Hopf subalgebra of U_{r, s}(G_{2}), which is generated by e_{j}, ω_{j}^{±1} (resp. f_{j}, ω_{j}^{′±1}) with 1 ≤ j ≤ 2.

Proposition 2.3. There exists a unique skew-dual pairing ⟨ , ⟩ : B′ × B ⟶ ℚ(r, s) of the Hopf subalgebras B and B′, such that

⟨f_{i}, e_{j}⟩ = δ_{ij} \frac{1}{s_{i} − r_{i}} , (1 ≤ i, j ≤ 2), (3)

⟨ω′_{1}, ω_{1}⟩ = rs^{−1}, (4_{1})

⟨ω′_{1}, ω_{2}⟩ = r^{−3}, (4_{2})

⟨ω′_{2}, ω_{1}⟩ = s^{3}, (4_{3})

⟨ω′_{2}, ω_{2}⟩ = r^{3}s^{−3}, (4_{4})

⟨ω_{i}^{′±1}, ω_{j}^{−1}⟩ = ⟨ω_{i}^{′±1}, ω_{j}⟩^{−1} = ⟨ω′_{i}, ω_{j}⟩^{∓1}, (1 ≤ i, j ≤ 2), (5)

and all other pairs of generators are 0. Moreover, we have ⟨S(a), S(b)⟩ = ⟨a, b⟩ for a ∈ B′, b ∈ B.

Proof. Since any skew-dual pairing of bialgebras is determined by the values on generators, the uniqueness assertion is clear. We proceed to prove the existence of the pairing.

We begin by defining a bilinear form ⟨ , ⟩ : B^{′cop} × B → ℚ(r, s) first on the generators satisfying (3), (4_{n}), and (5). Then extend it to a bilinear form on B^{′cop} × B by requiring that (1) and (2) hold for △_{B^{′cop}} = △_{B′}^{op}. We will verify that the relations in B and B′ are preserved, ensuring that the form is well-defined and then is a dual pairing of B and B^{′cop} by definition.

Firstly, it is direct to check that the bilinear form preserves all the relations among the ω_{i}^{±1} in B and the ω_{i}^{′±1} in B′. Secondly, we notice that the structure constants (4_{n}) ensure the compatibility of the form defined above with those relations of (G2) and (G3) in B or B′ respectively. We are left to verify that the form preserves the (r, s)-Serre relations in B and B′. It suffices to show that the form on B^{′cop} × B preserves the (r, s)-Serre relations in B because the verification for B^{′cop} is similar.

First, let us show that the form preserves the (r, s)-Serre relation of degree 2 in B, that is,

⟨X, e_{2}^{2}e_{1} − (r^{−3} + s^{−3}) e_{2}e_{1}e_{2} + r^{−3}s^{−3} e_{1}e_{2}^{2} ⟩ = 0,

where X is any word in the generators of B′. It suffices to consider three monomials: X = f_{2}^{2}f_{1}, f_{2}f_{1}f_{2}, f_{1}f_{2}^{2}. However, in the degree 2’s situation for type G_{2}, its proof is the same as that of type C_{2} (see [BGH1, (7C) and thereafter]).

Next, we verify that the (r, s)-Serre relation of degree 4 in B is preserved by the form, that is, to show the identity:

⟨X, e_{1}^{4}e_{2} − (r + s)(r^{2} + s^{2}) e_{1}^{3}e_{2}e_{1} + rs(r^{2} + s^{2})(r^{2} + rs + s^{2}) e_{1}^{2}e_{2}e_{1}^{2}
−(rs)^{3}(r + s)(r^{2} + s^{2}) e_{1}e_{2}e_{1}^{3} + (rs)^{6} e_{2}e_{1}^{4} ⟩ = 0,

where X is any word in the generators of B′. By definition, the left-hand side of the above identity equals

\begin{aligned}⟨△^{(4)}(X), & e_{1} ⊗ e_{1} ⊗ e_{1} ⊗ e_{1} ⊗ e_{2} \\ & − (r + s)(r^{2} + s^{2}) e_{1} ⊗ e_{1} ⊗ e_{1} ⊗ e_{2} ⊗ e_{1} \\ & + rs(r^{2} + s^{2})(r^{2} + rs + s^{2}) e_{1} ⊗ e_{1} ⊗ e_{2} ⊗ e_{1} ⊗ e_{1} \\ & − (rs)^{3}(r + s)(r^{2} + s^{2}) e_{1} ⊗ e_{2} ⊗ e_{1} ⊗ e_{1} ⊗ e_{1} \\ & + (rs)^{6} e_{2} ⊗ e_{1} ⊗ e_{1} ⊗ e_{1} ⊗ e_{1} ⟩, \end{aligned} ; (∗)

where △ in the left-hand side of the pairing ⟨ , ⟩ indicates △_{B′}^{op}. In order for any one of these terms to be nonzero, X must involve exactly four f_{1} factors, one f_{2} factor, and arbitrarily many ω_{j}^{′±1} factors (j = 1, 2).

It suffices to consider the following five key cases:

(i) X = f_{1}^{4}f_{2}, we have

△^{(4)}(X) = ( ω′_{1} ⊗ ω′_{1} ⊗ ω′_{1} ⊗ ω′_{1} ⊗ f_{1} + ω′_{1} ⊗ ω′_{1} ⊗ ω′_{1} ⊗ f_{1} ⊗ 1

+ ω′_{1} ⊗ ω′_{1} ⊗ f_{1} ⊗ 1 ⊗ 1 + ω′_{1} ⊗ f_{1} ⊗ 1 ⊗ 1 ⊗ 1 + f_{1} ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1 )^{4}·

( ω′_{2} ⊗ ω′_{2} ⊗ ω′_{2} ⊗ ω′_{2} ⊗ f_{2} + ω′_{2} ⊗ ω′_{2} ⊗ ω′_{2} ⊗ f_{2} ⊗ 1

+ ω′_{2} ⊗ ω′_{2} ⊗ f_{2} ⊗ 1 ⊗ 1 + ω′_{2} ⊗ f_{2} ⊗ 1 ⊗ 1 ⊗ 1 + f_{2} ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1 ).

Expanding △^{(4)}(X), we get 120 relevant terms having a non-zero contribution to (∗), together with their pairing-values listed in TABULAR 1 of Appendix, where a = ⟨f_{1}, e_{1}⟩^{4}⟨f_{2}, e_{2}⟩, x = ⟨ω′_{1}, ω_{1}⟩, x̄ = ⟨ω′_{1}, ω_{2}⟩, y = ⟨ω′_{2}, ω_{1}⟩. Thus,

the value of (∗) = (summing values in column 1)

− (summing values in column 2) · (r + s)(r^{2} + s^{2})

+ (summing values in column 3) · rs(r^{2} + s^{2})(r^{2} + rs + s^{2})

− (summing values in column 4) · (rs)^{3}(r + s)(r^{2} + s^{2})

+ (summing values in column 5) · (rs)^{6}.

If we sum all the pairing-values listed in each column of TABULAR 1, then we see that the paring-value of (∗) equals

a(1 + 3x + 5x^{2} + 6x^{3} + 5x^{4} + 3x^{5} + x^{6}) · [ 1 − (r + s)(r^{2} + s^{2})x̄

+ rs(r^{2} + s^{2}) · (r^{2} + rs + s^{2})x̄^{2} − (rs)^{3}(r + s)(r^{2} + s^{2})x̄^{3} + (rs)^{6}x̄^{4} ]

= a(1 + 3x + 5x^{2} + 6x^{3} + 5x^{4} + 3x^{5} + x^{6})(1−r^{3}x̄)(1−r^{2}sx̄)(1−rs^{2}x̄)(1−s^{3}x̄)

= 0. (because x̄ = ⟨ω′_{1}, ω_{2}⟩ = r^{−3} )

(ii) X = f_{2}f_{1}^{4}. By calculation, we get 120 relevant terms of △^{(4)}(X) in (∗) and their pairing-values listed as in TABULAR 2 of Appendix.

If we sum all the pairing-values listed in each column of TABULAR 2, then we see that the paring-values of (∗) equals

a(1 + 3x + 5x^{2} + 6x^{3} + 5x^{4} + 3x^{5} + x^{6}) · [ y^{4} − (s^{3} + rs^{2} + r^{2}s + r^{3}) · y^{3}

+ rs(s^{4} + rs^{3} + 2r^{2}s^{2} + r^{3}s + r^{4}) · y^{2} − (rs)^{3}(s^{3} + rs^{2} + r^{2}s + r^{3}) · y + (rs)^{6} ]

= a(1 + 3x + 5x^{2} + 6x^{3} + 5x^{4} + 3x^{5} + x^{6})(y − r^{3})(y − r^{2}s)(y − rs^{2})(y − s^{3})

= 0. (because y = ⟨ω′_{2}, ω_{1}⟩ = s^{3})

(iii) X = f_{1}^{2}f_{2}f_{1}^{2}. By calculation, we get 120 relevant of △^{(4)}(X) in (∗) and their pairing-values listed as in TABULAR 3 of Appendix.

If we sum all the pairing-values listed in each column of TABULAR 3, then we see that the paring-values of (∗) equals

ay^{2}(1 + 3x + 5x^{2} + 6x^{3} + 5x^{4} + 3x^{5} + x^{6})

− ay(s^{3} + rs^{2} + r^{2}s + r^{3}) · (1 + 3x + 4x^{2} + 3x^{3} + x^{4}) · (1 + x̄yx^{2})

+ ars(s^{4} + rs^{3} + 2r^{2}s^{2} + r^{3}s + r^{4}) · [ 1 + 2x + x^{2}

+ x̄xy(1 + 4x + 6x^{2} + 4x^{3} + x^{4}) + x̄^{2}y^{2}x^{4} · (1 + 2x + x^{2}) ]

− a(rs)^{3}(s^{3} + rs^{2} + r^{2}s + r^{3}) · (1 + 3x + 4x^{2} + 3x^{3} + x^{4}) · (x̄ + x̄^{2}x^{2}y)

+ (rs)^{6}ax̄^{2}(1 + 3x + 5x^{2} + 6x^{3} + 5x^{4} + 3x^{5} + x^{6})

= 2ar^{−1}s^{−1} · rs · (s^{4} + 3rs^{3} + 4r^{2}s^{2} + 3r^{3}s + r^{4}) · (s^{2} + r^{2})

+ 2ar^{−1}s^{−1} · (s^{4} + 3rs^{3} + 4r^{2}s^{2} + 3r^{3}s + r^{4}) · (s^{2} + rs + r^{2}) · (s^{2} + r^{2})

− 2ar^{−1}s^{−1} · (s^{4} + 3rs^{3} + 4r^{2}s^{2} + 3r^{3}s + r^{4}) · (s^{2} + r^{2}) · (r + s)^{2}

= 2ar^{−1}s^{−1} · (s^{4} + 3rs^{3} + 4r^{2}s^{2} + 3r^{3}s + r^{4}) · (s^{2} + r^{2}) · (r + s)^{2}

− 2ar^{−1}s^{−1} · (s^{4} + 3rs^{3} + 4r^{2}s^{2} + 3r^{3}s + r^{4}) · (s^{2} + r^{2}) · (r + s)^{2}

= 0. (because x = ⟨ω′_{1}, ω_{1}⟩ = rs^{−1}, x̄ = ⟨ω′_{1}, ω_{2}⟩ = r^{−3}, y = ⟨ω′_{2}, ω_{1}⟩ = s^{3}).

