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Poset binomials and rainbow characters

Abstract : This paper introduces a variation on the binomial coefficient that depends on a poset and interpolates between $q$-binomials and 1-binomials: a total order gives the usual $q$-binomial, and a poset with no relations gives the usual binomial coefficient. These coefficients arise naturally in the study of supercharacters of the finite groups of unipotent upper-triangular matrices, whose representation theory is dictated by the combinatorics of set partitions. In particular, we find a natural set of modules for these groups, whose characters have degrees given by $q$-binomials, and whose decomposition in terms of supercharacters are given by poset binomial coefficients. This results in a non-trivial family of formulas relating poset binomials to the usual $q$-binomials.
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https://hal.inria.fr/hal-01229709
Contributor : Alain Monteil <>
Submitted on : Tuesday, November 17, 2015 - 10:20:22 AM
Last modification on : Friday, July 26, 2019 - 1:44:05 PM
Long-term archiving on: : Thursday, February 18, 2016 - 11:42:27 AM

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Daniel Bragg, Nathaniel Thiem. Poset binomials and rainbow characters. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.217-228. ⟨hal-01229709⟩

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