Knuth relations, tableaux and MV-cycles (Q2870999)

The ``plactic monoid'' and the ``Young tableaux'' are remarkable combinatorial tools to study representations of \(\mathrm{GL}_n(C)\). The plactic monoid is the collection of words in an alphabet modulo Knuth equivalence. Semistandard Young tableaux pick one element in each equivalence class.NEWLINENEWLINEThe paper presents a geometric interpretation of the Knuth equivalence as follows. To a word \(w\) the authors associate a Bott-Samelson type variety \(\Sigma_w\), a cell in it \(C_w\), as well as a map \(\pi_w\) from \(\Sigma_w\) to a affine Grassmannian. They prove that the set \(\overline{\pi(C_w)}\) is a complete invariant of the Knuth equivalence class of \(w\).NEWLINENEWLINEThe work is motivated by earlier works of Mirkovic and Vilonen, who gave geometric interpretations of weight multiplicities of irreducible representations of semisimple algebraic groups.