Stephen Lewis ; Nathaniel Thiem
-
Nonzero coefficients in restrictions and tensor products of supercharacters of Un(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 Un(q) (extended abstract)Conference paper
Authors: Stephen Lewis 1; Nathaniel Thiem 2
NULL##NULL
Stephen Lewis;Nathaniel Thiem
1 Department of Mathematics [Seattle]
2 Department of Mathematics, University of Colorado
The standard supercharacter theory of the finite unipotent upper-triangular matrices Un(q) gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of Um(q)⊆Un(q) for m≤n lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of Un(q) is a nonnegative integer linear combination of supercharacters of Um(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 Un(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.