Crystal bases for the quantum superalgebra $U_q(\mathfrak {gl}(m,n))$

Parastatistics Algebra, Young Tableaux and the Super Plactic Monoid

“…The cyclic permutation of I 3 (V ) vanishes due to the super Jacobi idenity 3) and, counting the dimensions, we conclude that I 3 (V ) = V (2,1) . For the S 3 -representation I(3) the Jacobi identity implies…”

“…which is a deformation of the S 3 -module I(3) = eC[S 3 ] ∼ = S (2,1) . To this end we find an idempotent e(q) ∈ H 3 (q) which is a deformation of the Eulerian idempotent e, in the sense that e(1) = e.…”

“…It is spanned by two elements Γ We are now going to sketch how the Hecke algebra is related to the deformation U q (gl m|n ) of UEA U (gl m|n ) providing a generalization of the Schur-Weyl duality. For the defining relations of the quantum UEA U q (gl m|n ) we send the reader to [2].…”

Crystal graphs for Lie superalgebras and Cauchy decomposition

“…[15]). The purpose of this paper is to understand the decomposition (1.1) within a framework of (abstract) crystal graphs for Lie superalgebras which were developed by Benkart, Kang and Kashiwara [1]. For λ ∈ P m|n , we denote by B m|n (λ) the set of all (m, n)-hook semistandard tableaux of shape λ, which parameterizes the basis element of V m|n (λ) [2].…”

“…For λ ∈ P m|n , we denote by B m|n (λ) the set of all (m, n)-hook semistandard tableaux of shape λ, which parameterizes the basis element of V m|n (λ) [2]. According to the crystal base theory in [1], B m|n (λ) becomes a colored oriented graph, which we call a crystal graph for gl m|n or gl m|n -crystal. As in the case of symmetrizable Kac-Moody algebras, the crystal graphs for gl m|n have nice behaviors under tensor product, and we can decompose various finite dimensional representations of gl m|n in a purely combinatorial way (cf.…”