[Paper Review] Nonzero coefficients in restrictions and tensor products of supercharacters of $U_n(q)$

S. R. Lewis, Nathaniel Thiem|ArXiv.org|Dec 9, 2009

Advanced Combinatorial Mathematics11 references3 citations

TL;DR

This paper characterizes when supercharacter coefficients are nonzero in restrictions and tensor products for the unipotent upper-triangular group $U_n(q)$, using complete matchings in bipartite graphs constructed from set partitions. The key contribution is a uniform combinatorial criterion: a supercharacter appears in the decomposition if and only if an associated bipartite graph admits a complete matching.

ABSTRACT

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.

Motivation & Objective

• To understand the combinatorics of branching rule coefficients in the supercharacter theory of $U_n(q)$, which parallels the representation theory of the symmetric group $S_n$.
• To determine when a given supercharacter appears with nonzero coefficient in the restriction of another supercharacter to a subgroup $U_K \subseteq U_n$.
• To characterize when a supercharacter appears in the tensor product of two other supercharacters.
• To provide a uniform, combinatorial criterion—based on complete matchings in bipartite graphs—for nonzero coefficients in these decompositions.
• To lay foundational groundwork for computing explicit coefficients, extending known results on polynomiality of coefficients in $q$.

Proposed method

• Construct a bipartite graph $\Gamma_K(\lambda)$ from a set partition $\lambda$ and a subset $K \subseteq \{1,\dots,n\}$, encoding the structure of the embedding $U_K \subseteq U_n$.
• Define a bipartite graph $\Gamma(\lambda,\mu,\nu)$ for triplets of set partitions $\lambda, \mu, \nu$ to model tensor product decompositions.
• Use Theorem 4.2 to relate tensor products of supercharacters to restrictions of larger supercharacters, scaled by a power of $q$, enabling reduction to restriction problems.
• Apply a constructive algorithm based on the set partition structure and the $\Lambda_K$-labeling to determine the existence of complete matchings.
• Leverage the correspondence between matchings and nonzero coefficients: a nonzero coefficient occurs precisely when a complete matching exists.
• Generalize results to multisets $\lambda \in \mathcal{M}_K(q)$ to handle tensor products, using a modified version of the $\mathcal{C}_K(\lambda)$ set of crossing pairs.

Experimental results

Research questions

• RQ1 Under what conditions is a given supercharacter $\chi^\nu$ present with nonzero coefficient in the restriction $\mathrm{Res}^{U_n}_{U_K}(\chi^\lambda)$?
• RQ2 When does a supercharacter $\chi^\nu$ appear in the tensor product $\chi^\lambda \otimes \chi^\mu$ of two supercharacters of $U_n(q)$?
• RQ3 What combinatorial condition on the set partitions $\lambda$, $\mu$, and $\nu$ ensures the existence of a nonzero coefficient in the tensor product decomposition?
• RQ4 How can the structure of the embedding $U_K \subseteq U_n$ be encoded to determine the support of the restriction of a supercharacter?
• RQ5 Can the coefficient of the trivial character in such decompositions be explicitly computed, and under what conditions is it a power of $q$?

Key findings

• The trivial supercharacter appears in $\mathrm{Res}^{U_n}_{U_K}(\chi^\lambda)$ if and only if the bipartite graph $\Gamma_K(\lambda)$ admits a complete matching.
• A supercharacter $\chi^\nu$ appears in $\chi^\lambda \otimes \chi^\mu$ if and only if the associated bipartite graph $\Gamma(\lambda, \mu, \nu)$ has a complete matching.
• For the restriction $\mathrm{Res}^{U_n}_{U_K}(\chi^\lambda)$, the coefficient of $\chi^\mu$ is nonzero if and only if the graph $\Gamma_K(\lambda, \mu)$ has a complete matching.
• The coefficient of the trivial character in $\mathrm{Res}^{U_L}_{U_K}(\chi^\lambda)$ is $q^{r_K^L(\lambda) + |\mathcal{C}_K(\lambda)|}$, where $\mathcal{C}_K(\lambda)$ counts certain crossing pairs in the set partition.
• For tensor products, the coefficient of the trivial character is $q^{|\mathcal{C}_K(\lambda)|}$ when $\Lambda_K(\lambda) \in \{ (\circ,\bullet), (\bullet,\circ) \}$, generalizing to $q$-polynomial coefficients in general cases.
• Explicit examples show that coefficients need not be pure powers of $q$; for instance, $\langle \chi^{\{1\overset{a}{\frown}5, 1\overset{b}{\frown}5, 1\overset{c}{\frown}5\}}, {11} \rangle_{U_5(q)} = 3q - 2$ when $a + b + c = 0$.