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26
A super duality and KazhdanLusztig polynomials
, 2004
"... We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A), via the canonical and dual canonical bases on Fock spaces which in turn may be seen as a reformulation of the KazhdanLusztig theory. As a consequence, the usual parabolic Kazhdan ..."
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Cited by 27 (14 self)
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We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A), via the canonical and dual canonical bases on Fock spaces which in turn may be seen as a reformulation of the KazhdanLusztig theory. As a consequence, the usual parabolic KazhdanLusztig polynomials of type A compute the characters of finitedimensional irreducible modules of the general linear Lie superalgebra.
Howe Duality and Combinatorial Character Formula for Orthosymplectic Lie superalgebras
, 2003
"... We study the Howe dualities involving the reductive dual pairs (O(d), spo(2m2n)) and (Sp(d), osp(2m2n)) on the (super)symmetric tensor of C d ⊗ C mn. We obtain complete decompositions of this space with respect to their respective joint actions. We also use these dualities to derive a character ..."
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Cited by 23 (7 self)
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We study the Howe dualities involving the reductive dual pairs (O(d), spo(2m2n)) and (Sp(d), osp(2m2n)) on the (super)symmetric tensor of C d ⊗ C mn. We obtain complete decompositions of this space with respect to their respective joint actions. We also use these dualities to derive a character formula for these irreducible representations of spo(2m2n) and osp(2m2n) that appear in these decompositions.
Manifestly Conformal Descriptions and Higher Symmetries of Bosonic Singletons
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2010
"... The usual ambient space approach to conformal fields is based on identifying the ddimensional conformal space as the Dirac projective hypercone in a flat d + 2dimensional ambient space. In this work, we explicitly concentrate on singletons of any integer spin and propose an approach that allows o ..."
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Cited by 18 (8 self)
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The usual ambient space approach to conformal fields is based on identifying the ddimensional conformal space as the Dirac projective hypercone in a flat d + 2dimensional ambient space. In this work, we explicitly concentrate on singletons of any integer spin and propose an approach that allows one to have both locality and conformal symmetry manifest. This is achieved by using the ambient space representation in the fiber rather than in spacetime. This approach allows us to characterize a subalgebra of higher symmetries for any bosonic singleton, which is a candidate higherspin algebra for mixed symmetry gauge fields on anti de Sitter spacetime. Furthermore, we argue that this algebra actually exhausts all higher symmetries.
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 ..."
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Cited by 11 (5 self)
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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 ..."
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Cited by 11 (8 self)
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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
Infinitedimensional Lie superalgebras and . . .
 COMM. MATH. PHYS
, 2002
"... Making use of a Howe duality involving the infinitedimensional Lie superalgebra ̂ gl∞ ∞ and the finitedimensional group GLl of [CW3] we derive a character formula for a certain class of irreducible quasifinite representations of ̂ gl∞ ∞ in terms of hook Schur functions. We use the reduction p ..."
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Cited by 9 (5 self)
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Making use of a Howe duality involving the infinitedimensional Lie superalgebra ̂ gl∞ ∞ and the finitedimensional group GLl of [CW3] we derive a character formula for a certain class of irreducible quasifinite representations of ̂ gl∞ ∞ in terms of hook Schur functions. We use the reduction procedure of ̂ gl ∞ ∞ to ˆ gl nn to derive a character formula for a certain class of level 1 highest weight irreducible representations of ˆ gl nn, the affine Lie superalgebra associated to the finitedimensional Lie superalgebra gl nn. These modules turn out to form the complete set of integrable ˆ gl nnmodules of level 1. We also show that the characters of all integrable level 1 highest weight irreducible ˆ gl mnmodules may be written as a sum of products of hook Schur functions.
The Howe duality and Lie superalgebras
 Noncommutative Structures in Mathematics and Physics Proceedings of the NATO Advanced Research Workshop
"... Abstract. Howe’s duality is considered from a unifying point of view based on Lie superalgebras. New examples are offered. In particualr, we construct several simplest spinoroscillator representations and compute their highest weights for the “stringy ” Lie superalgebras (i.e., Lie superalgebras of ..."
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Cited by 9 (4 self)
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Abstract. Howe’s duality is considered from a unifying point of view based on Lie superalgebras. New examples are offered. In particualr, we construct several simplest spinoroscillator representations and compute their highest weights for the “stringy ” Lie superalgebras (i.e., Lie superalgebras of complex vector fields (or their nontrivial central extensions) on the supercircle S 1n and its twosheeted cover associated with the Möbius bundle). In our two lectures we briefly review, on the most elementary level, several results and problems unified by “Howe’s duality”. Details will be given elsewhere. The ground field in the lectures is C. In his famous preprint [24] R. Howe gave an inspiring explanation of what can be “dug out ” from H. Weyl’s “wonderful and terrible ” book [55], at least as far as invariant theory is concerned, from a certain unifying viewpoint. According to Howe, much is based on a remarkable correspondence between certain irreducible representations of Lie subalgebras Γ
Categorification of tensor powers of the vector representation
 of Uq(gl(1 ∣ 1)). arXiv:1305.6162
, 2013
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CHARACTER FORMULA FOR INFINITE DIMENSIONAL UNITARIZABLE MODULES OF THE GENERAL LINEAR SUPERALGEBRA
, 2003
"... Abstract. The Fock space of m+p bosonic and n+q fermionic quantum oscillators forms a unitarizable module of the general linear superalgebra gl m+pn+q. Its tensor powers decompose into direct sums of infinite dimensional irreducible highest weight gl m+pn+qmodules. We obtain an explicit decomposi ..."
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Cited by 8 (4 self)
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Abstract. The Fock space of m+p bosonic and n+q fermionic quantum oscillators forms a unitarizable module of the general linear superalgebra gl m+pn+q. Its tensor powers decompose into direct sums of infinite dimensional irreducible highest weight gl m+pn+qmodules. We obtain an explicit decomposition of any tensor power of this Fock space into irreducibles, and develop a character formula for the irreducible gl m+pn+qmodules arising in this way. Key words: Lie superalgebra, unitarizable representations, Howe duality, character formula.
Remarks on the SchurHoweSergeev Duality
, 2000
"... Abstract. We establish a new Howe duality between a pair of two queer Lie superalgebras (q(m), q(n)). This gives a representation theoretic interpretation of a wellknown combinatorial identity for Schur Qfunctions. We further establish the equivalence between this new Howe duality and the Schur–Se ..."
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Cited by 7 (5 self)
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Abstract. We establish a new Howe duality between a pair of two queer Lie superalgebras (q(m), q(n)). This gives a representation theoretic interpretation of a wellknown combinatorial identity for Schur Qfunctions. We further establish the equivalence between this new Howe duality and the Schur–Sergeev duality between q(n) and a central extension ˜ Hk of the hyperoctahedral group Hk. We show that the zeroweight space of a q(n)module with highest weight λ given by a strict partition of n is an irreducible module over the finite group ˜ Hn parameterized by λ. We also discuss some consequences of this Howe duality.