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Publication# Generic direct summands of tensor products for simple algebraic groups and quantum groups at roots of unity

Abstract

Let G be either a simple linear algebraic group over an algebraically closed field of characteristic l>0 or a quantum group at an l-th root of unity. The category Rep(G) of finite-dimensional G-modules is non-semisimple. In this thesis, we develop new techniques for studying Krull-Schmidt decompositions of tensor products of G-modules.More specifically, we use minimal complexes of tilting modules to define a tensor ideal of singular G-modules, and we show that, up to singular direct summands, taking tensor products of G-modules respects the decomposition of Rep(G) into linkage classes. In analogy with the classical translation principle, this allows us to reduce questions about tensor products of G-modules in arbitrary l-regular linkage classes to questions about tensor products of G-modules in the principal block of G. We then identify a particular non-singular indecomposable direct summand of the tensor product of two simple G-modules in the principal block (with highest weights in two given l-alcoves), which we call the generic direct summand because it appears generically in Krull-Schmidt decompositions of tensor products of simple G-modules (with highest weights in the given l-alcoves). We initiate the study of generic direct summands, and we use them to prove a necessary condition for the complete reducibility of tensor products of simple G-modules, when G is a simple algebraic group of type A_n.

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Tensor product

In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W

Linear algebraic group

In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example i

G-module

In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation

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We classify the spherical birational sheets in a complex simple simply-connected algebraic group. We use the classification to show that, when G is a connected reductive complex algebraic group with simply-connected derived subgroup, two conjugacy classes O-1, O-2 of G, with O-1 spherical, lie in the same birational sheet, up to a shift by a central element of G, if and only if the coordinate rings of O-1 and O-2 are isomorphic as G-modules. As a consequence, we prove a conjecture of Losev for the spherical subvariety of the Lie algebra of G. (C) 2021 Elsevier Inc. All rights reserved.

Henrik Densing Petersen, Alain Valette

We compute L-2-Betti numbers of postliminal, locally compact, unimodular groups in terms of ordinary dimensions of reduced cohomology with coefficients in irreducible unitary representations and the Plancherel measure. This allows us to compute the L-2-Betti numbers for semi-simple Lie groups with finite center, simple algebraic groups over local fields, and automorphism groups of locally finite trees acting transitively on the boundary. (C) 2013 Elsevier Inc. All rights reserved.

Let G be a simple algebraic group defined over an algebraically closed field k whose characteristic is either 0 or a good prime for G, and let u is an element of G be unipotent. We study the centralizer C-G(u), especially its centre Z(C-G(u)). We calculate the Lie algebra of Z(C-G(u)), in particular determining its dimension; we prove a succession of theorems of increasing generality, the last of which provides a formula for dim Z(C-G(u)) in terms of the labelled diagram associated to the conjugacy class containing u.

2011