(iv) X = f_{1}^{3}f_{2}f_{1}. By calculation, we get 120 relevant terms of △^{(4)}(X) in (∗) and their pairing-values listed as in TABULAR 4 of Appendix.

6 HU AND SHI

If we sum all the pairing-values listed in each column of TABULAR 4, then we see that the paring-values of (∗) equals ay(1+3x+5x^{2}+6x^{3}+5x^{4}+3x^{5}+x^{6}) − a(r+s)(r^{2}+s^{2})[ 1+2x+2x^{2}+x^{3}

+ xx̄y(1+3x+5x^{2}+5x^{3}+3x^{4}+x^{5}) ] + ars(r^{2}+s^{2})(r^{2}+rs+s^{2})·

[ x̄(1+3x+4x^{2}+3x^{3}+x^{4}) + x̄^{2}yx^{2}(1+3x+4x^{2}+3x^{3}+x^{4}) ] − a(rs)^{3}(r+s)(r^{2}+s^{2})[ x̄^{2}(1+3x+5x^{2}+5x^{3}+3x^{4}+x^{5}) + x̄^{3}y(x^{3}+2x^{4}+2x^{5}+x^{6}) ] + ar^{6}s^{6}x̄^{3}(1+3x+5x^{2}+6x^{3}+5x^{4}+3x^{5}+x^{6})

= as^{−3}(r+s)^{2}(r^{2}+s^{2})(r^{2}+rs+s^{2})

− as^{−3}r^{−3}(r+s)^{2}(r^{2}+s^{2})(r^{2}+rs+s^{2})(r^{2}s+2r^{3}+rs^{2})

+ as^{−3}r^{−3}(r^{2}+s^{2})(r^{2}+rs+s^{2})^{2}(r+s)^{3} − ar^{−3}s^{−2}(r+s)^{2}(r^{2}+s^{2})(r^{2}+rs+2s^{2}) + ar^{−3}(r+s)^{2}(r^{2}+s^{2})(r^{2}+rs+s^{2})

= a(r+s)^{2}(r^{2}+s^{2})(r^{2}+rs+s^{2})[s^{−3} − s^{−3}r^{−3}(r^{2}s+rs^{2}+2r^{3})

+ r^{−3}s^{−3}(r^{2}+rs+s^{2})(r+s) − r^{−3}s^{−2}(r^{2}+rs+2s^{2}) + r^{−3}] = 0.

(v) X = f_{1}f_{2}f_{1}^{3}. By calculation, we get 120 relevant terms of △^{(4)}(X) in (∗) and their pairing-values of (∗) listed as in TABULAR 5 of Appendix.

If we sum all the pairing-values in TABULAR 5, then we see that the paringvalue of (∗) equals

ay^{3}(1+3x+5x^{2}+6x^{3}+5x^{4}+3x^{5}+x^{6}) − a(r+s)(r^{2}+s^{2})[ x̄y^{3}x^{3}(1+2x

+2x^{2}+x^{3})+y^{2}(1+3x+5x^{2}+5x^{3}+3x^{4}+x^{5}) ] + ars(r^{2}+s^{2})(r^{2}+rs+s^{2})·

[ y(1+3x+4x^{2}+3x^{3}+x^{4}) + x̄y^{2}x^{2}(1+3x+4x^{2}+3x^{3}+x^{4}) ]

− a(rs)^{3}(r+s)(r^{2}+s^{2})[ x̄yx(1+3x+5x^{2}+5x^{3}+3x^{4}+x^{5})+1+2x+2x^{2}+x^{3} ] + ar^{6}s^{6}x̄(1+3x+5x^{2}+6x^{3}+5x^{4}+3x^{5}+x^{6})

= a(r+s)^{3}(r^{2}+s^{2})(r^{2}+rs+s^{2}) − as(r+s)^{2}(r^{2}+s^{2})(r^{2}+rs+s^{2})(r^{2}+rs+2s^{2})

+ a(r^{2}+s^{2})(r^{2}+rs+s^{2})^{2}(r+s)^{3} − ar(r+s)^{2}(r^{2}+s^{2})(r^{2}+rs+s^{2})(2r^{2}+rs+s^{2}) = a(r+s)^{2}(r^{2}+s^{2})(r^{2}+rs+s^{2})[ r^{3}+s^{3}−s(r^{2}+rs+2s^{2}) +(r+s)(r^{2}+rs+s^{2})−r(2r^{2}+rs+s^{2}) ] = 0. Up to now, the five cases of Δ^{(4)}(X) have been checked. The same verification

shows that the relations in B^{′cop} are preserved for G_{2}. So we complete the proof. □

Definition 2.4. For any two skew-paired Hopf algebras A and U by a skewdual pairing ⟨ , ⟩ one may form the Drinfel’d quantum double D(A, U) as in [KS, 3.2], which is a Hopf algebra whose underlying coalgebra is A⊗U with the tensor product coalgebra structure, whose algebra structure is defined by

(6)  (a ⊗ f)(a′ ⊗ f′) = ∑⟨S_{U}(f_{(1)}), a′_{(1)}⟩⟨(f_{(3)}), a′_{(3)}⟩aa′_{(2)} ⊗ f_{(2)}f′,

for a, a′ ∈ A and f, f′ ∈ U, and whose antipode S is given by

(7)  S(a ⊗ f) = (1 ⊗ S_{U}(f))(S_{A}(a) ⊗ 1).

Clearly, both mappings A ∋ a ↦ a⊗1 ∈ D(A, U) and U ∋ f ↦ 1⊗f ∈ D(A, U) are injective Hopf algebra homomorphisms. Let us denote the image a ⊗ 1 (resp. 1 ⊗ f) of a (resp. f) in D(A, U) by â (resp. f̂). By (6), we have the following cross relations between elements â (for a ∈ A) and f̂ (for f ∈ U) in the algebra D(A, U):

f̂â = ∑⟨S_{U}(f_{(1)}), a_{(1)}⟩⟨(f_{(3)}), a_{(3)}⟩â_{(2)}f̂_{(2)}, (8)

∑⟨f_{(1)}, a_{(1)}⟩f̂_{(2)}â_{(2)} = ∑ â_{(1)}f̂_{(1)}⟨f_{(2)}, a_{(2)}⟩. (9)

In fact, as an algebra the double D(A, U) is the universal algebra generated by the algebras A and U with cross relations (8) or, equivalently, (9).

Theorem 2.5. The two-parameter quantum group U_{r, s}(G_{2}) is isomorphic to the Drinfel’d quantum double D(B, B′).

The proof is the same as that of [BGH1, Theorem 2.5].

Remark 2.6. Both proofs of Proposition 2.3 and Theorem 2.5 show the compatibility of the defining relations of U_{r,s}(G_{2}), where the proof of Theorem 2.5 indicates that the cross relations between B and B′ are just the half ones appearing in (G1)–(G4), while the proof of Proposition 2.3 then shows the compatibility of the remaining relations appearing respectively in B and B′ including another half ones in (G1)–(G4) and those (r, s)-Serre relations (G5)–(G6).

3. Lusztig’s Symmetries from U_{r, s}(G_{2}) to U_{s^{−1}, r^{−1}}(G_{2})

As did in [BGH1] for the classical types A, B, C, D, we say (U_{s^{−1},r^{−1}}(G_{2}), ⟨ | ⟩) the associated quantum group corresponding to (U_{r,s}(G_{2}), ⟨ , ⟩), where the pairing ⟨ω′_{i}| ω_{j}⟩ is defined via substituting (r, s) by (s^{−1}, r^{−1}) in the defining formula for ⟨ω′_{i}, ω_{j}⟩. Obviously, we have

⟨ω′_{i}| ω_{j}⟩ = ⟨ω′_{j}, ω_{i}⟩.

In what follows, we will study the Lusztig’s symmetry property lying between (U_{r,s}(G_{2}), ⟨ , ⟩) and its associated object (U_{s^{−1},r^{−1}}(G_{2}), ⟨ | ⟩), which indeed indicates the difference in structures between the two-parameter quantum group we introduced above and the usual one-parameter quantum group of Drinfel’d-Jimbo type.

In order to define the Lusztig’s symmetries, we introduce the notation of divided-power elements (in (U_{s^{−1},r^{−1}}(G_{2}), ⟨ | ⟩) ): for any nonnegative integer k ∈ ℕ, set

⟨k⟩_{i} = \frac{s_{i}^{−k} − r_{i}^{−k}}{s_{i}^{−1} − r_{i}^{−1}} , ⟨k⟩_{i}! = ⟨1⟩_{i}⟨2⟩_{i} ⋯ ⟨k⟩_{i},

and for any element e_{i}, f_{i} ∈ (U_{s^{−1},r^{−1}}(G_{2}), ⟨ | ⟩), we define divided-power elements:

e_{i}^{(k)} = e_{i}^{k}/⟨k⟩_{i}! , f_{i}^{(k)} = f_{i}^{k}/⟨k⟩_{i}! .

