A Plactic Algebra for Semisimple Lie Algebras

Abstract

A plactic algebra can be thought of as a (non-commutative) model for the representation ring of a semisimple Lie algebra g. This algebra was introduced by Lascoux and Schutzenberger in [13], [18] in order to study the representation theory of GLn(C) and Sn. This new tool enabled them for example to give the first rigouros proof of the Littlewood-Richardson rule to determine the decomposition of tensor products into direct sums of irreducible representations. Using a case by case analysis, such a plactic algebra has been constructed also for some other simple groups, see [1], [8], [19], [20], [21]. Recently, two constructions of isomorphic plactic algebras have been given for symmetrisable Kac-Moody algebras. From the point of view of quantum groups, this algebra is the algebra of crystal bases ([5], [6], [7], [16], [17], [19]). The second construction realizes this algebra as the algebra ZP of equivalence classes of paths in the space XQ of rational weights ([5], [14], [15]). For simplicity, assume that G is a simple, simply connected algebraic group. To give a description of ZP which is more in the spirit of the original work of Lascoux and Schutzenberger, let V = Vλ1 ⊕ . . .⊕Vλr be a faithful representation and let D be the associated set of L-S paths, i.e. D is a basis of the corresponding model of V in ZP. Let Z{D} be the free associative algebra generated by D. If λ = ∑ aωω is a dominant weight, then let |λ| denote the sum ∑ aω. The canonical projection which maps a monomial to the concatenation: