Factorization of classical characters twisted by roots of unity: II

2022/12/23 by Kumari, Nishu
#05A15 #05E05 #05E10 #20G05 #20G20 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2212.12477

Abstract

Fix natural numbers n ≥ 1, t ≥ 2 and a primitive t^th root of unity ω. In previous work with A. Ayyer (J. Alg., 2022), we studied the factorization of specialized irreducible characters of GL_tn, SO_2tn+1, Sp_2tn and O_2tn evaluated at elements to ω^j x_i for 0 ≤ j ≤ t-1 and 1 ≤ i ≤ n. In this work, we extend the results to the groups GL_tn+m (0 ≤ m ≤ t-1), SO_2tn+3, Sp_2tn+2 and O_2tn+2 evaluated at similar specializations: (1) for the GL_tn+m(ℂ) case, we set the first tn elements to ω^j x_i for 0 ≤ j ≤ t-1 and 1 ≤ i ≤ n and the remaining m to y, ωy, …, ω^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 Spin_2n+1 and SL_2n, SL_2n+1 and Sp_2n as well as Spin_2n+2 and 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.

Related

• Further results for classical and universal characters twisted by roots of unity
• Character factorizations for representations of GL(n,C)
• Character factorisations, z-asymmetric partitions and plethysm
• Restriction of characters to subgroups of wreath products and basic sets for the symmetric group
• Factorial characters of classical Lie groups and their combinatorial realisations