Definition 3.1. To every i (i = 1, 2), there corresponds a ℚ-linear mapping T_{i} : (U_{r,s}(G_{2}), ⟨ , ⟩) ⟶ (U_{s^{−1}, r^{−1}}(G_{2}), ⟨ | ⟩) such that T_{i}(r) = s^{−1}, T_{i}(s) = r^{−1}, which acts on the generators ω_{j}, ω′_{j}, e_{j}, f_{j} (1 ≤ j ≤ 2) as

T_{i}(ω_{j}) = ω_{j} ω_{i}^{−a_{ij}}, T_{i}(ω′_{j}) = ω′_{j} ω′_{i}^{−a_{ij}},
T_{i}(e_{i}) = − ω′_{i}^{−1}f_{i}, T_{i}(f_{i}) = −(r_{i}s_{i}) e_{i} ω_{i}^{−1},

and for i ≠ j,

T_{i}(e_{j}) = ∑_{ν=0}^{−a_{ij}}(−1)^{ν}(rs)^{\frac{ν}{2} (−a_{ij}−ν)}⟨ω′_{j}, ω_{i}⟩^{−ν}⟨ω′_{i}, ω_{i}⟩^{\frac{ν}{2} (1+a_{ij})}e_{i}^{(ν)}e_{j} e_{i}^{(−a_{ij}−ν)},

T_{i}(f_{j}) = (r_{j}s_{j})^{δ_{ij}^{+}} ∑_{ν=0}^{−a_{ij}}(−1)^{ν}(rs)^{\frac{ν}{2} (−a_{ij}−ν)}⟨ω′_{i}, ω_{j}⟩^{ν}⟨ω′_{i}, ω_{i}⟩^{− \frac{ν}{2} (1+a_{ij})}f_{i}^{(−a_{ij}−ν)}f_{j} f_{i}^{(ν)},

here (a_{ij}) is the Cartan matrix of the simple Lie algebra 𝔤 of type G_{2}, and for any i ≠ j,

δ_{ij}^{+} = \begin{cases}2, & if i < j, & a_{ij} ≠ 0, \\ 1, & otherwise . \end{cases}

Lemma 3.2. T_{i} (i = 1, 2) preserves the defining relations (G1)–(G3) of U_{r,s}(G_{2}) into its associated object U_{s^{−1},r^{−1}}(G_{2}).

Proof. For G_{2}, we have

⟨ω′_{1}, ω_{1}⟩ = rs^{−1} = ⟨ω′_{1}| ω_{1}⟩, ⟨ω′_{1}, ω_{2}⟩ = r^{−3} = ⟨ω′_{2}| ω_{1}⟩,
⟨ω′_{2}, ω_{1}⟩ = s^{3} = ⟨ω′_{1}| ω_{2}⟩, ⟨ω′_{2}, ω_{2}⟩ = r^{3}s^{−3} = ⟨ω′_{2}| ω_{2}⟩.

We show that T_{1}, T_{2} preserve the defining relations (G1)–(G3). (G1) are automatically satisfied. To check (G2) and (G3): first of all, by direct calculation, we have T_{k}(⟨ω′_{i}, ω_{j}⟩) = ⟨T_{k}(ω′_{i}), T_{k}(ω_{j})⟩ = ⟨ω′_{j}, ω_{i}⟩ = ⟨ω′_{i}| ω_{j}⟩, for i, j, k ∈ {1, 2}. This fact ensures that T_{k}(k = 1, 2) preserve (G2) and (G3), that is,

T_{k}(ω_{j})T_{k}(e_{i})T_{k}(ω_{j})^{−1} = ⟨ω′_{i}| ω_{j}⟩T_{k}(e_{i}), T_{k}(ω_{j})T_{k}(f_{i})T_{k}(ω_{j})^{−1} = ⟨ω′_{i}| ω_{j}⟩^{−1}T_{k}(f_{i}),

T_{k}(ω′_{j})T_{k}(e_{i})T_{k}(ω′_{j})^{−1} = ⟨ω′_{j}| ω_{i}⟩^{−1}T_{k}(e_{i}), T_{k}(ω′_{j})T_{k}(f_{i})T_{k}(ω′_{j})^{−1} = ⟨ω′_{j}| ω_{i}⟩T_{k}(f_{i}),

where checking other three identities is equivalent to checking the first one. □

Lemma 3.3. T_{i} (i = 1, 2) preserves the defining relations (G4) into its associated object U_{s^{−1},r^{−1}}(G_{2}).

Proof. Put Δ = r^{2} + rs + s^{2}. To check (G4): for i = 1, 2, we have
[T_{i}(e_{i}), T_{i}(f_{i})] = (r_{i}s_{i})ω_{i}^{′−1}(f_{i}e_{i} − e_{i}f_{i})ω_{i}^{−1} = T_{i}([e_{i}, f_{i}]),
[T_{2}(e_{1}), T_{2}(f_{1})] = [e_{1}e_{2} − r^{3}e_{2}e_{1}, rs(f_{2}f_{1} − s^{3}f_{1}f_{2})]

= rs{f_{2}[e_{1}, f_{1}]e_{2} + e_{1}[e_{2}, f_{2}]f_{1} − r^{3}([e_{2}, f_{2}]f_{1}e_{1} + e_{2}f_{2}[e_{1}, f_{1}])

− s^{3}([e_{1}, f_{1}]f_{2}e_{2} + e_{1}f_{1}[e_{2}, f_{2}]) + (rs)^{3}(e_{2}[e_{1}, f_{1}]f_{2} + f_{1}[e_{2}, f_{2}]e_{1})}

= \frac{ω_{2}ω_{1} − ω′_{2}ω′_{1}}{s^{−1} − r^{−1}} = \frac{T_{2}(ω_{1}) − T_{2}(ω′_{1})}{s^{−1} − r^{−1}} = T_{2}([e_{1}, f_{1}]),

TWO-PARAMETER QUANTUM GROUP OF EXCEPTIONAL TYPE G_{2} 9

and as for

\begin{aligned}[T_{1}(e_{2}), T_{1}(f_{2})] & = \frac{r^{3}s^{3}}{(r + s)^{2}Δ^{2}} [ (rs^{2})^{3}e_{2}e_{1}^{3} − rs^{3}Δe_{1}e_{2}e_{1}^{2} + sΔe_{1}^{2}e_{2}e_{1} − e_{1}^{3}e_{2}, \\ & (r^{2}s)^{3}f_{1}^{3}f_{2} − sr^{3}Δf_{1}^{2}f_{2}f_{1} + rΔf_{1}f_{2}f_{1}^{2} − f_{2}f_{1}^{3} ], \end{aligned}

we have to show that the above bracket in the right hand side is equal to

Δ(r + s)^{2} · \frac{ω_{2}ω_{1}^{3} − ω′_{2}ω_{1}^{′3}}{r − s} .

To do so, let us introduce the notations of “quantum root vectors” in terms of adjoint actions as follows

E_{12} = (ad_{l}e_{1})(e_{2}) = e_{1}e_{2} − s^{3}e_{2}e_{1},
F_{12} = (ad_{r}f_{1})(f_{2}) = f_{2}f_{1} − r^{3}f_{1}f_{2},
E_{112} = (ad_{l}e_{1})^{2}(e_{2}) = e_{1}E_{12} − rs^{2}E_{12}e_{1},
F_{112} = (ad_{r}f_{1})^{2}(f_{2}) = F_{12}f_{1} − r^{2}sf_{1}F_{12},
E_{1112} = (ad_{l}e_{1})^{3}(e_{2}) = e_{1}^{3}e_{2} − sΔe_{1}^{2}e_{2}e_{1} + rs^{3}Δe_{1}e_{2}e_{1}^{2} − (rs^{2})^{3}e_{2}e_{1}^{3},
F_{1112} = (ad_{r}f_{1})^{3}(f_{2}) = f_{2}f_{1}^{3} − rΔf_{1}f_{2}f_{1}^{2} + sr^{3}Δf_{1}^{2}f_{2}f_{1} − (r^{2}s)^{3}f_{1}^{3}f_{2}.

That is, we need to verify

[ E_{1112}, F_{1112} ] = Δ(r + s)^{2} · \frac{ω_{2}ω_{1}^{3} − ω′_{2}ω_{1}^{′3}}{r − s} .

By direct calculation (using the Leibniz rule), we have [e_{1}, F_{12}] = −Δω_{1}f_{2}, [e_{2}, F_{12}] = f_{1}ω′_{2},

[E_{12}, f_{1}] = −Δe_{2}ω′_{1}, [E_{12}, f_{2}] = ω_{2}e_{1},

[E_{12}, F_{12}] = \frac{ω_{1}ω_{2} − ω′_{1}ω′_{2}}{r − s} ,

[e_{1}, F_{112}] = −(r + s)^{2}ω_{1}F_{12}, [e_{2}, F_{112}] = s(s^{2} − r^{2})f_{1}^{2}ω′_{2}, [E_{112}, f_{1}] = −(r + s)^{2}E_{12}ω′_{1}, [E_{112}, f_{2}] = r(r^{2} − s^{2})ω_{2}e_{1}^{2}, [E_{112}, F_{12}] = (r + s)^{2}ω_{1}ω_{2}e_{1}, [E_{12}, F_{112}] = (r + s)^{2}f_{1}ω′_{1}ω′_{2},

[E_{112}, F_{112}] = (r + s)^{2} · \frac{ω_{1}^{2}ω_{2} − ω_{1}^{′2}ω′_{2}}{r − s} ,

as well as

[e_{1}, F_{1112}] = [e_{1}, F_{112}f_{1} − rs^{2}f_{1}F_{112}] = −Δω_{1}F_{112},
[E_{112}, F_{1112}] = [E_{112}, F_{112}f_{1} − rs^{2}f_{1}F_{112}]

= [E_{112}, F_{112}]f_{1} − rs^{2}f_{1}[E_{112}, F_{112}] + F_{112}[E_{112}, f_{1}] − rs^{2}[E_{112}, f_{1}]F_{112}

= Δ(r + s)^{2}f_{1}ω_{1}^{′2}ω′_{2},

10 HU AND SHI

[E_{1112}, F_{1112}] = [e_{1}E_{112} − r^{2}sE_{112}e_{1}, F_{1112}]

= [e_{1}, F_{1112}]E_{112} − r^{2}sE_{112}[e_{1}, F_{1112}] + e_{1}[E_{112}, F_{1112}] − r^{2}s[E_{112}, F_{1112}]e_{1} = Δω_{1}[E_{112}, F_{112}] + Δ(r + s)^{2}[e_{1}, f_{1}]ω_{1}^{′2}ω′_{2}

= Δ(r + s)^{2} · \frac{ω_{2}ω_{1}^{3} − ω′_{2}ω_{1}^{′3}}{r − s} .

Thus, we arrive at [T_{1}(e_{2}), T_{1}(f_{2})] = T_{1}([e_{2}, f_{2}]) ∈ U_{s^{−1}, r^{−1}}(G_{2}). □ Lemma 3.4. T_{2} preserves the (r, s)-Serre relations (G5)_{1}, (G6)_{1} into its asso-

ciated object U_{s^{−1},r^{−1}}(G_{2}), i.e., T_{2}(e_{2})^{2}T_{2}(e_{1}) − (r^{3} + s^{3})T_{2}(e_{2})T_{2}(e_{1})T_{2}(e_{2}) + (rs)^{3}T_{2}(e_{1})T_{2}(e_{2})^{2} = 0, (1) T_{2}(f_{1})T_{2}(f_{2})^{2} − (r^{3} + s^{3})T_{2}(f_{2})T_{2}(f_{1})T_{2}(f_{2}) + (rs)^{3}T_{2}(f_{1})T_{2}(f_{2})^{2} = 0. (2)

Proof. For the degree 2 (r, s)-Serre relation (G5)_{1}

e_{2}^{2}e_{1} − (r^{−3} + s^{−3})e_{2}e_{1}e_{2} + r^{−3}s^{−3}e_{1}e_{2}^{2} = 0,

observe that

(3) T_{2}(e_{1})T_{2}(e_{2}) = r^{−3}T_{2}(e_{2})T_{2}(e_{1}) − r^{−3}e_{1},  T_{2}(e_{2})e_{1} = s^{3}e_{1}T_{2}(e_{2}).

