# 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.

• 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? – darij grinberg Oct 30 '18 at 20:49
• I think so, although I am not yet too familiar with hopf algebras. I will read your overview and Zelevinsky though both seem useful. – Sheel Stueber Oct 30 '18 at 21:51

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.