Littlewood Richardson Rule for general linear group over finite field I just finished reading Green's 1955 paper on characters of general linear groups and have also been reading Macdonald's Symmetric Functions and Hall Polynomials. I see that there is a recursive formula for the characters of $GL_n(\mathbb{F}_q)$ but there does not seem to be any literature on a Littlewood-Richardson type rule for $GL_n(\mathbb{F}_q)$ . Namely, given two irreducible representations, I am interested in studying the computational complexity of decomposing the tensor product into irreducibles. This generally means I need very explicit formulas such as the Littlewood Richardson rule. I saw a few questions about this a few years ago, but I was wondering if there had been any update in the literature.
 A: The symmetric group $S_n$ is the $q=1$ "limit" of $\mathrm{GL}_n(q)$. We expect the $q=1$ case to be simpler than $q>1$. But decomposing tensor products of irreducible $S_n$-characters is difficult (see Ikenmeyer, Mulmuley, and Walter
 - On vanishing of Kronecker coefficients and Pak and Panova - On the complexity of computing Kronecker coefficients), so the same should be true for  $\mathrm{GL}_n(q)$. Moreover, it is a notorious open problem to find a combinatorial interpretation for the Kronecker coefficients of $S_n$, i.e., the multiplicities of irreducible characters in the tensor product of irreducible characters. In addition, irreducible characters obtained as linear combinations of the character of the action of $\mathrm{GL}_n(q)$ on (left) cosets of  $\mathrm{GL}_{\lambda_1}(q)\times \mathrm{GL}_{\lambda_2}(q)
\times\dotsb$ are perfect $q$-analogues of the irreducible characters of $S_n$, so I would expect that decomposing their tensor products would give a nice refinement of the Kronecker coefficients for  $S_n$. I don't know whether someone has already worked this out.
