Insertion Scheme for the Classical Lie Algebras

Abstract

In this paper, we give an insertion scheme for the tableaux of KashiwaraNakashima realizing the crystal bases of the irreducible highest weight modules over the classical Lie algebras. It gives an explicit combinatorial description of the decomposition of the tensor product V ðlÞ V ðmÞ into irreducible modules.

Crystal Bases for Quantum Classical Algebras and Nakajima’s Monomials

“…Finally, we close this section with a bijection between the monomial realization and the Kashiwara-Nakashima tableau realization. Proposition 3.12 [9,10].…”

Correspondence between Young walls and Young tableaux and its application

“…Now, we have a tensor product decomposition rule, so-called Littlewood-Richardson rule, using tableaux given by Kashiwara and Nakashima. That is, Proposition 3.17 [1,[12][13][14]. Let U q (g) (g = A n , C n , B n , D n ) be a quantum classical Lie algebra.…”

“…Here, the notation T ← U for the tableaux T , U consisting of one column is referred to [12]. Then by [12,Theorem 4.14], it is clear that the map ψ is a crystal isomorphism.…”

“…Then the map φ : B(λ) ⊗ B(µ) → k B(τ k ) sending T ⊗ U to T ← U is the crystal isomorphism over U q (g)-modules. Here, the notation T ← U for the tableaux T , U is referred to [12].…”

“…The basis vector is parameterized by certain tableaux with given shape which is different from generalized Young diagram given by Kashiwara and Nakashima. Moreover, motivated by the fact that the tableau realization of B(λ) given by Kashiwara and Nakashima also can be derived from the Young wall realization of B(λ) corresponding to the connected component having the largest number of blocks, we give a crystal isomorphism from our new realization to that of Kashiwara and Nakashima. Indeed, it will be a part of insertion scheme for the crystal of the classical Lie algebras given in [1,[12][13][14]. Finally, as an application of this 1-1 correspondence, we give an insertion scheme for the reduced proper Young walls for the classical Lie algebras of type A n and C n .…”