Characters of classical groups twisted by roots of unity

Abstract

This thesis focuses on the study of specialized characters of irreducible polynomial representations of the complex classical Lie groups of types A, B, C and D. We study various specializations where the characters are evaluated at elements twisted by roots of unity. The details of the results are as follows.

Throughout the thesis, we fix an integer t >= 2 and a primitive t'th root of unity ω. We first consider the irreducible characters of polynomial representations of the general linear group, the symplectic group and the orthogonal group evaluated at elements ω^k x_i for 0 ≤ k ≤ t-1 and 1 ≤ i ≤n. The case of the general linear group was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). 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. This characterization turns out to depend on the t-core of the indexed partition. Furthermore, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. We also give product formulas for general z-asymmetric partitions and z-asymmetric t-cores, and show that there are infinitely many z-asymmetric t-cores for t >= z+2.

We extend the above results for the irreducible characters of the classical groups evaluated at similar specializations. For the general linear case, we set the first tn elements to ω^j x_i for 0 ≤ j ≤ t-1 and 1 ≤ i ≤ n and the last m to y, ω y, ..., ω^{m-1} y. For the other families, we take the same specializations but with m=1. Our motivation for studying these are the conjectures of Wagh--Prasad (Manuscripta Math., 2020) relating the irreducible representations of classical groups.

The hook Schur polynomials are the characters of covariant and contravariant irreducible representations of the general linear Lie superalgebra. These are a supersymmetric analogue of the characters of irreducible polynomial representations of the general linear group and are indexed by two families of variables. We consider similarly specialized skew hook Schur polynomials evaluated at ω^p x_i and ω^q y_j, for 0 ≤ p, q ≤ t-1, 1 ≤ i ≤ n and 1 ≤ j ≤ m. We characterize the skew shapes for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials.

For certain combinatorial objects, the number of fixed points under a cyclic group action turns out to be the evaluation of a nice function at the roots of unity. This is known as the cyclic sieving phenomenon (CSP) and has been the focus of several studies. We use the factorization result for the above hook Schur polynomial to prove the CSP on the set of semistandard supertableaux of skew shapes for odd t. Using a similar proof strategy, we give a complete generalization of a result of Lee-Oh (Electron. J. Combin., 2022) for the CSP on the set of skew SSYT conjectured by Alexandersson--Pfannerer--Rubey--Uhlin (Forum Math. Sigma, 2021).