# Representation theory of the general linear group over a finite prime field

I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained.

The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely classified and well-understood via Schur-Weyl duality, the algebraic Peter-Weyl theorem and the entire theory of reductive groups in characteristic zero.

I am looking for a good reference on what is known about the representation theory of $\operatorname{GL}_n(\mathbb F_p)$ over $\mathbb F_p$, i.e. the study of $\mathbb F_p$-vector spaces with an action of $\operatorname{GL}_n(\mathbb F_p)$. Here, $\mathbb F_p$ is the field with $p$ elements, $p$ a prime number. Are the irreducible modules completely classified? Can they somehow be indexed by certain partitions, similar to the characteristic zero case?

I am particularly interested in whether or not there is some equivalent of the Pieri rule, i.e. decomposing the tensor product of an irreducible representation with a symmetric power of $\mathbb F_p^n$. However, I suppose that question only makes sense when it is possible to classify irreducibles in some combinatorial way.

• The number of such inequivalent irreducible modules is well known to be $p^{n}-p^{n-1}$. These are in bijection with monic polynomials of degree $n$ with non-zero constant term in $\mathbb{F}_{p}[x].$ However, explicit description of the simple modules is, I believe, notoriously difficult. – Geoff Robinson Nov 17 '14 at 13:13

## 1 Answer

One classical source for the case $n=2$, with somewhat old-fashioned notation for some of the related groups, is a paper by Richard Brauer and his student Cecil Nesbitt: On the modular characters of groups, Ann. of Math. (2) 42, (1941). 556–590. In this very special case, the actual modules are not too difficult to describe in terms of symmetric powers of the natural 2-dimensional module. But for arbitrary $n$ there are still mostly unknowns.

For a modern survey which includes most of the relevant references (especially to work of Steinberg, Jantzen, Lusztig), see my LMS Lecture Notes 326: Modular Representations of Finite Groups of Lie Type (Cambridge, 2006), in particular Chapter 19. As in the study of the ambient algebraic groups, a parametrization of modular irreducibles by highest weights is readily given in terms of highest weights, which can be translated if desired into the language of partitions for general and special linear groups. See Jantzen's book Representations of Algebraic Groups (2nd ed., AMS, 2003). (Steinberg's twisted tensor product theorem from 1963 then shows how to deduce results for finite fields larger than the prime field.) While these particular groups are often much easier to study than arbitrary groups of Lie type, we still seem to be far from a complete understanding. And the partition viewpoint may not be too helpful.

But the details about dimensions and possible constructions of the modules are still largely unknown, apart from $n=2,3,4$. As Jantzen's book demonstrates, the dimension problems are well-organized and for large primes such data can in principle be computed from Lusztig's viewpoint in terms of Hecke algebras for certain affine Weyl groups. But as Geoff remarks, construction of the actual modules is quite problematic, while even the computations of degrees and such is usually beyond reach in practice.

Concerning tensor products and other constructions, little has been worked out except for very small $n$.