Factorization of classical characters twisted by roots of unity

For a fixed integer $t \geq 2$, we consider the irreducible characters of representations of the classical groups of types A, B, C and D, namely $\text{GL}_{tn}, \text{SO}_{2tn+1}, \text{Sp}_{2tn}$ and $\text{O}_{2tn}$, evaluated at elements $ω^k x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$, where $ω$ is a primitive $t$'th root of unity. The case of $\text{GL}_{tn}$ 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. In this article, we give a uniform approach for all cases. We also look at $\text{GL}_{tn+1}$ 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 $t \geq z+2$.

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