Linear algebraic group

Summary

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 is the orthogonal group, defined by the relation M^TM = I_n where M^T is the transpose of M. Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and . In the 1950s, Armand Borel constructed mu

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https://en.wikipedia.org/wiki/Linear_algebraic_group

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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.

EPFL2022

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

Jonathan Gruber

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.

EPFL2022

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.

EPFL2022