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【短课程】Duality ABC in representation theory

Tuesday, June 9th, 2020, 5:13 PM  
摘要 (abstract):

This mini-course intends to cover the following topics (1)-(4) and some of the additional topics (depending on the time and interest). The prerequisites include an introductory course on semisimple Lie algebras and one on symmetric groups over C.

(1) Schur duality Schur duality relates representations of general linear Lie algebra gl(n) (or group GL(n)) and of the symmetric group S_d.

(2) Howe duality Howe duality (of type A) relates representations of GL(m) and GL(n). In addition, Howe duality can also be formulated between a pair of classical groups (or classical Lie algebras) of type BCD. Howe duality is closely related to classical invariant theory.

(3) q-Schur duality of type A (or Schur-Jimbo duality) Schur-Jimbo duality relates a quantum group U_q(gl(n)) and Hecke algebra H_q(S_d).

(4) q-Schur duality of type BCD (or i-Schur duality) i-Schur duality relates an “i-quantum group” U^i and Hecke algebra H_q(B_d) of type B. The pair (U_q(gl(n)),U^i) forms a quantum symmetric pair.

Some possible additional topics include:

(CB) Canonical basis and q-Schur dualities For the q-Schur dualities in (3)-(4), the canonical bases on the q-tensor space from the Hecke algebra or quantum algebra viewpoint coincide.

(AFF) q-Schur dualities of a?ne type

(FL) q-Schur dualities and geometry of flag varieties
授课时间 (schedule):


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