Abstract: In this talk, we will show that, in the type A family — the family of quantum gln, glm|n and ˆgln, there i s a new basis for every member of the family, which contains a set of generators, such that the structure constants associated with multiplying a basis element by a generator are completely determined by the labelling matrices.
The first type of such bases was constructed two decades ago by Beilinson-Lusztig- MacPherson for quantum gln, using a geometric setting (i.e., the partial flag varieties) for quantum Schur algebras. We now use the Hecke algebras of symmetric groups or affine symmetric groups and their associated Schur (super)algebras to construct such bases for quantum glm|n and ˆgln. The construction is purely algebraic and combinatorial. I will mainly focus on the affine case in this talk. An interesting application of this construction is some new multiplication formulas in the Hall algebra of a cyclic quiver.
This is joint work with Qiang Fu (the affine case) and with Haixia Gu (the super case).