Acting T_{2} algebraically on the left-hand side of (G5)_{1}, we have

T_{2}(e_{2})^{2}T_{2}(e_{1}) − (r^{3} + s^{3})T_{2}(e_{2})T_{2}(e_{1})T_{2}(e_{2}) + (rs)^{3}T_{2}(e_{1})T_{2}(e_{2})^{2}
= T_{2}(e_{2})r^{3}(T_{2}(e_{1})T_{2}(e_{2}) + r^{−3}e_{1}) − (r^{3} + s^{3})T_{2}(e_{2})T_{2}(e_{1})T_{2}(e_{2})
+ (rs)^{3}(r^{−3}T_{2}(e_{2})T_{2}(e_{1}) − r^{−3}e_{1})T_{2}(e_{2})
= 0,

that is, we have proved (1).

The proof of (2) is similar to (1). □ In order to prove Lemma 3.8 below, it needs first to prepare three auxiliary

Lemmas 3.5—3.7.

Lemma 3.5. With the notation in Lemma 3.3, we have

[E_{1112}E_{112} − r^{3}E_{112}E_{1112}, f_{2}] = 0. Proof. Since e_{1}E_{1112} − r^{3}E_{1112}e_{1} = ad_{l}(e_{1})^{4}(e_{2}) = 0 (Serre relation), and

\begin{aligned}[E_{1112}, f_{2}] & = [e_{1}E_{112} − r^{2}sE_{112}e_{1}, f_{2}] = e_{1}[E_{112}, f_{2}] − r^{2}s[E_{112}, f_{2}]e_{1} \\ & = r^{3}(r−s)(r^{2}−s^{2}) ω_{2}e_{1}^{3}, \end{aligned}

we obtain

[E_{1112}E_{112} − r^{3}E_{112}E_{1112}, f_{2}] = E_{1112}[E_{112}, f_{2}] + [E_{1112}, f_{2}]E_{112}
− r^{3}(E_{112}[E_{1112}, f_{2}] + [E_{112}, f_{2}]E_{1112})
= r^{3}(r−s)(r^{2}−s^{2}) ω_{2} (e_{1}^{3}E_{112} − rΔe_{1}E_{1112}e_{1} − (r^{2}s)^{3}E_{112}e_{1}^{3})

= r^{3}(r−s)(r^{2}−s^{2}) ω_{2} (e_{1}^{3}E_{112} − rΔe_{1}^{2}E_{112}e_{1} + r^{3}sΔe_{1}E_{112}e_{1}^{2} − (r^{2}s)^{3}E_{112}e_{1}^{3})

= r^{3}(r−s)(r^{2}−s^{2}) ω_{2} (e_{1} · (SR) − rs^{2}(SR) · e_{1})
= 0,

where (SR) denotes the left-hand-side presentation of the (r, s)-Serre relation (G5)_{2} e_{1}^{2}E_{112} − r^{2}(r + s)e_{1}E_{112}e_{1} + r^{5}sE_{112}e_{1}^{2} = 0,

TWO-PARAMETER QUANTUM GROUP OF EXCEPTIONAL TYPE G_{2} 11

and we used the replacement E_{1112} = e_{1}E_{112} − r^{2}sE_{112}e_{1} in the third equality. □

Lemma 3.6. With the notation in Lemma 3.3, we have

[E_{1112}E_{112} − r^{3}E_{112}E_{1112}, f_{1}] = 0.

Proof. Easy to check [E_{1112}, f_{1}] = −ΔE_{112}ω′_{1}. So we have

[E_{1112}E_{112} − r^{3}E_{112}E_{1112}, f_{1}] = E_{1112}[E_{112}, f_{1}] + [E_{1112}, f_{1}]E_{112}
− r^{3}(E_{112}[E_{1112}, f_{1}] + [E_{112}, f_{1}]E_{1112})
= (r + s) [(r + s)((rs)^{3}E_{12}E_{1112} − E_{1112}E_{12}) + r(r − s)ΔE_{112}^{2}]ω′_{1}.

It suffices to show

(4)  E_{1112}E_{12} = (rs)^{3}E_{12}E_{1112} + r(r − s)(r + s)^{−1}ΔE_{112}^{2}.

At first, we note that the (r, s)-Serre relation (G5)_{1} is equivalent to

E_{12}e_{2} = r^{3}e_{2}E_{12}.

As e_{1}e_{2} = E_{12} + s^{3}e_{2}e_{1}, we get

E_{112}e_{2} = (e_{1}E_{12} − rs^{2}E_{12}e_{1})e_{2} = r^{3}e_{1}e_{2}E_{12} − rs^{2}E_{12}e_{1}e_{2}
= r^{3}(E_{12} + s^{3}e_{2}e_{1})E_{12} − rs^{2}E_{12}(E_{12} + s^{3}e_{2}e_{1})
= r(r^{2}−s^{2})E_{12}^{2} + (rs)^{3}e_{2}(e_{1}E_{12} − rs^{2}E_{12}e_{1})
= r(r^{2}−s^{2})E_{12}^{2} + (rs)^{3}e_{2}E_{112}.

Next, we claim E_{1112}e_{2} = (rs^{2})^{3}e_{2}E_{1112} − r(rs−r^{2}+s^{2})E_{112}E_{12} + (rs)^{2}(r^{2}+rs−s^{2})E_{12}E_{112}. Indeed, as E_{1112} = e_{1}E_{112} − r^{2}sE_{112}e_{1}, E_{112} = e_{1}E_{12} − rs^{2}E_{12}e_{1}, e_{1}e_{2} = E_{12} +

s^{3}e_{2}e_{1}, we have

E_{1112}e_{2} = e_{1}(E_{112}e_{2}) − r^{2}sE_{112}(e_{1}e_{2})
= r(r^{2} − s^{2})e_{1}E_{12}^{2} + (rs)^{3}(e_{1}e_{2})E_{112} − r^{2}sE_{112}(e_{1}e_{2})
= r(r^{2} − s^{2})e_{1}E_{12}^{2} + (rs)^{3}E_{12}E_{112} + (rs^{2})^{3}e_{2}e_{1}E_{112} − r^{2}sE_{112}E_{12}
− (rs^{2})^{2}(E_{112}e_{2})e_{1}
= r(r^{2} − s^{2})E_{112}E_{12} + (rs)^{2}(r^{2} − s^{2})E_{12}e_{1}E_{12} + (rs)^{3}E_{12}E_{112}

+ (rs^{2})^{3}e_{2}e_{1}E_{112} − r^{2}sE_{112}E_{12} − r^{3}s^{4}(r^{2} − s^{2})E_{12}^{2}e_{1} − r^{5}s^{7}e_{2}E_{112}e_{1}

= (rs^{2})^{3}e_{2}E_{1112} − r(rs−r^{2}+s^{2})E_{112}E_{12} + (rs)^{2}(r^{2}+rs−s^{2})E_{12}E_{112}.

In order to prove (4), we firstly note that

[(r+s)((rs)^{3}E_{12}E_{1112} − E_{1112}E_{12}) + r(r − s)ΔE_{112}^{2}, f_{1}]

= (r + s)(rs)^{3}(E_{12}[E_{1112}, f_{1}] + [E_{12}, f_{1}]E_{1112}) − (r + s)(E_{1112}[E_{12}, f_{1}] + [E_{1112}, f_{1}]E_{12}) + r(r − s)Δ(E_{112}[E_{112}, f_{1}] + [E_{112}, f_{1}]E_{112}) = −(r + s)(rs)^{3}Δ(E_{12}E_{112} + s^{3}e_{2}E_{1112})ω′_{1} + (r + s)Δ(E_{1112}e_{2} + r^{2}sE_{112}E_{12})ω′_{1} − r(r − s)(r + s)^{2}Δ(E_{112}E_{12} + rs^{2}E_{12}E_{112})ω′_{1}

= 0 (by the above identity),

12 HU AND SHI

secondly, instead of f_{1} by f_{2} in the above formula, we get

[(r+s)((rs)^{3}E_{12}E_{1112} − E_{1112}E_{12}) + r(r − s)ΔE_{112}^{2}, f_{2}]

= (r + s)(rs)^{3}(r^{3}(r^{2} − s^{2})(r − s)E_{12}ω_{2}e_{1}^{3} + ω_{2}e_{1}E_{1112})

− (r + s)(E_{1112}ω_{2}e_{1} + r^{3}(r^{2} − s^{2})(r − s)ω_{2}e_{1}^{3}E_{12})

+ r^{2}(r − s)(r^{2} − s^{2})Δ(E_{112}ω_{2}e_{1}^{2} + ω_{2}e_{1}^{2}E_{112}) = (r + s)(rs)^{3}ω_{2}((rs)^{3}(r^{2} − s^{2})(r − s)E_{12}e_{1}^{3} + e_{1}E_{1112})

− (r + s)ω_{2}((r^{2}s)^{3}E_{1112}e_{1} + r^{3}(r^{2} − s^{2})(r − s)e_{1}^{3}E_{12})

+ r^{2}(r − s)(r^{2} − s^{2})Δω_{2}((rs)^{3}E_{112}e_{1}^{2} + e_{1}^{2}E_{112}) = r^{2}(r − s)(r^{2} − s^{2})Δω_{2}((rs)^{3}E_{112}e_{1}^{2} + e_{1}^{2}E_{112})

− r^{3}(r^{2} − s^{2})^{2}ω_{2}(e_{1}^{3}E_{12} − (rs^{2})^{3}E_{12}e_{1}^{3}) = r^{2}(r − s)(r^{2} − s^{2})Δω_{2}((rs)^{3}E_{112}e_{1}^{2} + e_{1}^{2}E_{112})

− r^{3}(r^{2} − s^{2})^{2}ω_{2}(e_{1}^{2}E_{112} + rs^{2}e_{1}E_{112}e_{1} + (rs^{2})^{2}E_{112}e_{1}^{2}) = (rs)^{2}(r − s)(r^{2} − s^{2})ω_{2}(e_{1}^{2}E_{112} − r^{2}(r + s)e_{1}E_{112}e_{1} + r^{5}sE_{112}e_{1}^{2}) = (rs)^{2}(r − s)(r^{2} − s^{2})ω_{2}(ad_{l}e_{1})^{2}(E_{112}) = (rs)^{2}(r − s)(r^{2} − s^{2})ω_{2}(ad_{l}e_{1})^{4}(e_{2}) = 0.

