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Given unipotent characters U 1, . . . , U  k  of GL n  \( \left( {{{\mathbb{F}}_q}} \right) \), we prove that \( \left\langle {{U_1} \otimes \cdots \cdots \otimes {U_k},1} \right\rangle \) is a polynomial in q with non-negative integer coefficients. We study the degree of this polynomial and give a necessary and sufficient condition in terms of the representation theory of symmetric groups and root systems for this polynomial to be non-zero.

Tensor products of unipotent characters of general linear groups over finite fields