A plactic algebra of extremal weight crystals and the Cauchy identity for Schur operators

Abstract

We give a new bijective interpretation of the Cauchy identity for Schur operators which is a commutation relation between two formal power series with operator coefficients. We introduce a plactic algebra associated with the Kashiwara's extremal weight crystals over the Kac-Moody algebra of type A +∞ , and construct a Knuth type correspondence preserving the plactic relations. This bijection yields the Cauchy identity for Schur operators as a homomorphic image of its associated identity for plactic characters of extremal weight crystals, and also recovers Sagan and Stanley's correspondence for skew tableaux as its restriction.Keywords Plactic algebra · Crystal · Schur operator where s λ and s ⊥ λ are linear operators on Λ induced from the left multiplication by s λ (x) and its adjoint with respect to the Hall inner product on Λ, respectively. One may regard s λ and s ⊥ λ as operators on QP = λ∈P Qλ, where λ is identified with 428 J Algebr Comb (2011) 34:427-449 s λ (x). Moreover s λ and s ⊥ λ can be given as Schur functions in certain locally noncommutative operators on QP called Schur operators by Fomin, while P (x) and Q(x) can be written as Cauchy products in Schur operators and x [3,4].Let y = {y 1 , y 2 , . . .} be another set of formal commuting variables. It is well known that the following commutation relation holds:1.1) called generalized Cauchy identity or Cauchy identity for Schur operators. Considering both sides as operators with coefficients in Λ x ⊗ Λ y and then equating each entry of their matrix forms, we obtain a Cauchy identity for skew Schur functions [16],where α, β are given partitions. A bijective interpretation of the Cauchy identity for skew Schur functions was given by Sagan and Stanley [17], and it was extended to a bijection in a more general framework by Fomin [3] including various analogues of Knuth correspondence.Recently, a new representation theoretic interpretation of the Cauchy identity for Schur operators was given by the author [11] using the notion of Kashiwara's extremal weight crystals [8] over the quantized enveloping algebra associated with the Kac-Moody algebra of type A +∞ , say gl >0 . It is proved that a Schur operator can be realized as a functor of tensoring by an extremal weight crystal element and (1.1) can be understood as a non-commutative character identity corresponding to the decomposition of the crystal graph of the Fock space with infinite positive level, which is an infinite analogue of the level n fermionic Fock space decomposition due to Frenkel [5].Motivated by a categorification of Schur operators in [11], we give a new combinatorial way to explain both the Cauchy identities for Schur operators and skew Schur functions in terms of a single bijection. More precisely, the main result in this paper is to construct a Knuth type correspondence, which gives a bijective interpretation of the identity (1.1) or its dual form, as the usual Knuth correspondence does for the Cauchy product, and also recovers the Sagan and Stanley's correspondence as its restrictio...

Combinatorial representation theory of Lie algebras. Richard Stanley’s work and the way it was continued

“…A representation-theoretic interpretation of (2), also based on crystals, is given in [19]. Furthermore, an application of the skew dual RSK correspondence in [32] to the construction of crystal bases of certain modules for the quantum superalgebra U q (gl(m|n)) is given in [20].…”