So similar to the deduction of [BKL, Lemma 3.4], we get (4). □ By [BKL, Lemma 3.4], Lemmas 3.5 and 3.6 imply Lemma 3.7.

E_{1112}E_{112} − r^{3}E_{112}E_{1112} = 0.

Lemma 3.8. T_{1} preserves the (r, s)-Serre relations (G5)_{1}, (G6)_{1} into its associated object U_{s^{−1},r^{−1}}(G_{2}), i.e., T_{1}(e_{2})^{2}T_{1}(e_{1}) − (r^{3} + s^{3})T_{1}(e_{2})T_{1}(e_{1})T_{1}(e_{2}) + (rs)^{3}T_{1}(e_{1})T_{1}(e_{2})^{2} = 0, (5) T_{1}(f_{1})T_{1}(f_{2})^{2} − (r^{3} + s^{3})T_{1}(f_{2})T_{1}(f_{1})T_{1}(f_{2}) + (rs)^{3}T_{1}(f_{2})^{2}T_{1}(f_{1}) = 0. (6)

Proof. By direct calculation, we have

\begin{aligned}T_{1}(e_{2})T_{1}(e_{1}) & = [ − \frac{1}{s^{3}(r + s)Δ}E_{1112}] · (−ω_{1}^{′−1}f_{1}) \\ & = s^{3}T_{1}(e_{1})T_{1}(e_{2}) − \frac{1}{rs^{2}(r + s)}E_{112}. \end{aligned} ; (7)

Hence, to prove (5) is equivalent to prove

T_{1}(e_{2})E_{112} − r^{3}E_{112}T_{1}(e_{2}) = 0.

However, the latter is given by Lemma 3.7.

The proof of (6) is analogous. □ In order to prove Lemma 3.13 below, it needs first to prepare four auxiliary

Lemmas 3.9—3.12. Write E_{21} := (ad_{l}e_{2})(e_{1}) = e_{2}e_{1} − r^{−3}e_{1}e_{2}, and note that (G5)_{1} is equivalent to (ad_{l}e_{2})(E_{21}) = e_{2}E_{21} − s^{−3}E_{21}e_{2} = 0, i.e., E_{21}e_{2} = s^{3}e_{2}E_{21}.

Lemma 3.9.

[ e_{1}E_{21}^{3} − sΔE_{21}e_{1}E_{21}^{2} + rs^{3}ΔE_{21}^{2}e_{1}E_{21} − (rs^{2})^{3}E_{21}^{3}e_{1}, f_{1} ] = 0.

Proof. Since

[E_{21}, f_{1}] = r^{−3}Δe_{2}ω_{1}, ω_{1}E_{21} = rs^{2}E_{21}ω_{1},

[E_{21}^{2}, f_{1}] = r^{−3}s^{−1}(r + s)ΔE_{21}e_{2}ω_{1}, ω′_{1}E_{21} = r^{2}sE_{21}ω′_{1}, (8)

[E_{21}^{3}, f_{1}] = r^{−3}s^{−2}Δ^{2}E_{21}^{2}e_{2}ω_{1},

we get

∑1 = \frac{ω_{1}−ω′_{1}}{r−s} E_{21}^{3} − (rs^{2})^{3}E_{21}^{3} \frac{ω_{1}−ω′_{1}}{r−s} − sΔE_{21} \frac{ω_{1}−ω′_{1}}{r−s} E_{21}^{2} + rs^{3}ΔE_{21}^{2} \frac{ω_{1}−ω′_{1}}{r−s} E_{21}
= −(rs)^{3}ΔE_{21}^{3}ω′_{1} + rs^{2}ΔE_{21}^{2}ω′_{1}E_{21}
= 0,

and

[ e_{1}E_{21}^{3} − sΔE_{21}e_{1}E_{21}^{2} + rs^{3}ΔE_{21}^{2}e_{1}E_{21} − (rs^{2})^{3}E_{21}^{3}e_{1}, f_{1} ]

= e_{1}[E_{21}^{3}, f_{1}] + [e_{1}, f_{1}]E_{21}^{3} − (rs^{2})^{3}(E_{21}^{3}[e_{1}, f_{1}] + [E_{21}^{3}, f_{1}]e_{1})

− sΔ(E_{21}e_{1}[E_{21}^{2}, f_{1}] + E_{21}[e_{1}, f_{1}]E_{21}^{2} + [E_{21}, f_{1}]e_{1}E_{21}^{2})

+ rs^{3}Δ(E_{21}^{2}e_{1}[E_{21}, f_{1}] + E_{21}^{2}[e_{1}, f_{1}]E_{21} + [E_{21}^{2}, f_{1}]e_{1}E_{21})

= r^{−3}s^{−2}Δ^{2}e_{1}E_{21}^{2}e_{2}ω_{1} + \frac{ω_{1}−ω′_{1}}{r−s} E_{21}^{3}

− sΔ[r^{−3}s^{−1}(r+s)ΔE_{21}e_{1}E_{21}e_{2}ω_{1} + E_{21} \frac{ω_{1}−ω′_{1}}{r−s} E_{21}^{2} + r^{−3}Δe_{2}ω_{1}e_{1}E_{21}^{2}]

+ rs^{3}Δ[r^{−3}ΔE_{21}^{2}e_{1}e_{2}ω_{1} + E_{21}^{2} \frac{ω_{1}−ω′_{1}}{r−s} E_{21} + r^{−3}s^{−1}(r+s)ΔE_{21}e_{2}ω_{1}e_{1}E_{21}]

− (rs^{2})^{3}(E_{21}^{3} \frac{ω_{1}−ω′_{1}}{r−s} + r^{−3}s^{−2}Δ^{2}E_{21}^{2}e_{2}ω_{1}e_{1})

= _{∑1} + (r^{−3}s^{−2}Δ^{2})_{∑2} ω_{1} = (r^{−3}s^{−2}Δ^{2})_{∑2} ω_{1},

where

\begin{aligned}∑2 & = e_{1}E_{21}^{2}e_{2} − s^{2}(r + s)E_{21}e_{1}E_{21}e_{2} − (rs^{2})^{3}e_{2}e_{1}E_{21}^{2} \\ & + rs^{5}E_{21}^{2}e_{1}e_{2} + r^{3}s^{5}(r + s)E_{21}e_{2}e_{1}E_{21} − r^{4}s^{5}E_{21}^{2}e_{2}e_{1}. \end{aligned}

It remains to show ∑_{2} = 0. As E_{21}e_{2} = s^{3}e_{2}E_{21} and e_{2}e_{1} − r^{−3}e_{1}e_{2} = E_{21}, we get

∑2 = (e_{1}E_{21}^{2}e_{2} − (rs^{2})^{3}e_{2}e_{1}E_{21}^{2}) + (rs^{5}E_{21}^{2}e_{1}e_{2} − r^{4}s^{5}E_{21}^{2}e_{2}e_{1})
− s^{2}(r + s)E_{21}e_{1}E_{21}e_{2} + r^{3}s^{5}(r + s)E_{21}e_{2}e_{1}E_{21}
= −(rs^{2})^{3}E_{21}^{3} − r^{4}s^{5}E_{21}^{3} + r^{3}s^{5}(r + s)E_{21}^{3}
= 0.

Hence, we complete the proof. □

14 HU AND SHI

Lemma 3.10.

[ e_{1}E_{21}^{3} − sΔE_{21}e_{1}E_{21}^{2} + rs^{3}ΔE_{21}^{2}e_{1}E_{21} − (rs^{2})^{3}E_{21}^{3}e_{1}, f_{2} ] = 0. Proof. Noting that

[E_{21}, f_{2}] = −r^{−3}ω′_{2}e_{1}, E_{21}ω′_{2} = r^{3}ω′_{2}E_{21},

[E_{21}^{2}, f_{2}] = −r^{−3}ω′_{2}(e_{1}E_{21} + r^{3}E_{21}e_{1}), (9)

[E_{21}^{3}, f_{2}] = −r^{−3}ω′_{2}(e_{1}E_{21}^{2} + r^{3}E_{21}e_{1}E_{21} + r^{6}E_{21}^{2}e_{1}),

we obtain

[ e_{1}E_{21}^{3} − sΔE_{21}e_{1}E_{21}^{2} + rs^{3}ΔE_{21}^{2}e_{1}E_{21} − (rs^{2})^{3}E_{21}^{3}e_{1}, f_{2} ]
= e_{1}[E_{21}^{3}, f_{2}] − sΔ(E_{21}e_{1}[E_{21}^{2}, f_{2}] + [E_{21}, f_{2}]e_{1}E_{21}^{2})
+ rs^{3}Δ(E_{21}^{2}e_{1}[E_{21}, f_{2}] + [E_{21}^{2}, f_{2}]e_{1}E_{21}) − (rs^{2})^{3}[E_{21}^{3}, f_{2}]e_{1}
= −r^{−3}ω′_{2} {s^{3}e_{1}(e_{1}E_{21}^{2} + r^{3}E_{21}e_{1}E_{21} + r^{6}E_{21}^{2}e_{1})
− sΔ[(rs)^{3}E_{21}e_{1}(e_{1}E_{21} + r^{3}E_{21}e_{1}) + e_{1}^{2}E_{21}^{2}]
+ rs^{3}Δ[(r^{2}s)^{3}E_{21}^{2}e_{1}^{2} + (e_{1}E_{21} + r^{3}E_{21}e_{1})e_{1}E_{21}]
−(rs^{2})^{3}(e_{1}E_{21}^{2} + r^{3}E_{21}e_{1}E_{21} + r^{6}E_{21}^{2}e_{1})e_{1}}
= −r^{−2}s ω′_{2} S.

where

\begin{aligned}S & = (rs)^{2}(r^{3}−s^{3})(e_{1}E_{21}^{2}e_{1} + E_{21}e_{1}^{2}E_{21}) + s^{2}(2r^{2}+rs+s^{2})(e_{1}E_{21})^{2} \\ & − r^{5}s^{3}(2s^{2}+rs+r^{2})(E_{21}e_{1})^{2} − (r+s)(e_{1}^{2}E_{21}^{2} − (rs)^{6}E_{21}^{2}e_{1}^{2}). \end{aligned}

It remains to prove S = 0, which is equivalent to show S_{i} := [S, f_{i}] = 0 for i = 1, 2, according to [BKL, Lemma 3.4].

