Factorization of Classical Characters twisted by Roots of Unity

Abstract: In representation theory, Schur polynomials are the characters of the irreducible polynomial representations of the classical groups of type A, namely $GL_n(\mathbb{C})$. Motivated by a celebrated result of Kostant, D. Prasad considered factorization of Schur polynomials in $tn$ variables, for $t \geq 2$, a fixed positive integer, specialized to $(\exp(2 \pi \iota k/t) x_j)_{0 \leq k \leq t-1, 1 \leq j \leq n}$ (Israel J. Math., 2016). He characterized partitions for which these Schur polynomials are nonzero and showed that if the Schur polynomial is nonzero, it factorizes into characters of smaller classical groups of Type A.

We generalize Prasad's result to the irreducible characters of classical groups of type B, C and D, namely $O_{2tn+1}(\mathbb{C}), \Sp_{2tn}(\mathbb{C})$ and $O_{2tn}(\mathbb{C})$, with the same specialization. We give a uniform approach for all cases. The proof uses Cauchy-type determinant formulas for these characters and involves a careful study of the beta sets of partitions. This is joint work with Arvind Ayyer and is available at arxiv.org/abs/2109.11310.

combinatoricsrings and algebrasrepresentation theory

Audience: researchers in the topic

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ARCSIN - Algebra, Representations, Combinatorics and Symmetric functions in INdia

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