Stephen Lewis ; Nathaniel Thiem - Nonzero coefficients in restrictions and tensor products of supercharacters of $U_n(q)$ (extended abstract)

dmtcs:2840 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) - https://doi.org/10.46298/dmtcs.2840
Nonzero coefficients in restrictions and tensor products of supercharacters of $U_n(q)$ (extended abstract)Article

Authors: Stephen Lewis 1; Nathaniel Thiem 2

  • 1 Department of Mathematics [Seattle]
  • 2 Department of Mathematics, University of Colorado

The standard supercharacter theory of the finite unipotent upper-triangular matrices $U_n(q)$ gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of $U_m(q) \subseteq U_n(q)$ for $m \leq n$ lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of $U_n(q)$ is a nonnegative integer linear combination of supercharacters of $U_m(q)$ (in fact, it is polynomial in $q$). In a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of $U_n(q)$, this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters. These conditions are given uniformly in terms of complete matchings in bipartite graphs.


Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: unipotent upper-triangular matrices,supercharacters,set-partitions,matching,bipartite graphs,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • FRG: Collaborative Research: Characters, Liftings, and Types: Investigations in p-adic Representation Theory; Funder: National Science Foundation; Code: 0854893

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