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Unit# The Testerman Group

Group

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Algebraic group

In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs bot

Reductive group

In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a represen

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

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Martin W. Liebeck, Donna Testerman

We continue our work, started in [9], on the program of classifying triples (X, Y, V), where X, Yare simple algebraic groups over an algebraically closed field of characteristic zero with X < Y, and Vis an irreducible module for Y such that the restriction V down arrow X is multiplicity-free. In this paper we handle the case where X is of type A, and is irreducibly embedded in Y of type B, C or D. It turns out that there are relatively few triples for X of arbitrary rank, but a number of interesting exceptional examples arise for small ranks. (c) 2021 Elsevier Inc. All rights reserved.

Let k be an algebraically closed field of arbitrary characteristic, let G be a simple simply connected linear algebraic group and let V be a rational irreducible tensor-indecomposable finite-dimensional kG-module. For an element g of G we denote by $V_{g}(x)$ the eigenspace corresponding to the eigenvalue x of g on V. We define N to be the minimum difference between the dimension of V and the dimension of $V_{g}(x)$, where g is a non-central element of G. In this thesis we identify pairs (G,V) with the property that $N\leq \sqrt{\dim(V)}$. This problem is an extension of the classification result obtained by Guralnick and Saxl for the condition $N\leq \max\bigg\{2,\frac{\sqrt{\dim(V)}}{2}\bigg\}$. Moreover, for all the pairs (G,V) we had to consider in our classification, we will determine the value of N.

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.