The representation of the symmetric group on m-Tamari intervals

By Mireille Bousquet-M'elou, Guillaume Chapuy, Louis-Franccois Pr'eville-Ratelle. 2012

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An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0;0), ending at (mn;n), and never going below the line fx = myg. The set of these paths can be equipped with a lattice structure, called them-Tamari lattice and denoted by T (m) n , which generalizes the usual Tamari lattice Tn obtained when m = 1. This lattice was introduced by F. Bergeron in connection with the study of diagonally coinvariant spaces in three sets ofn variables. The representation of Sn on these spaces is conjectured to be closely related to the natural representation of Sn on (labelled) intervals of them-Tamari lattice studied in this paper. An interval (P;Q) of T (m) n is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group Sn acts on labelled intervals of T (m) n by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of Sn. In particular, the dimension of the representation, that is, the number of labelledm-Tamari intervals of sizen, is found to be (m + 1) n (mn + 1) n 2 : These results are new, even whenm = 1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. The form of this equation is highly non- standard: it involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect tox. The hardest part of the proof consists in solving it, and we develop original techniques to do so. R´

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8718a3b184e68fdc615f155f618cb31888b700f7