Decomposition of Tensor Products Involving a Steinberg Module

作者：Kildetoft Tobias^*

来源：Algebras and Representation Theory, 2017, 20(4): 951-975.

DOI：10.1007/s10468-017-9670-7

• Algebraic groups
• Frobenius kernels
• Finite Chevalley groups
• Tilting modules

We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G (r) or a finite Chevalley group . In all three cases, we give formulas reducing this to standard character data for G. Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G (r) or . Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin's tilting conjecture.

• 出版日期2017-8