Super duality and Kazhdan-Lusztig polynomials

  title={Super duality and Kazhdan-Lusztig polynomials},
  author={Shun-Jen Cheng and Weiqiang Wang and R. B. Zhang},
  journal={Transactions of the American Mathematical Society},
We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional irreducible modules of the general linear Lie superalgebra are computed by the usual parabolic Kazhdan-Lusztig polynomials of type A. In addition, we establish closed formulas for canonical and dual canonical bases for the tensor product of any two fundamental… 

Figures from this paper

The brundan-kazhdan-lusztig conjecture for general linear lie superalgebras

In the framework of canonical and dual canonical bases of Fock spaces, Brundan in 2003 formulated a Kazhdan-Lusztig type conjecture for the characters of the irreducible and tilting modules in the

Brundan-Kazhdan-Lusztig and super duality conjectures

We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their

Super duality and irreducible characters of ortho-symplectic Lie superalgebras

We formulate and establish a super duality which connects parabolic categories O for the ortho-symplectic Lie superalgebras and classical Lie algebras of BCD types. This provides a complete and

Dualities for Lie superalgebras

We explain how Lie superalgebras of types gl and osp provide a natural framework generalizing the classical Schur and Howe dualities. This exposition includes a discussion of super duality, which

Irreducible Characters of General Linear Superalgebra and Super Duality

We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finite-dimensional modules, by directly relating

Super duality for general linear Lie superalgebras and applications

We apply the super duality formalism recently developed by the authors to obtain new equivalences of various module categories of general linear Lie superalgebras. We establish the correspondence of


We provide a new proof of the super duality equivalence between infinite-rank parabolic BGG categories of general linear Lie (super) algebras conjectured by Cheng and Wang and first proved by Cheng

A New Approach to Kazhdan-lusztig Theory of Type $b$ Via Quantum Symmetric Pairs

We show that Hecke algebra of type B and a coideal subalgebra of the type A quantum group satisfy a double centralizer property, generalizing the Schur-Jimbo duality in type A. The quantum group of

Representations of the general linear Lie superalgebra in the BGG category O

This is a survey of some recent developments in the highest weight repesentation theory of the general linear Lie superalgebra gln|m(C). The main focus is on the analog of the Kazhdan-Lusztig



Howe Duality for Lie Superalgebras

We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense)

Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra q(n)

Tilting Modules for Lie Superalgebras

Abstract We develop a general theory of tilting modules for graded Lie superalgebras, extending work of Soergel for graded Lie algebras. The main result of the article gives a twisted version of BGG

UniversalR-matrix for quantized (super)algebras

For quantum deformations of finite-dimensional contragredient Lie (super)algebras we give an explicit formula for the universalR-matrix. This formula generalizes the analogous formulae for quantized

Kazhdan-Lusztig polynomials and canonical basis

AbstractIn this paper we show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the groupSn coincide with the coefficients of the canonical basis innth tensor


Let be the tensor algebra of the identity representation of the Lie superalgebras in the series and . The method of Weyl is used to construct a correspondence between the irreducible representations

Kazhdan-Lusztig polynomials and character formula for the Lie superalgebragI(m/n)

We find the character formula for irreducible finite-dimensionalgl(m/n)-modules. Also multiplicities of the composition factors in a Kac module are calculated.

A Littlewood-Richardson rule for evaluation representations of Uq(ŝln)

We give a combinatorial description of the composition fact ors of the induction product of two evaluation modules of the affine Iwahori-Hecke algebra o f typeGLm. Using quantum affine Schur-Weyl

Kazhdan-Lusztig conjecture and holonomic systems

In [7], D. Kazhdan and G. Lusztig gave a conjecture on the multiplicity of simple modules which appear in a Jordan-H61der series of the Verma modules. This multiplicity is described in the terms of