Knuth relations, tableaux and MV-cycles

Stéphane Gaussent (1, 2) , Peter Littelmann (3) , An Hoa Nguyen (3)

1 ICJ - Institut Camille Jordan
2 IECL - Institut Élie Cartan de Lorraine
3 Mathematisches Institut, Universität zu Köln

We give a geometric interpretation of the Knuth equivalence relations in terms of the affine Graß mann variety. The Young tableaux are seen as sequences of coweights, called galleries. We show that to any gallery corresponds a Mirković-Vilonen cycle and that two galleries are equivalent if, and only if, their associated MV cycles are equal. Words are naturally identified to some galleries. So, as a corollary, we obtain that two words are Knuth equivalent if, and only if, their associated MV cycles are equal.

Domaines: Théorie des représentations [math.RT]

Stéphane Gaussent, Peter Littelmann, An Hoa Nguyen. Knuth relations, tableaux and MV-cycles. Journal of the Ramanujan Mathematical Society, 2013, 28A, pp.191-219. ⟨hal-00738213⟩