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Schur algebras and representation theory

Martin, Stuart, (1964-)

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Schur algebras and representation theory

Record Type:	Electronic resources : Monograph/item
Title/Author:	Schur algebras and representation theory/ Stuart Martin.
remainder title:	Schur Algebras & Representation Theory
Author:	Martin, Stuart,
Published:	Cambridge :Cambridge University Press, : 1993.,
Description:	xv, 232 p. :ill., digital ;24 cm.
Notes:	Title from publisher's bibliographic system (viewed on 05 Oct 2015).
[NT 15003449]:	1. Polynomial functions and combinatorics. 1.1. Introductory remarks. 1.2. Schur's thesis. 1.3. The polynomial algebra. 1.4. Combinatorics. 1.5. Character theory and weight spaces. 1.6. Irreducible objects in P[subscript K](n, r) -- 2. The Schur algebra. 2.1. Definition. 2.2. First properties. 2.3. The Schur algebra S[subscript K](n, r). 2.4. Bideterminants and codeterminants. 2.5. The Straightening Formula. 2.6. The Desarmenien matrix and independence -- 3. Representation theory of the Schur algebra. 3.1. Modules for [Alpha subscript r] and S[subscript r]. 3.2. Schur modules as induced modules. 3.3. Heredity chains. 3.4. Schur modules and Weyl modules. 3.5. Modular representation theory for Schur algebras -- 4. Schur functors and the symmetric group. 4.1. The Schur functor. 4.2. Applying the Schur functor. 4.3. Hom functors for quasi-hereditary algebras. 4.4. Decomposition numbers for G and [Gamma]. 4.5. [Delta]-[actual symbol not reproducible]-good filtrations. 4.6. Young modules -- 5. Block theory.
[NT 15003449]:	5.1. Summary of block theory. 5.2. Return of the Hom functors. 5.3. Primitive blocks. 5.4. General blocks. 5.5. The finiteness theorem. 5.6. Examples -- 6. The q-Schur algebra. 6.1. Quantum matrix space. 6.2. The q-Schur algebra, first visit. 6.3. Weights and polynomial modules. 6.4. Characters and irreducible [Alpha subscript q](n)-modules. 6.5. R-forms for q-Schur algebras. 6.6. The q-Schur algebra, second visit -- 7. Representation theory of S[subscript q](n, r). 7.1. q-Weyl modules. 7.2. The q-determinant in [Alpha subscript q](n, r). 7.3. A quantum GL[subscript n]. 7.4. The category P[subscript q](n, r). 7.5. P[subscript q](n, r) is a highest weight category. 7.6. Representations of GL[subscript n](q) and the q-Young modules. 7.7. Conclusion -- Appendix: a review of algebraic groups -- A.1 Linear algebraic groups: definitions -- A.2 Examples of linear algebraic groups -- A.3 The weight lattice -- A.4 Root systems -- A.5 Weyl groups -- A.6 The affine Weyl group.
[NT 15003449]:	A.7 Simple modules for reductive groups -- A.8 General linear group schemes.
Subject:	Representations of groups. -
Online resource:	https://doi.org/10.1017/CBO9780511470899
ISBN:	9780511470899

Schur algebras and representation theory
Martin, Stuart,1964-

Schur algebras and representation theory[electronic resource] /Schur Algebras & Representation TheoryStuart Martin. - Cambridge :Cambridge University Press,1993. - xv, 232 p. :ill., digital ;24 cm. - Cambridge tracts in mathematics ;112. - Cambridge tracts in mathematics ;112..

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

1. Polynomial functions and combinatorics. 1.1. Introductory remarks. 1.2. Schur's thesis. 1.3. The polynomial algebra. 1.4. Combinatorics. 1.5. Character theory and weight spaces. 1.6. Irreducible objects in P[subscript K](n, r) -- 2. The Schur algebra. 2.1. Definition. 2.2. First properties. 2.3. The Schur algebra S[subscript K](n, r). 2.4. Bideterminants and codeterminants. 2.5. The Straightening Formula. 2.6. The Desarmenien matrix and independence -- 3. Representation theory of the Schur algebra. 3.1. Modules for [Alpha subscript r] and S[subscript r]. 3.2. Schur modules as induced modules. 3.3. Heredity chains. 3.4. Schur modules and Weyl modules. 3.5. Modular representation theory for Schur algebras -- 4. Schur functors and the symmetric group. 4.1. The Schur functor. 4.2. Applying the Schur functor. 4.3. Hom functors for quasi-hereditary algebras. 4.4. Decomposition numbers for G and [Gamma]. 4.5. [Delta]-[actual symbol not reproducible]-good filtrations. 4.6. Young modules -- 5. Block theory.

