Séminaire Lotharingien de Combinatoire, 86B.38 (2022), 12 pp.

Arvind Ayyer and Nishu Kumari

Factorization of Classical Characters Twisted by Roots of Unity: Extended Abstract

Abstract. For a fixed integer t≥2, we consider the irreducible characters of representations of the classical groups of types A, B, C and D, namely GLtn, SO2tn+1, Sp2tn and OE2tn, evaluated at elements ωkxi for 0≤kt-1 and 1≤in, where ω is a primitive t'th root of unity. The case of GLtn was considered by D. Prasad (Israel J. Math., 2016). In this article, we give a uniform approach for all cases. In each case, we characterize partitions for which the character value is nonzero in terms of what we call z-asymmetric partitions, where z is an integer that depends on the group. Moreover, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. The proof uses Cauchy-type determinant formulas for these characters and involves a careful study of the beta sets of partitions. We also give product formulas for general z-asymmetric partitions and z-asymmetric t-cores. Lastly, we show that there are infinitely many z-asymmetric t-cores for |z|≤t-2.

Received: November 25, 2021. Accepted: March 4, 2022. Final version: April 1, 2022.

The following versions are available: