ON TENSORING WITH THE STEINBERG REPRESENTATION
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Springer Science+Business Media New York (2019)
ON TENSORING WITH THE STEINBERG REPRESENTATION C. P. BENDEL∗
D. K. NAKANO∗∗
Department of Mathematics, Statistics and Computer Science University of Wisconsin-Stout Menomonie WI 54751, USA [email protected]
Department of Mathematics University of Georgia Athens GA 30602, USA [email protected]
C. PILLEN∗∗∗
P. SOBAJE∗∗∗∗
Department of Mathematics and Statistics University of South Alabama Mobile, AL 36688, USA [email protected]
Department of Mathematics Georgia Southern University Statesboro, GA 30458, USA [email protected]
Abstract. Let G be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic p > 0. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of Donkin: one on tilting modules and the lifting of projective modules for Frobenius kernels of G and another on the existence of certain filtrations of G-modules. A key question related to these conjectures is whether the tensor product of the rth Steinberg module with a simple module with pr th restricted highest weight admits a good filtration. In this paper we verify this statement (i) when p ≥ 2h − 4 (h is the Coxeter number), (ii) for all rank two groups, (iii) for p ≥ 3 when the simple module corresponds to a fundamental weight and (iv) for a number of cases when the rank is less than or equal to five.
1. Introduction 1.1. Representations and filtrations Let G be a simple, simply connected algebraic group scheme over the algebraically closed field k of characteristic p > 0. Let X be the set of integral weights and DOI: 10.1007/s00031.019-09530-x
∗ Supported in part by Simons Foundation Collaboration Grant 317062. ∗∗ Supported in part by NSF grant DMS-1701768. ∗∗∗ Supported in part by Simons Foundation Collaboration Grant 245236. ∗∗∗∗
Supported in part by NSF RTG grant DMS-1344994. Received July 12, 2018. Accepted November 14, 2018. Corresponding Author: D. K. Nakano, e-mail: [email protected]
C. P. BENDEL, D. K. NAKANO, C. PILLEN, P. SOBAJE
X+ denote the dominant integral weights (relative to a fixed choice of a Borel subgroup). For any λ ∈ X+ , one can construct a non-zero module ∇(λ) = indG Bλ and the Weyl module ∆(λ). The character of these modules is given by Weyl’s character formula. The finite-dimensional simple modules L(λ) are indexed by dominant integral weights X+ and can be realized as the socle of ∇(λ) (and the head of ∆(λ)). A central idea in this area has been the concept of good and Weyl filtrations. A G-module admits a good filtration (resp. Weyl filtration) if and only if it admits a G-filtration with sections of the form ∇(µ) (resp. ∆(µ)) where µ ∈ X+ . Cohomological criteria have been proved by Donkin and Scott which give necessary and sufficient conditions for a module to admit a good filtration (resp. Weyl filtration). A module which admits both a good and Weyl filtration is called a tilting module. Ringel [Rin] and Donkin [Don2] proved that (
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