The quantum general linear supergroup, canonical bases and Kazhdan-Lusztig polynomials

Abstract

Canonical bases of the tensor powers of the natural Uq(gl m|n )-module V are constructed by adapting the work of Frenkel, Khovanov and Kirrilov to the quantum supergroup setting. This result is generalized in several directions. We first construct the canonical bases of the Z2-graded symmetric algebra of V and tensor powers of this superalgebra; then construct canonical bases for the superalgebra Oq(M m|n ) of a quantum (m, n) × (m, n)-supermatrix; and finally deduce from the latter result the canonical basis of every irreducible tensor module for Uq(gl m|n ) by applying a quantum analogue of the Borel-Weil construction.

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

“…A brief review of early works on the theory and applications of quantum supergroups can be found in [63]. Partially successful constructions of crystal and canonical bases for quantum supergroups were given in [1,12,37,53,55,54,72,73].…”

Categorification of tensor powers of the vector representation of $$U_q({\mathfrak {gl}}(1|1))$$ U q ( gl ( 1 | 1 ) )

“…Our categorification relies on a very careful analysis of the representation theory of gl(1|1) and its canonical basis (based on [Zha09]). In the categorification, indecomposable projective modules correspond to canonical basis elements, that we can compute explicitly via a diagram calculus, analogous to the diagram calculus developed in [FK97] for sl 2 .…”

First fundamental theorems of invariant theory for quantum supergroups

“…A brief review of early works on the theory and applications of quantum supergroups can be found in [63]. Partially successful constructions of crystal and canonical bases for quantum supergroups were given in [1,12,37,53,55,54,72,73].…”