To this end, we first observe the following fact Lemma 3.11.

e_{1}^{3}E_{21} − sΔe_{1}^{2}E_{21}e_{1} + rs^{3}Δe_{1}E_{21}e_{1}^{2} − (rs^{2})^{3}E_{21}e_{1}^{3} = 0.

Proof. It is easy to see that

e_{1}^{3}E_{21} − sΔe_{1}^{2}E_{21}e_{1} + rs^{3}Δe_{1}E_{21}e_{1}^{2} − (rs^{2})^{3}E_{21}e_{1}^{3} = r^{−3}(ad_{l}e_{1})^{4}(e_{2}), which is in fact the (r, s)-Serre relation (G5)_{2} up to a factor r^{−3}. □

TWO-PARAMETER QUANTUM GROUP OF EXCEPTIONAL TYPE G_{2} 15

Using (9), we obtain

S_{2} = (rs)^{2}(r^{3}−s^{3})(e_{1}[E_{21}^{2}, f_{2}]e_{1} + [E_{21}, f_{2}]e_{1}^{2}E_{21} + E_{21}e_{1}^{2}[E_{21}, f_{2}])

+ s^{2}(2r^{2}+rs+s^{2})(e_{1}[E_{21}, f_{2}]e_{1}E_{21} + e_{1}E_{21}e_{1}[E_{21}, f_{2}]) − r^{5}s^{3}(2s^{2}+rs+r^{2})([E_{21}, f_{2}]e_{1}E_{21}e_{1} + E_{21}e_{1}[E_{21}, f_{2}]e_{1}) − (r + s)(e_{1}^{2}[E_{21}^{2}, f_{2}] − (rs)^{6}[E_{21}^{2}, f_{2}]e_{1}^{2})

= (−r^{−3}) {(rs)^{2}(r^{3}−s^{3})[e_{1}ω′_{2}(e_{1}E_{21} + r^{3}E_{21}e_{1})e_{1} + ω′_{2}e_{1}^{3}E_{21} + E_{21}e_{1}^{2}ω′_{2}e_{1}] + s^{2}(2r^{2}+rs+s^{2})(e_{1}ω′_{2}e_{1}^{2}E_{21} + e_{1}E_{21}e_{1}ω′_{2}e_{1}) − r^{5}s^{3}(2s^{2}+rs+r^{2})(ω′_{2}e_{1}^{2}E_{21}e_{1} + E_{21}e_{1}ω′_{2}e_{1}^{2}) − (r + s)[e_{1}^{2}ω′_{2}(e_{1}E_{21} + r^{3}E_{21}e_{1}) − (rs)^{6}ω′_{2}(e_{1}E_{21} + r^{3}E_{21}e_{1})e_{1}^{2}]}

= (−r^{−3}) ω′_{2} {(rs)^{2}(r^{3}−s^{3})[s^{3}e_{1}(e_{1}E_{21} + r^{3}E_{21}e_{1})e_{1} + e_{1}^{3}E_{21} + (rs^{2})^{3}E_{21}e_{1}^{3}] + s^{2}(2r^{2}+rs+s^{2})(s^{3}e_{1}^{3}E_{21} + (rs^{2})^{3}e_{1}E_{21}e_{1}^{2}) − r^{5}s^{3}(2s^{2}+rs+r^{2})(e_{1}^{2}E_{21}e_{1} + (rs)^{3}E_{21}e_{1}^{3}) − (r + s)[s^{6}e_{1}^{2}(e_{1}E_{21} + r^{3}E_{21}e_{1}) − (rs)^{6}(e_{1}E_{21} + r^{3}E_{21}e_{1})e_{1}^{2}]}

= (−r^{−3})(rs)^{2}(r^{3}+s^{3}) ω′_{2} [ e_{1}^{3}E_{21} − sΔe_{1}^{2}E_{21}e_{1} + rs^{3}Δe_{1}E_{21}e_{1}^{2} − (rs^{2})^{3}E_{21}e_{1}^{3} ] = 0. (by Lemma 3.11)

Now we turn to prove S_{1} = 0. Using (8) and noting that [e_{1}^{2}, f_{1}] = \frac{r+s}{rs} · \frac{sω_{1}−rω′_{1}}{r−s} e_{1}, we get

S_{1} = (rs)^{2}(r^{3}−s^{3})([e_{1}, f_{1}]E_{21}^{2}e_{1} + e_{1}[E_{21}^{2}, f_{1}]e_{1} + e_{1}E_{21}^{2}[e_{1}, f_{1}]
+ [E_{21}, f_{1}]e_{1}^{2}E_{21} + E_{21}[e_{1}^{2}, f_{1}]E_{21} + E_{21}e_{1}^{2}[E_{21}, f_{1}])

+ s^{2}(2r^{2}+rs+s^{2})([e_{1}, f_{1}]E_{21}e_{1}E_{21} + e_{1}[E_{21}, f_{1}]e_{1}E_{21}

+ e_{1}E_{21}[e_{1}, f_{1}]E_{21} + e_{1}E_{21}e_{1}[E_{21}, f_{1}])

− r^{5}s^{3}(2s^{2}+rs+r^{2})([E_{21}, f_{1}]e_{1}E_{21}e_{1} + E_{21}[e_{1}, f_{1}]E_{21}e_{1}

+ E_{21}e_{1}[E_{21}, f_{1}]e_{1} + E_{21}e_{1}E_{21}[e_{1}, f_{1}])

− (r + s)([e_{1}^{2}, f_{1}]E_{21}^{2} + e_{1}^{2}[E_{21}^{2}, f_{1}] − (rs)^{6}[E_{21}^{2}, f_{1}]e_{1}^{2} − (rs)^{6}E_{21}^{2}[e_{1}^{2}, f_{1}])

= (rs)^{2}(r^{3}−s^{3}) (\frac{ω_{1}−ω′_{1}}{r−s} E_{21}^{2}e_{1} + r^{−3}s^{−1}(r+s)Δe_{1}E_{21}e_{2}ω_{1}e_{1} + e_{1}E_{21}^{2} \frac{ω_{1}−ω′_{1}}{r−s}

+ r^{−3}Δe_{2}ω_{1}e_{1}^{2}E_{21} + \frac{r+s}{rs}E_{21} \frac{sω_{1}−rω′_{1}}{r−s} e_{1}E_{21} + r^{−3}ΔE_{21}e_{1}^{2}e_{2}ω_{1})

+ s^{2}(2r^{2}+rs+s^{2}) (\frac{ω_{1}−ω′_{1}}{r−s} E_{21}e_{1}E_{21} + r^{−3}Δe_{1}e_{2}ω_{1}e_{1}E_{21}

+ e_{1}E_{21} \frac{ω_{1}−ω′_{1}}{r−s} E_{21} + r^{−3}Δe_{1}E_{21}e_{1}e_{2}ω_{1})

− r^{5}s^{3}(2s^{2}+rs+r^{2}) (r^{−3}Δe_{2}ω_{1}e_{1}E_{21}e_{1} + E_{21} \frac{ω_{1}−ω′_{1}}{r−s} E_{21}e_{1}

+ r^{−3}ΔE_{21}e_{1}e_{2}ω_{1}e_{1} + E_{21}e_{1}E_{21} \frac{ω_{1}−ω′_{1}}{r−s} )

− (r+s) (\frac{r+s}{rs} \frac{sω_{1}−rω′_{1}}{r−s} e_{1}E_{21}^{2} + r^{−3}s^{−1}(r+s)Δe_{1}^{2}E_{21}e_{2}ω_{1}

−r^{3}s^{5}(r+s)ΔE_{21}e_{2}ω_{1}e_{1}^{2} − (rs)^{5}(r+s)E_{21}^{2} \frac{sω_{1}−rω′_{1}}{r−s} e_{1})

= A + B + C + D,

where A, B, C, D are given as follows.

Noting that ω_{1}E_{21} = rs^{2}E_{21}ω_{1} and ω′_{1}E_{21} = r^{2}sE_{21}ω′_{1}, we have

A : = (rs)^{2}Δ(ω_{1}−ω′_{1})E_{21}^{2}e_{1} − r^{5}s^{3}(2s^{2}+rs+r^{2})E_{21} \frac{ω_{1}−ω′_{1}}{r−s} E_{21}e_{1}

+ (rs)^{5} \frac{(r+s)^{2}}{r−s} E_{21}^{2}(sω_{1}−rω′_{1})e_{1}

= −r^{5}s^{4}(r^{3}−s^{3})E_{21}^{2}e_{1}ω_{1},

B : = (rs)(r+s)ΔE_{21}(sω_{1}−rω′_{1})e_{1}E_{21} + s^{2}(2r^{2}+rs+s^{2})\frac{ω_{1}−ω′_{1}}{r−s} E_{21}e_{1}E_{21}

− r^{5}s^{3}(2s^{2}+rs+r^{2})E_{21}e_{1}E_{21} \frac{ω_{1}−ω′_{1}}{r−s} = 0,

C : = (rs)^{2}Δe_{1}E_{21}^{2}(ω_{1}−ω′_{1}) + s^{2}(2r^{2}+rs+s^{2})e_{1}E_{21} \frac{ω_{1}−ω′_{1}}{r−s} E_{21}

− \frac{(r+s)^{2}}{rs} \frac{sω_{1}−rω′_{1}}{r−s} e_{1}E_{21}^{2}

= rs^{2}(r^{3}−s^{3})e_{1}E_{21}^{2}ω_{1}, furthermore, using E_{21}e_{2} = s^{3}e_{2}E_{21}, r^{−3}e_{1}e_{2} = e_{2}e_{1} − E_{21} and e_{2}e_{1} = E_{21} + r^{−3}e_{1}e_{2}, we get

D : = r^{−3}Δ {(rs)^{2}(r^{3}−s^{3})[s^{−1}(r+s)e_{1}E_{21}e_{2}ω_{1}e_{1} + e_{2}ω_{1}e_{1}^{2}E_{21} + E_{21}e_{1}^{2}e_{2}ω_{1}]

+ s^{2}(2r^{2}+rs+s^{2})[e_{1}e_{2}ω_{1}e_{1}E_{21} + e_{1}E_{21}e_{1}e_{2}ω_{1}] − r^{5}s^{3}(2s^{2}+rs+r^{2})[e_{2}ω_{1}e_{1}E_{21}e_{1} + E_{21}e_{1}e_{2}ω_{1}e_{1}] −(r+s)^{2}s^{−1}[e_{1}^{2}E_{21}e_{2}ω_{1} − (rs)^{6}E_{21}e_{2}ω_{1}e_{1}^{2}]}

