Homological Theory of Representations

2021 ◽  
Author(s):  
Henning Krause
Keyword(s):  
Representation Theory ◽  
Derived Categories ◽  
Schur Algebras ◽  
General Linear Groups ◽  
Morita Theory ◽  
Polynomial Representations ◽  
Hereditary Algebras ◽  
Homological Theory ◽  
Tilting Objects ◽  
Finitely Presented

Modern developments in representation theory rely heavily on homological methods. This book for advanced graduate students and researchers introduces these methods from their foundations up and discusses several landmark results that illustrate their power and beauty. Categorical foundations include abelian and derived categories, with an emphasis on localisation, spectra, and purity. The representation theoretic focus is on module categories of Artin algebras, with discussions of the representation theory of finite groups and finite quivers. Also covered are Gorenstein and quasi-hereditary algebras, including Schur algebras, which model polynomial representations of general linear groups, and the Morita theory of derived categories via tilting objects. The final part is devoted to a systematic introduction to the theory of purity for locally finitely presented categories, covering pure-injectives, definable subcategories, and Ziegler spectra. With its clear, detailed exposition of important topics in modern representation theory, many of which were unavailable in one volume until now, it deserves a place in every representation theorist's library.

scholarly journals Young modules for symmetric groups

2001 ◽  
Vol 71 (2) ◽  
pp. 201-210 ◽  
Author(s):  
Karin Erdmann
Keyword(s):  
Representation Theory ◽  
Finite Groups ◽  
Symmetric Groups ◽  
General Linear ◽  
Linear Groups ◽  
Schur Algebras ◽  
General Linear Groups ◽  
Green Correspondence

AbstractLet K be a field of characteristic p. The permutation modules associated to partitions of n, usually denoted as Mλ, play a central role not only for symmetric groups but also for general linear groups, via Schur algebras. The indecomposable direct summands of these Mλ were parametrized by James; they are now known as Young modules; and Klyachko and Grabmeier developed a ‘Green correspondence’ for Young modules. The original parametrization used Schur algebras; and James remarked that he did not know a proof using only the representation theory of symmetric groups. We will give such proof, and we will at the same time also prove the correspondence result, by using only the Brauer construction, which is valid for arbitrary finite groups.

Semisimple Schur Algebras

1998 ◽  
Vol 124 (1) ◽  
pp. 15-20 ◽  
Author(s):  
STEPHEN R. DOTY ◽  
DANIEL K. NAKANO
Keyword(s):  
Representation Theory ◽  
General Linear ◽  
Linear Groups ◽  
Infinite Field ◽  
Polynomial Representation ◽  
Schur Algebras ◽  
Symmetric Powers ◽  
General Linear Groups ◽  
Finite Dimensional ◽  
Frobenius Kernels

Schur algebras are certain finite-dimensional algebras that completely control the polynomial representation theory of the general linear groups over an infinite field. Infinitesimal Schur algebras are truncated versions of the classical Schur algebras which control the polynomial representation theory of the Frobenius kernels of general linear groups. In this paper we use some elementary results on symmetric powers to classify the semisimple Schur algebras. We then classify the semisimple infinitesimal Schur algebras as well.

Finite Repetitive Generalized Cluster Complexes and d-Cluster Categories

2013 ◽  
Vol 20 (01) ◽  
pp. 123-140
Author(s):  
Teng Zou ◽  
Bin Zhu
Keyword(s):  
Simplicial Complex ◽  
Root System ◽  
Cluster Complex ◽  
Cluster Complexes ◽  
Endomorphism Algebras ◽  
Tilted Algebras ◽  
Cluster Categories ◽  
Hereditary Algebras ◽  
Cluster Tilting ◽  
Tilting Objects

For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of d-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) d-cluster tilted algebras. Moreover, we prove that the generalized d-cluster categories of hereditary algebras of finite representation type provide a category model for the n-repetitive generalized cluster complexes.

scholarly journals Completing perfect complexes

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1387-1427 ◽  
Author(s):  
Henning Krause
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Direct Construction ◽  
Coherent Sheaves ◽  
Perfect Complexes ◽  
Coherent Ring ◽  
Abelian Categories ◽  
Finitely Presented

Abstract This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

Cellular bases of generalised q-Schur algebras

2016 ◽  
Vol 162 (3) ◽  
pp. 533-560
Author(s):  
STEPHEN DOTY ◽  
ANTHONY GIAQUINTO
Keyword(s):  
Representation Theory ◽  
Finite Type ◽  
Basic Structure ◽  
Schur Algebras ◽  
Generators And Relations

AbstractStarting from their defining presentation by generators and relations, we develop the basic structure and representation theory of generalised q-Schur algebras of finite type.

scholarly journals Burnside rings for Real 2-representation theory: The linear theory

2020 ◽  
pp. 2050012 ◽  
Author(s):  
Dmitriy Rumynin ◽  
Matthew B. Young
Keyword(s):  
Representation Theory ◽  
Fundamental Group ◽  
Linear Theory ◽  
Character Theory ◽  
Burnside Ring ◽  
Grothendieck Ring ◽  
Fundamental Study ◽  
The Real ◽  
Morita Theory ◽  
Burnside Rings

This paper is a fundamental study of the Real 2-representation theory of 2-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a 2-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real 2-representations as a Real variant of the Burnside ring of the fundamental group of the 2-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to 2-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real 2-representation theory of 2-groups, as studied by the second author.

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