Decomposition of tensor products of modular irreducible representations for SL3

TY - JOUR

T1 - Decomposition of tensor products of modular irreducible representations for SL3

AU - Bowman-Scargill, Chris

AU - Doty, Stephen

AU - Martin, S

PY - 2011/1/1

Y1 - 2011/1/1

N2 - 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.

AB - 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.

M3 - Article

SN - 1306-6048

VL - 9

SP - 177

EP - 219

JO - International Electronic Journal of Algebra

JF - International Electronic Journal of Algebra

ER -