# Branching rule for classical Lie algebras in positive characteristic

The restriction of an irreducible $\mathfrak{sl}_n(\mathbb{C})$-module to $\mathfrak{sl}_{n-1}(\mathbb{C})$ is described by a branching rule which says that if $L(\lambda)$ is the simple $\mathfrak{sl}_n(\mathbb{C})$-module corresponding to a partition $\lambda$ then its restriction is a direct sum of $L(\mu)$s where the $\mu$ are obtained by removing some boxes from $\lambda$.

Now consider the analogous problem for $\mathfrak{sl}_n(k)$, where $k$ is a field of characteristic $p>0$. It is no longer generally true that simple modules restrict to semisimple modules. My question is whether a branching rule determining the indecomposable summands of the restriction of a simple module is known. A general answer may be too much to ask, but I would be interested in references for nontrivial special cases, or for small $n$, or for other classical Lie algebras.

• It's asking a lot to get even the indecomposable summands in most cases, since indecomposable modules are largely unknown. The literature on special cases is scattered, but I'll try to check further. – Jim Humphreys Nov 14 '13 at 19:15

As my comment indicated, there is currently little hope of writing down general branching rules in characteristic $p$. In fact, given the history of work on Lusztig's conjecture about formal characters of simple modules, it's unclear whether good "formulas" as such will exist in this subject; perhaps one has to settle for recursive calculations, which may or may not be feasible except in small cases.
In a more definite direction, it's useful here to switch from Lie algebras to algebraic groups: the Lie algebra irredudibles with "restricted" highest weights relative to $p$ yield all irreducibles for the group as twisted tensor products (Steinberg). Here the foundations are well organized in Jantzen's book. For special linear groups as for others, only in small ranks or for small $p$ can one determine explicitly the irreducibles. The Weyl modules have the advantage of giving known weight combinatorics, but the disadvantage of usually being far from irreducible.
There isn't a lot of explicit study of such questions, I think, but interesting work in special cases has been done by people associated (now or formerly) with Minsk: A.S. Kleshchev, I.D. Suprunenko, and others. An example is the recent work of Anna Osinovskaya, On the restrictions of modular irreducible representations of algebraic groups of type $A_n$ to naturally embedded subgroups of type $A_2$. J. Group Theory 8 (2005), no. 1, 43–92.