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arXiv · 2202.00708

0-Hecke modules for row-strict dual immaculate functions

Elizabeth Niese·Sheila Sundaram·Stephanie van Willigenburg·Julianne Vega·Shiyun Wang
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Abstract

We introduce a new basis of quasisymmetric functions, the row-strict dual immaculate functions. We construct a cyclic, indecomposable 0-Hecke algebra module for these functions. 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 of quasisymmetric functions. We give an explicit description of the effect of $\psi$ on the associated 0-Hecke modules, via the poset induced by the 0-Hecke action on standard immaculate tableaux. This remarkable poset reveals other 0-Hecke submodules and quotient modules, often cyclic and indecomposable, notably for a row-strict analogue of the extended Schur functions studied in Assaf-Searles (2019). Like the dual immaculate function, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, corresponding to a specific descent set. We give a complete combinatorial and representation-theoretic picture by constructing 0-Hecke modules for the remaining variations on descent sets, and showing that \emph{all} the possible variations for generating functions of tableaux occur as characteristics of the 0-Hecke modules determined by these descent sets.

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Elizabeth Niese, Sheila Sundaram, Stephanie van Willigenburg, Julianne Vega, Shiyun Wang. 2022-02-01. 0-Hecke modules for row-strict dual immaculate functions. https://doi.org/10.1090/tran/9006

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