A Plactic Algebra for Semisimple Lie Algebras

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