= Δ {(rs)^{2}(r^{3}−s^{3})[(rs)^{−2}(r+s)e_{1}E_{21}e_{2}e_{1} + (e_{2}e_{1})e_{1}E_{21} + E_{21}e_{1}(r^{−3}e_{1}e_{2})] + s^{2}(2r^{2}+rs+s^{2})[r^{−1}se_{1}e_{2}e_{1}E_{21} + e_{1}E_{21}(r^{−3}e_{1}e_{2})] − r^{5}s^{3}(2s^{2}+rs+r^{2})[(e_{2}e_{1})E_{21}e_{1} + r^{−2}s^{−1}E_{21}e_{1}e_{2}e_{1}] −(r+s)^{2}s^{2}[e_{1}(r^{−3}e_{1}e_{2})E_{21} − r^{5}sE_{21}(e_{2}e_{1})e_{1}]} ω_{1}

= Δ {(rs)^{2}(r^{3}−s^{3})[(rs)^{−2}(r+s)e_{1}E_{21}e_{2}e_{1} + r^{−3}e_{1}e_{2}e_{1}E_{21} + E_{21}e_{1}e_{2}e_{1}] + s^{2}(2r^{2}+rs+s^{2})[r^{−1}se_{1}e_{2}e_{1}E_{21} + e_{1}E_{21}e_{2}e_{1} − e_{1}E_{21}^{2}] − r^{5}s^{3}(2s^{2}+rs+r^{2})[E_{21}^{2}e_{1} + (rs)^{−3}e_{1}E_{21}e_{2}e_{1} + r^{−2}s^{−1}E_{21}e_{1}e_{2}e_{1}] −(r+s)^{2}s^{2}[e_{1}e_{2}e_{1}E_{21} − e_{1}E_{21}^{2} − r^{5}sE_{21}^{2}e_{1} − r^{2}sE_{21}e_{1}e_{2}e_{1}]} ω_{1}

= (r^{3}−s^{3}) (r^{5}s^{4}E_{21}^{2}e_{1} − rs^{2}e_{1}E_{21}^{2}) ω_{1}. Thus, we show S_{1} = A + B + C + D = 0. Up to now, we complete the proof of Lemma 3.10. □ By [BKL, Lemma 3.4], Lemmas 3.9 and 3.10 give rive to the following identity Lemma 3.12.

e_{1}E_{21}^{3} − sΔE_{21}e_{1}E_{21}^{2} + rs^{3}ΔE_{21}^{2}e_{1}E_{21} − (rs^{2})^{3}E_{21}^{3}e_{1} = 0.

Lemma 3.13. T_{2} preserves the (r, s)-Serre relations (G5)_{2}, (G6)_{2} into its associated object U_{s^{−1},r^{−1}}(G_{2}).

Proof. For the degree 4 (r, s)-Serre relation (G5)_{2}, we have to prove

(rs)^{6}T_{2}(e_{1})^{4}T_{2}(e_{2}) − (rs)^{3}(r+s)(r^{2}+s^{2})T_{2}(e_{1})^{3}T_{2}(e_{2})T_{2}(e_{1})
+ (rs)(r^{2}+s^{2})(r^{2}+rs+s^{2})T_{2}(e_{1})^{2}T_{2}(e_{2})T_{2}(e_{1})^{2}
− (r+s)(r^{2}+s^{2})T_{2}(e_{1})T_{2}(e_{2})T_{2}(e_{1})^{3} + T_{2}(e_{2})T_{2}(e_{1})^{4} = 0.

By virtue of the commutative relation in (3), this equality is reduced to

e_{1}T_{2}(e_{1})^{3} − sΔT_{2}(e_{1})e_{1}T_{2}(e_{1})^{2} + rs^{3}ΔT_{2}(e_{1})^{2}e_{1}T_{2}(e_{1}) − (rs^{2})^{3}T_{2}(e_{1})^{3}e_{1} = 0.

However, as T_{2}(e_{1}) = e_{1}e_{2} − r^{3}e_{2}e_{1} = (−r^{3})E_{21}, the above identity is exactly given by Lemma 3.12.

Similarly, we can verify that T_{2} preserves the (r, s)-Serre relation (G6)_{2} into its associated object U_{s^{−1},r^{−1}}(G_{2}). □

Lemma 3.14. T_{1} preserves the (r, s)-Serre relations (G5)_{2}, (G6)_{2} into its associated object U_{s^{−1},r^{−1}}(G_{2}).

Proof. For the degree 4 (r, s)-Serre relation (G5)_{2}, we have to prove

(rs)^{6}T_{1}(e_{1})^{4}T_{1}(e_{2}) − (rs)^{3}(r+s)(r^{2}+s^{2})T_{1}(e_{1})^{3}T_{1}(e_{2})T_{1}(e_{1})
+ (rs)(r^{2}+s^{2})(r^{2}+rs+s^{2})T_{1}(e_{1})^{2}T_{1}(e_{2})T_{1}(e_{1})^{2}
− (r+s)(r^{2}+s^{2})T_{1}(e_{1})T_{1}(e_{2})T_{1}(e_{1})^{3} + T_{1}(e_{2})T_{1}(e_{1})^{4} = 0.

Using the commutative relation in (7), this equality is reduced to (10) E_{112}T_{1}(e_{1})^{3}−rΔT_{1}(e_{1})E_{112}T_{1}(e_{1})^{2}+r^{3}sΔT_{1}(e_{1})^{2}E_{112}T_{1}(e_{1})−(r^{2}s)^{3}T_{1}(e_{1})^{3}E_{112} = 0.

Observing [E_{112} f_{1}] = −(r + s)^{2}E_{12}ω′_{1} and ω′_{1}E_{112} = rs^{2}E_{112}ω′_{1}, we get the following commutative relation

E_{112}T_{1}(e_{1}) = rs^{2}T_{1}(e_{1})E_{112} + r^{−1}s(r + s)^{2}E_{12},

by which, (10) is further reduced to

(11)  E_{12}T_{1}(e_{1})^{2} − r^{2}(r + s)T_{1}(e_{1})E_{12}T_{1}(e_{1}) + r^{5}sT_{1}(e_{1})^{2}E_{12} = 0.
Again, as [E_{12}, f_{1}] = −Δe_{2}ω′_{1}, we have
E_{12}T_{1}(e_{1}) = r^{2}sT_{1}(e_{1})E_{12} + r^{−1}sΔe_{2},

by which, (11) is finally reduced to the evident relation

e_{2}T_{1}(e_{1}) = r^{3}T_{1}(e_{1})e_{2},

owing to T_{1}(e_{1}) = −ω_{1}^{′−1}f_{1}.

The proof of the second part is similar. □

Theorem 3.15. T_{i}(i = 1, 2) are the Lusztig’s symmetries from U_{r,s}(G_{2}) to its associated quantum group U_{s^{−1},r^{−1}}(G_{2}) as ℚ-isomorphisms, which induce the usual Lusztig’s symmetries as ℚ(q)-automorphisms on the standard quantum group U_{q}(G_{2}) of Drinfel’d-Jimbo type only when r = q = s^{−1}. □

4. Appendix: Some Calculations in the proof of Proposition 2.3

For X = f_{1}^{4}f_{2}, the relevant terms of Δ^{(4)}(X) in (∗) and their paring-values are as follows:

TABULAR 1

SUMMANDS  1
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2} a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  xa
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{2}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{3}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  xa
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2} x^{2}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2} x^{3}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{4}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{2}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{3}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{4}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{5}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  xa
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{2}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{3}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{4}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{2}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2} x^{3}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2} x^{4}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{5}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{3}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{4}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{5}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  x^{6}a
SUMMANDS  2
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄xa
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{2}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{3}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄xa
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1} x̄x^{2}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1} x̄x^{3}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{4}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{2}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{3}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{4}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{5}a

SUMMANDS  2
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄xa
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{2}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{3}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{4}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{2}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1} x̄x^{3}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1} x̄x^{4}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{5}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{3}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{4}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{5}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  x̄x^{6}a
SUMMANDS  3
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{2}xa
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{2}x^{2}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{2}x^{3}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{2}x^{4}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{2}x^{2}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{2}x^{3}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{2}x^{4}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{2}x^{5}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{2}x^{3}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{2}x^{4}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{2}x^{5}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{2}x^{6}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{2}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{2}xa
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{2}x^{2}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{2}x^{3}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{2}xa
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1} x̄^{2}x^{2}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1} x̄^{2}x^{3}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{2}x^{4}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{2}x^{2}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{2}x^{3}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{2}x^{4}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{2}x^{5}a
SUMMANDS  4
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{3}xa
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{3}x^{2}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{3}x^{3}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{3}x^{4}a

SUMMANDS  4
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{3}x^{2}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{3}x^{3}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{3}x^{4}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{3}x^{5}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{3}x^{3}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{3}x^{4}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{3}x^{5}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{3}x^{6}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{3}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{3}xa
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{3}x^{2}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{3}x^{3}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{3}xa
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} x̄^{3}x^{2}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} x̄^{3}x^{3}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{3}x^{4}a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{3}x^{2}a
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{3}x^{3}a
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} x̄^{3}x^{4}a
ω_{1}^{′3}f_{1}ω′_{2} ⊗ ω_{1}^{′3}f_{2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{3}x^{5}a

SUMMANDS  5
ω_{1}^{′4}f_{2} ⊗ f_{1}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{4}xa
ω_{1}^{′4}f_{2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{4}x^{2}a
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{4}x^{3}a
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′3}f_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{4}x^{4}a
ω_{1}^{′4}f_{2} ⊗ f_{1}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{4}x^{2}a
ω_{1}^{′4}f_{2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{4}x^{3}a
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} x̄^{4}x^{4}a
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′3}f_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{4}x^{5}a
ω_{1}^{′4}f_{2} ⊗ f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{4}x^{3}a
ω_{1}^{′4}f_{2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{4}x^{4}a
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{4}x^{5}a
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′3}f_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  x̄^{4}x^{6}a
ω_{1}^{′4}f_{2} ⊗ f_{1}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{4}a
ω_{1}^{′4}f_{2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{4}xa
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{4}x^{2}a
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′3}f_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{4}x^{3}a
ω_{1}^{′4}f_{2} ⊗ f_{1}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{4}xa
ω_{1}^{′4}f_{2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} x̄^{4}x^{2}a
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} x̄^{4}x^{3}a
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′3}f_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{4}x^{4}a

SUMMANDS  5
ω_{1}^{′4}f_{2} ⊗ f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} x̄^{4}x^{2}a
ω_{1}^{′4}f_{2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗′ f_{1}ω_{1} ⊗ f_{1} x̄^{4}x^{3}a
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} x̄^{4}x^{4}a
ω_{1}^{′4}f_{2} ⊗ ω_{1}^{′3}f_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  x̄^{4}x^{5}a

The relevant terms and the pairing-values of Δ^{(4)}(X) for X = f_{2}f_{1}^{4} in (∗) are as follows:

