Factorization of classical characters twisted by roots of unity: II

Abstract

Fix natural numbers $n\ge 1$, $t\ge 2$ and a primitive $t^{\text{th}}$ root of unity $\omega$. In previous work with A. Ayyer (J. Alg., 2022), we studied the factorization of specialized irreducible characters of ${\text{GL}}_{tn}$, ${\text{SO}}_{2tn+1},$ ${\text{Sp}}_{2tn}$ and ${\text{O}}_{2tn}$ evaluated at elements to ${\omega }^j{x}_i$ for $0\le j\le t-1$ and $1\le i\le n$. In this work, we extend the results to the groups ${\text{GL}}_{tn+m}$ $(0\le m\le t-1)$, ${\text{SO}}_{2tn+3}$, ${\text{Sp}}_{2tn+2}$ and ${\text{O}}_{2tn+2}$ evaluated at similar specializations: (1) for the ${\text{GL}}_{tn+m}(\mathbb{C})$ case, we set the first $tn$ elements to ${\omega }^j{x}_i$ for $0\le j\le t-1$ and $1\le i\le n$ and the remaining $m$ to $y,\omega y,\dots ,{\omega }^{m-1}y$; (2) for the other three families, the same specializations but with $m=1$. The main results of this paper are a characterization of partitions for which these characters vanish and a factorization of nonzero characters into those of smaller classical groups. Our motivation is the conjectures of Wagh and Prasad (Manuscripta Math., 2020) relating the irreducible representations of ${\text{Spin}}_{2n+1}$ and ${\text{SL}}_{2n}$, ${\text{SL}}_{2n+1}$ and ${\text{Sp}}_{2n}$ as well as ${\text{Spin}}_{2n+2}$ and ${\text{Sp}}_{2n}$. Our proofs use the Weyl character formulas and the beta-sets of $t$-core partitions. Lastly, we give a bijection to prove that there are infinitely many $t$-core partitions for which these characters are nonzero.

43 pages, 1 figure

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• Universal characters twisted by roots of unity