Factorization of classical characters twisted by roots of unity: II

Abstract

Fix natural numbers n⩾1, t⩾2 and a primitive tth root of unity ω. In previous work with A. Ayyer (2022) [5], we studied the factorization of specialized irreducible characters of GLtn, SO2tn+1, Sp2tn and O2tn evaluated at elements to ωjxi for 0⩽j⩽t−1 and 1⩽i⩽n. In this work, we extend the results to the groups GLtn+m(0⩽m⩽t−1), SO2tn+3, Sp2tn+2 and O2tn+2 evaluated at similar specializations: (1) for the GLtn+m case, we set the first tn elements to ωjxi for 0⩽j⩽t−1 and 1⩽i⩽n and the remaining m to y,ωy,…,ωm−1y; (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 (2020) [18] relating the irreducible representations of Spin2n+1 and SL2n, SL2n+1 and Sp2n as well as Spin2n+2 and Sp2n. Our proofs use the Weyl character formulas and the beta-sets of t-core partitions. Lastly, we give a bijection to prove that there infinitely many t-core partitions for which these characters are nonzero.