DECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL3 (WITH AN APPENDIX BY C.M. RINGEL)

Year 2011, Volume: 9 Issue: 9, 177 - 219, 01.06.2011

C. Bowman S. R. Doty S. Martin

Abstract

We give an algorithm for working out the indecomposable direct
summands in a Krull–Schmidt decomposition of a tensor product of two simple
modules for G = SL3 in characteristics 2 and 3. It is shown that there is a
finite family of modules such that every such indecomposable summand is
expressible as a twisted tensor product of members of that family.
Along the way we obtain the submodule structure of various Weyl and
tilting modules. Some of the tilting modules that turn up in characteristic
3 are not rigid; these seem to provide the first example of non-rigid tilting
modules for algebraic groups. These non-rigid tilting modules lead to examples
of non-rigid projective indecomposable modules for Schur algebras, as shown
in the Appendix.
Higher characteristics (for SL3) will be considered in a later paper.

Keywords

Algebraic groups, tilting modules, Weyl modules, quivers