Factorization of classical characters twisted by roots of unity

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 O2tn, evaluated at elements ωkxi for 0≤k≤t−1 and 1≤i≤n, where ω is a primitive t'th root of unity. The case of GLtn was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In this article, we give a uniform approach for all cases. We also look at GLtn+1 where we specialize the elements as before and set the last variable to 1. 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 which 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 t≥z+2.