TABULAR 2

SUMMANDS  1
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2} ay^{4}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  axy^{4}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{2}y^{4}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{3}y^{4}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  axy^{4}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{2}y^{4}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{3}y^{4}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{4}y^{4}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{2}y^{4}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{3}y^{4}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{4}y^{4}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{5}y^{4}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  axy^{4}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{2}y^{4}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{3}y^{4}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{4}y^{4}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{2}y^{4}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{3}y^{4}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{4}y^{4}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{5}y^{4}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{3}y^{4}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{4}y^{4}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{5}y^{4}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{6}y^{4}
SUMMANDS  2
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1} ay^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  axy^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{2}y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{3}y^{3}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  axy^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{2}y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{3}y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{4}y^{3}

SUMMANDS  2
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{2}y^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{3}y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{4}y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{5}y^{3}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  axy^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{2}y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{3}y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{4}y^{3}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{2}y^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{3}y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{4}y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{5}y^{3}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{3}y^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{4}y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{5}y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{6}y^{3}

SUMMANDS  3
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} axy^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}y^{2}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}y^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}y^{2}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}y^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}y^{2}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ay^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axy^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}y^{2}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axy^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}y^{2}
ω′_{2}f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}y^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{5}y^{2}

SUMMANDS  4
ω′_{2}f_{1}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} axy
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}y
ω′_{2}f_{1}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}y
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}y
ω′_{2}f_{1}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}y
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}y
ω′_{2}f_{1}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ay
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axy
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}y
ω′_{2}f_{1}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axy
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}y
ω′_{2}f_{1}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}y
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{5}y

SUMMANDS  5
f_{2}ω_{1}^{′4} ⊗ f_{1}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax
f_{2}ω_{1}^{′4} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′3}f_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}
f_{2}ω_{1}^{′4} ⊗ f_{1}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}
f_{2}ω_{1}^{′4} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′3}f_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}
f_{2}ω_{1}^{′4} ⊗ f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}
f_{2}ω_{1}^{′4} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′3}f_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}
f_{2}ω_{1}^{′4} ⊗ f_{1}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  a
f_{2}ω_{1}^{′4} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′3}f_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}

SUMMANDS  5
f_{2}ω_{1}^{′4} ⊗ f_{1}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax
f_{2}ω_{1}^{′4} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′3}f_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}
f_{2}ω_{1}^{′4} ⊗ f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}
f_{2}ω_{1}^{′4} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}
f_{2}ω_{1}^{′4} ⊗ ω_{1}^{′3}f_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{5}

The relevant terms and the pairing-values of Δ^{(4)}(X) for X = f_{1}^{2}f_{2}f_{1}^{2} in (∗) are as follows:

TABULAR 3

SUMMANDS  1
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2} ay^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  axy^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2} axy^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{2}y^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2} axy^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{2}y^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{2}y^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{3}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{2}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{3}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{3}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{4}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{4}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{5}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{5}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{6}y^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{2}y^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{3}y^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{3}y^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{4}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{3}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{4}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{4}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{5}y^{2}
SUMMANDS  2
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ay
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  axy
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1} axy

SUMMANDS  2
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{2}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  axy
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{2}y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{3}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{2}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{3}y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{3}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{4}y
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{2}x̄y^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{3}x̄y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{3}x̄y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{4}x̄y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{4}x̄y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{5}x̄y^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{3}x̄y^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{4}x̄y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{4}x̄y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{5}x̄y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{5}x̄y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{6}x̄y^{2}

SUMMANDS  3
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄y
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}x̄y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}x̄y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄^{2}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{5}x̄^{2}y^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄y
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}x̄y

SUMMANDS  3
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}x̄y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}x̄y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}x̄^{2}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}x̄^{2}y^{2}
SUMMANDS  4
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax̄
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄^{2}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}x̄
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}x̄
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{4}x̄^{2}y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄^{2}y
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  axx̄
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄^{2}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}x̄
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}x̄
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{5}x̄^{2}y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}x̄^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}x̄^{2}y
SUMMANDS  5
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′3}f_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′3}f_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄^{2}

SUMMANDS  5
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′3}f_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{5}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  axx̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′3}f_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′3}f_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{5}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}x̄^{2}
ω_{1}^{′2}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′3}f_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}x̄^{2}

The relevant terms and the pairing-values of Δ^{(4)}(X) for X = f_{1}^{3}f_{2}f_{1} in (∗) are as follows:

TABULAR 4

SUMMANDS  1
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{2}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ay
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  axy
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{2}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  axy
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{2}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{2}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{2}y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{3}y
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2}  axy
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{2}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{2}y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{3}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{3}y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{4}y
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{3}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{3}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{4}y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{4}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{5}y
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{3}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{4}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{4}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{5}y

SUMMANDS  1
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{5}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{6}y
SUMMANDS  2
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1}  a
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{2}
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{3}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1}  axx̄y
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{2}x̄y
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{2}x̄y
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{3}x̄y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{2}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{3}x̄y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{3}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{4}x̄y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{3}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{4}x̄y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{4}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{5}x̄y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{3}x̄y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{4}x̄y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{4}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{5}x̄y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{5}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{2} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{6}x̄y
SUMMANDS  3
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax̄
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄^{2}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄^{2}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}x̄
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄^{2}y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄^{2}y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{5}x̄^{2}y

SUMMANDS  3
f_{1}ω_{1}^{′3}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  axx̄
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄^{2}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄^{2}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}x̄
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄^{2}y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}x̄^{2}y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}x̄^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω_{1}^{′2}f_{2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}x̄^{2}y
SUMMANDS  4
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax̄^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄^{2}
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′3}f_{2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄^{3}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}x̄^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}x̄^{2}
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′3}f_{2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{4}x̄^{3}y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  axx̄^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄^{2}
f_{1}ω_{1}^{′3}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄^{2}
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′3}f_{2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄^{3}y
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}x̄^{2}
ω′_{1}f_{1}ω_{1}^{′2}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}x̄^{2}
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′3}f_{2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{5}x̄^{3}y
ω_{1}^{′2}f_{1}ω′_{1}ω′_{2} ⊗ ω_{1}^{′2}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}x̄^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′3}f_{2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{5}x̄^{3}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′3}f_{2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}x̄^{3}y
SUMMANDS  5
ω_{1}^{′3}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} axx̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄^{3}

SUMMANDS  5
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′3}f_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′3}f_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{4}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′3}f_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{5}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  axx̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′3}f_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′3}f_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{5}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}x̄^{3}
ω_{1}^{′3}f_{2}ω′_{1} ⊗ ω_{1}^{′3}f_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}x̄^{3}

The relevant terms and the results of calculations of Δ^{(4)}(X) for X = f_{1}f_{2}f_{1}^{3} in (∗) are as follows:

TABULAR 5

SUMMANDS  1
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ay^{3}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  axy^{3}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  axy^{3}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{2}y^{3}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{2}y^{3}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{3}y^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  axy^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{2}y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{2}y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{3}y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{3}y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{4}y^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{2}y^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{3}y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{3}y^{3}

SUMMANDS  1
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{4}y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2}  ax^{4}y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ ω′_{2}f_{1} ⊗ f_{2} ax^{5}y^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{3}y^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{4}y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{4}y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2} ax^{5}y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{5}y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{1}ω′_{2} ⊗ f_{2}  ax^{6}y^{3}
SUMMANDS  2
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ay^{2}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  axy^{2}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  axy^{2}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{2}y^{2}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{2}y^{2}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{3}y^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  axy^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{2}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{2}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{3}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{3}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{4}y^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{2}y^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{3}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{3}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{4}y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1}  ax^{4}y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{1}ω′_{1}ω′_{2} ⊗ f_{2}ω′_{1} ⊗ f_{1} ax^{5}y^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{3}x̄y^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{4}x̄y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{4}x̄y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1} ax^{5}x̄y^{3}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{5}x̄y^{3}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1} ⊗ ω′_{1}f_{2} ⊗ f_{1}  ax^{6}x̄y^{3}
SUMMANDS  3
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ay
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  axy
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1} axy
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}y
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{2}y
f_{1}ω′_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}y

SUMMANDS  3
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axy
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ f_{1}ω′_{2}ω_{1}^{′2} ⊗ f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}y
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}x̄y^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}x̄y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{4}x̄y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{5}x̄y^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}x̄y^{2}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}x̄y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{5}x̄y^{2}
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}x̄y^{2}
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω_{1}^{′2}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}x̄y^{2}

SUMMANDS  4
f_{1}ω′_{2}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  a
f_{1}ω′_{2}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax
f_{1}ω′_{2}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax
f_{1}ω′_{2}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}
f_{1}ω′_{2}ω_{1}^{′3} ⊗ f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄y
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄y
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}x̄y
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}x̄y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{4}x̄y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{5}x̄y
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}x̄y
ω′_{1}ω′_{2}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}x̄y
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}x̄y

SUMMANDS  4
ω_{1}^{′2}ω′_{2}f_{1}ω′_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{5}x̄y
ω_{1}^{′3}ω′_{2}f_{1} ⊗ ω′_{1}f_{2}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}x̄y
SUMMANDS  5
ω′_{1}f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  axx̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ f_{1}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  axx̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{2}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{2}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′3}f_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{3}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′3}f_{1} ⊗ f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{2}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{3}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{3}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′3}f_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{4}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1}  ax^{4}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′3}f_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ f_{1}ω′_{1} ⊗ f_{1} ax^{5}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{3}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω′_{1}f_{1}ω_{1}^{′2} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{4}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{4}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′3}f_{1} ⊗ ω′_{1}f_{1}ω′_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1} ax^{5}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′2}f_{1}ω′_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{5}x̄
ω′_{1}f_{2}ω_{1}^{′3} ⊗ ω_{1}^{′3}f_{1} ⊗ ω_{1}^{′2}f_{1} ⊗ ω′_{1}f_{1} ⊗ f_{1}  ax^{6}x̄

References

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[BGH2] N. Bergeron, Y. Gao and N. Hu, Representations of two-parameter quantum orthogonal groups and symplectic groups, math.QA/0510124, Contemp. Math. of AMS (to appear), in 2006.

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[BW3] G. Benkart and S. Witherspoon, Restricted two-parameter quantum groups, Fields Institute Communications, “Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry”, vol. 40, Amer. Math. Soc., Providence, RI, 2004, pp. 293–318.

[Dr] V.G. Drinfel’d, Quantum groups, in “Proceedings ICM”, Berkeley, Amer. Math. Soc. 1987, pp. 798–820.

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Department of Mathematics, East China Normal University, Shanghai 200062, PR China

    E-mail address: nhhu@euler.math.ecnu.edu.cn

Department of Mathematics, East China Normal University, Shanghai 200062, PR China