A Combinatorial Bijection Between Standard Young Tableaux and Reduced Words of Grassmannian Permutations

Rudolf Winkel

Séminaire lotharingien de combinatoire, Tome 36 (1996)

Résumé

For every partition \lambda we construct a very simple combinatorial bijection between the set of standard Young tableaux of shape \lambda and the set of reduced words for the Grassmannian permutation \pi(\lambda) associated to \lambda. The basic tools in setting up this bijection are partial orders on the respective sets. These partial orders are interesting in their own right, and we give some first results about them: (1) the poset of standard tableaux for an arbitrary shape D is isomorphic to an order ideal in left weak Bruhat order, (2) for hook shapes the Poincaré polynomial is the q-binomial coefficient, (3) for general Ferrer shapes a recursion formula for the Poincaré polynomials is given, (4) the poset of reduced words for a Grassmannian permutation is anti-isomorphic to the poset of reduced words for its ``conjugate'' and inverse permutation, (5) for the Grassmannian and dominant permutation associated to a hook shape the respective posets of reduced words are isomorphic.