Journal of the Ramanujan Mathematical Society

Volume 28A, Issue SPL, July – Special Issue 2013 pp. 191–219.

Knuth relations, tableaux and MV-cycles

Authors: Stéephane Gaussent, Peter Littelmann and An Hoa Nguyen
Author institution: Université de Lyon, Institut Camille Jordan (UMR 5208) Université Jean Monnet, 23, rue du Docteur Michelon 42023, Saint-Etienne Cedex 2, France

Summary: We give a geometric interpretation of the Knuth equivalence relations in terms of the geometry of the affine Graßmann variety G. Given a word w, we associate to w, in a canonical way, a Bott-Samelson type variety Σw, a cell C ⊂ Σw, a dominant coweight λ and a map πw: Σ w → G such that w can naturally be identified with the center of the cell. We show that the closure π (Cw) is an MV-cycle in the affine Schubert variety X λ w. The Knuth equivalence of words gets the following geometric reformulation: two words w,w' are Knuth equivalent if and only if π w(Cw) = π w' (Cw'). In fact, the construction works not only for words but also for Young tableau, and, more generally, key tableaux.

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