Mathematics Colloquia and Seminars

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We introduce a new combinatorial construction, called a dual equivalence graph, based on Haiman's 1992 discovery of an equivalence relation on tableaux which is "dual" to jeu-de-taquin. We define a generating function on the vertices of such graphs and show that it is always Schur positive. We outline the construction of a graph on $k$-tuples of standard young tableaux which we conjecture to be a dual equivalence graph and prove this conjecture for $k \leq 3$. This gives a combinatorial description of the Schur coefficients of the ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. Recalling Haglund's recent monomial expansion for Macdonald polynomials, we conclude with a combinatorial formula for the $q,t$-Kostka polynomials.