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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.

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    $\begingroup$ The representation ring of $\operatorname{GL}_\bullet\left(\mathbb{F}_q\right)$, as a PSH-algebra, is isomorphic to a tensor product of many copies of the symmetric functions. See, e.g., Chapter 4 in arXiv:1409.8356v5, or the original book of Zelevinsky's for much more. Thus you should get the structure constants for Harish-Chandra induction of these representations. Is that what you want? $\endgroup$ – darij grinberg Oct 30 '18 at 20:49
  • $\begingroup$ I think so, although I am not yet too familiar with hopf algebras. I will read your overview and Zelevinsky though both seem useful. $\endgroup$ – Sheel Stueber Oct 30 '18 at 21:51
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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.

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  • $\begingroup$ Ah thank you. I'm very familiar with the first paper, which is why I was wondering if we can somehow use the symmetric group characters to do the decompositon. $\endgroup$ – Sheel Stueber Oct 30 '18 at 21:50
  • $\begingroup$ So, it may be interesting to try and find out if this problem can be reduced to the Kronecker problem. If that were the case, then it should be easy to say, for starters, what is the decomposition of the tensor product of two cuspidal representations. $\endgroup$ – Amritanshu Prasad Oct 31 '18 at 3:39

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