Results 1  10
of
26
Highest weight categories arising from Khovanov's diagram algebra II: Koszulity
"... This is the second of a series of four articles studying various generalisations of Khovanov’s diagram algebra. In this article we develop the general theory of Khovanov’s diagrammatically defined “projective functors” in our setting. As an application, we give a direct proof of the fact that the ..."
Abstract

Cited by 103 (12 self)
 Add to MetaCart
(Show Context)
This is the second of a series of four articles studying various generalisations of Khovanov’s diagram algebra. In this article we develop the general theory of Khovanov’s diagrammatically defined “projective functors” in our setting. As an application, we give a direct proof of the fact that the
Super duality and irreducible characters of orthosymplectic Lie superalgebras
 Invent. Math
"... ar ..."
(Show Context)
Howe Duality for Lie Superalgebras
 COMPOSITIO MATH
, 2000
"... We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicityfree decomposition of a symmetric and skewsymmetric algebra (in the super sense) under the action of the dual pair and present explicit formulas for the highest weight vectors in each ..."
Abstract

Cited by 26 (7 self)
 Add to MetaCart
(Show Context)
We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicityfree decomposition of a symmetric and skewsymmetric algebra (in the super sense) under the action of the dual pair and present explicit formulas for the highest weight vectors in each isotypic subspace of the symmetric algebra. We give an explicit multiplicityfree decomposition into irreducible gl(mn)modules of the symmetric and skewsymmetric algebras of the symmetric square of the natural representation of gl(mn). In the former case we find as well explicit formulas for the highest weight vectors. Our work unifies and generalizes the classical results in symmetric and skewsymmetric models and admits several applications.
Dual canonical bases and KazhdanLusztig polynomials
"... Abstract. We derive a formula for the entries of the (unitriangular) transition matrices between the standard monomial and dual canonical bases of the irreducible polynomial representations of Uq(gl n) in terms of KazhdanLusztig polynomials. 1. ..."
Abstract

Cited by 21 (6 self)
 Add to MetaCart
Abstract. We derive a formula for the entries of the (unitriangular) transition matrices between the standard monomial and dual canonical bases of the irreducible polynomial representations of Uq(gl n) in terms of KazhdanLusztig polynomials. 1.
Irreducible characters of general linear superalgebra and super duality
, 2009
"... 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 finitedimensional modules, by directly relating the problem to the classical KazhdanLusztig theory. We further verify a parabolic version of a conj ..."
Abstract

Cited by 20 (6 self)
 Add to MetaCart
(Show Context)
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 finitedimensional modules, by directly relating the problem to the classical KazhdanLusztig theory. We further verify a parabolic version of a conjecture of Brundan on the irreducible characters in the BGG category O of the general linear superalgebra. We also prove the super duality conjecture.
BrundanKazhdanLusztig and super duality conjectures
"... Abstract. 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 KazhdanLusztig theories which was initiated by Brundan. We show that the BrundanKazhdanLus ..."
Abstract

Cited by 13 (6 self)
 Add to MetaCart
(Show Context)
Abstract. 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 KazhdanLusztig theories which was initiated by Brundan. We show that the BrundanKazhdanLusztig (BKL) polynomials for gl(mn) in our parabolic setup can be identified with the usual parabolic KazhdanLusztig polynomials. We establish some special cases of the BKL conjecture on the parabolic category O of gl(mn)modules and additional results which support the BKL conjecture and super duality conjecture. Contents
KOSTANT HOMOLOGY FORMULAS FOR OSCILLATOR MODULES OF LIE SUPERALGEBRAS
, 2009
"... We provide a systematic approach to obtain formulas for characters and Kostant uhomology groups of the oscillator modules of the finite dimensional general linear and orthosymplectic superalgebras, via Howe dualities for infinite dimensional Lie algebras. Specializing these Lie superalgebras to L ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
(Show Context)
We provide a systematic approach to obtain formulas for characters and Kostant uhomology groups of the oscillator modules of the finite dimensional general linear and orthosymplectic superalgebras, via Howe dualities for infinite dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the abovementioned uhomology computation are worked out.
Howe duality and Kostant’s homology formula for infinitedimensional Lie superalgebras
 Int. Math. Res. Not. 2008 (2008), Art. ID rnn
"... Abstract. Using Howe duality we compute explicitly Kostanttype homology groups for a wide class of representations of the infinitedimensional Lie superalgebra b gl ∞ ∞ and its classical subalgebras at positive integral levels. We also obtain Kostanttype homology formulas for the Lie algebra b gl ..."
Abstract

Cited by 11 (8 self)
 Add to MetaCart
(Show Context)
Abstract. Using Howe duality we compute explicitly Kostanttype homology groups for a wide class of representations of the infinitedimensional Lie superalgebra b gl ∞ ∞ and its classical subalgebras at positive integral levels. We also obtain Kostanttype homology formulas for the Lie algebra b gl ∞ at negative integral levels. We further construct resolutions in terms of generalized Verma modules for these representations. Contents
BrundanKazhdanLusztig Conjectures for the general linear Lie superalgebras
"... ar ..."
(Show Context)
A Fock space approach to representation theory of osp(22n), arXiv:math.QA/0510106
, 2006
"... Abstract. A Fock space is introduced that admits an action of a quantum group of type A supplemented with some extra operators. The canonical and dual canonical basis of the Fock space are computed and then used to derive the finitedimensional tilting and irreducible characters for the Lie superalg ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
Abstract. A Fock space is introduced that admits an action of a quantum group of type A supplemented with some extra operators. The canonical and dual canonical basis of the Fock space are computed and then used to derive the finitedimensional tilting and irreducible characters for the Lie superalgebra osp(22n). We also determine all the composition factors of the symmetric tensors of the natural osp(22n)module.