Lusztig's q-analogue of weight multiplicity and one-dimensional sums for affine root systems

Affine crystals, one-dimensional sums and parabolic Lusztig q-analogues

This paper is concerned with one-dimensional sums in classical affine types. We prove a conjecture of the third author and Zabrocki by showing they all decompose in terms of one-dimensional sums related to affine type A provided the rank of the root system considered is sufficiently large. As a consequence, any one-dimensional sum associated to a classical affine root system with sufficiently large rank can be regarded as a parabolic Lusztig q-analogue.

On the X=M=K Conjecture

In the large rank limit, for any nonexceptional affine algebra, the graded branching multiplicities known as one-dimensional sums, are conjectured to have a simple relationship with those of type A, which are known as generalized Kostka polynomials. This is called the X=M=K conjecture. It is proved for tensor products of the symmetric power Kirillov-Reshetikhin modules for all nonexceptional affine algebras except those whose Dynkin diagrams are isomorphic to that of untwisted affine type D near the zero node. Combined with results of Lecouvey, this realizes the above one-dimensional sums of a…

Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n)

The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra $\mathfrak{gl}(m|n)$ over $\C$ was solved a few years ago by V. Serganova. In this article, we present an entirely different approach. One consequence is a direct and elementary proof of a conjecture made by van der Jeugt and Zhang for the composition multiplicities of Kac modules. This does not seem to follow easily from Serganova's formula, since that involves certain alternating sums. We also compute Ext's between simple modules in the category of finite dimensional represen…

通过Koornwinder多项式和X = K现象的量子化Howe型对偶性

Quantized Howe-type dualities via Koornwinder polynomials and the X=K phenomenon

利用Mimachi的Koornwinder多项式对偶柯西公式等工具，推导经典仿射型Kirillov-Reshetikhin列晶体张量积相关一维和与Lusztig q-重数的等式，证明除\(B_n^{(1)}\)和\(D_n^{(1)}\)外的对偶性。

Extended graphical calculus for categorified quantum sl(2)

A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. We obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements. These formulas have integral coefficients and imply that o…

Quantum affine algebras at roots of unity and equivariant K-theory

In this short note, we show that the Ginzburg-Vasserot map between the quantum affine algebra of type A_(n-1) and the equivariant K-theory group of the Steinberg Variety (of n-step flags in C^d) restricts and remains surjective at the level of the integral forms. In particular, this shows that the parametrization of irreducible, finite-dimensional modules and the Kazhdan-Lusztig multiplicity formulas are valid for the (restricted) specialization at roots of unity.