Algebraic Combinatorics Seminar - Sheila Sundaram

Title: Quasisymmetric functions,  descent sets, immaculate tableaux, and 0-Hecke modules

Abstract:

The first half of this talk will be expository and devoted to a discussion of (quasi)symmetric functions and tableaux.

We define new families of quasisymmetric functions, in particular the new basis of row-strict dual immaculate functions, with an associated cyclic, indecomposable 0-Hecke algebra module. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution \psi on the ring Qsym of quasisymmetric functions. We uncover the remarkable properties of the immaculate Hecke poset induced by the 0-Hecke action on standard immaculate tableaux, revealing other submodules and quotient modules, often cyclic and indecomposable.

As in the dual immaculate case, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, defined by a specific descent set.

We complete the combinatorial and representation-theoretic picture by constructing 0-Hecke modules for the remaining variations on descent sets. We show that the generating functions of all the possible variations  of tableaux are characteristics of these 0-Hecke modules, captured in the immaculate Hecke poset.

This talk is based on joint work with Elizabeth Niese, Stephanie van Willigenburg, Julianne Vega and Shiyun Wang.