Let $V$ and $W$ be complex irreducible representations of $GL_n(F_q)$ where $F_q$ is finite field. Is the decomposition of $V \otimes W$ into irreducible representations known?

PS

Same question: Decomposing tensor products of irreducible representations of reductive groups over a finite field

Symmetric Functions and Hall Polynomialsby I.G. Macdonald (2nd ed., 1995, Oxford). As in Green's paper, the goal is an efficient description of irreducible characters using lots of combinatorics. But it remains technically challenging to compute an actual character table of any size, or to decompose tensor products, or to carry out branching rules. $\endgroup$ – Jim Humphreys Jul 28 '10 at 16:33`$p$`

, the "patterns" of decomposition you get depend on the Weyl group but not essentially on`$p$`

even though the characters involve parameters depending on`$p$`

. Look at GL`$(2,p)$`

or SL`$(2,p)$`

for`$p$`

large: "most" irreducible characters have degree`$p-1$`

or`$p+1$`

(roughly half of each), while a typical tensor product decomposes as a sum of roughly`$p$`

of these in limited patterns. Bookkeeping is tedious even in this simple case, of course, and hopeless in general. Each power`$p^m$`

expands this list of patterns in a way depending on`$m$`

. $\endgroup$ – Jim Humphreys Jul 29 '10 at 13:053more comments