The Schur algebra is an algebraic system providing a link between the representation theory of the symmetric and general linear groups (both finite and infinite). In the text Dr Martin gives a full, self-contained account of this algebra and these links, covering both the basic theory of Schur algebras and related areas. He discusses the usual representation-theoretic topics such as constructions of irreducible modules, the blocks containing them, their modular characters and the problem of computing decomposition numbers; moreover deeper properties such as the quasi-hereditariness of the Schur algebra are discussed. The opportunity is taken to give an account of quantum versions of Schur algebras and their relations with certain q-deformations of the coordinate rings of the general linear group. The approach is combinatorial where possible, making the presentation accessible to graduate students. This is the first comprehensive text in this important and active area of research; it will be of interest to all research workers in representation theory.

ISBN: 9780511470899Subjects--Topical Terms:

519598
Representations of groups.

LC Class. No.: QA176 / .M37 1993

Dewey Class. No.: 512.55

Schur algebras and representation theory
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246 3 $a Schur Algebras & Representation Theory
260 $a Cambridge : $b Cambridge University Press, $c 1993.
300 $a xv, 232 p. : $b ill., digital ; $c 24 cm.
490 1 $a Cambridge tracts in mathematics ; $v 112
500 $a Title from publisher's bibliographic system (viewed on 05 Oct 2015).
505 0 $a 1. Polynomial functions and combinatorics. 1.1. Introductory remarks. 1.2. Schur's thesis. 1.3. The polynomial algebra. 1.4. Combinatorics. 1.5. Character theory and weight spaces. 1.6. Irreducible objects in P[subscript K](n, r) -- 2. The Schur algebra. 2.1. Definition. 2.2. First properties. 2.3. The Schur algebra S[subscript K](n, r). 2.4. Bideterminants and codeterminants. 2.5. The Straightening Formula. 2.6. The Desarmenien matrix and independence -- 3. Representation theory of the Schur algebra. 3.1. Modules for [Alpha subscript r] and S[subscript r]. 3.2. Schur modules as induced modules. 3.3. Heredity chains. 3.4. Schur modules and Weyl modules. 3.5. Modular representation theory for Schur algebras -- 4. Schur functors and the symmetric group. 4.1. The Schur functor. 4.2. Applying the Schur functor. 4.3. Hom functors for quasi-hereditary algebras. 4.4. Decomposition numbers for G and [Gamma]. 4.5. [Delta]-[actual symbol not reproducible]-good filtrations. 4.6. Young modules -- 5. Block theory.
505 0 $a 5.1. Summary of block theory. 5.2. Return of the Hom functors. 5.3. Primitive blocks. 5.4. General blocks. 5.5. The finiteness theorem. 5.6. Examples -- 6. The q-Schur algebra. 6.1. Quantum matrix space. 6.2. The q-Schur algebra, first visit. 6.3. Weights and polynomial modules. 6.4. Characters and irreducible [Alpha subscript q](n)-modules. 6.5. R-forms for q-Schur algebras. 6.6. The q-Schur algebra, second visit -- 7. Representation theory of S[subscript q](n, r). 7.1. q-Weyl modules. 7.2. The q-determinant in [Alpha subscript q](n, r). 7.3. A quantum GL[subscript n]. 7.4. The category P[subscript q](n, r). 7.5. P[subscript q](n, r) is a highest weight category. 7.6. Representations of GL[subscript n](q) and the q-Young modules. 7.7. Conclusion -- Appendix: a review of algebraic groups -- A.1 Linear algebraic groups: definitions -- A.2 Examples of linear algebraic groups -- A.3 The weight lattice -- A.4 Root systems -- A.5 Weyl groups -- A.6 The affine Weyl group.
505 0 $a A.7 Simple modules for reductive groups -- A.8 General linear group schemes.
520 $a The Schur algebra is an algebraic system providing a link between the representation theory of the symmetric and general linear groups (both finite and infinite). In the text Dr Martin gives a full, self-contained account of this algebra and these links, covering both the basic theory of Schur algebras and related areas. He discusses the usual representation-theoretic topics such as constructions of irreducible modules, the blocks containing them, their modular characters and the problem of computing decomposition numbers; moreover deeper properties such as the quasi-hereditariness of the Schur algebra are discussed. The opportunity is taken to give an account of quantum versions of Schur algebras and their relations with certain q-deformations of the coordinate rings of the general linear group. The approach is combinatorial where possible, making the presentation accessible to graduate students. This is the first comprehensive text in this important and active area of research; it will be of interest to all research workers in representation theory.
650 0 $a Representations of groups. $3 519598
650 0 $a Representations of algebras. $3 572480
830 0 $a Cambridge tracts in mathematics ; $v 112. $3 3470710
856 4 0 $u https://doi.org/10.1017/CBO9780511470899