Schur algebras and representation 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 algeb...
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
1993
|
| Series: | Cambridge tracts in mathematics
112 |
| Subjects: | |
| Online Access: | BSB01 FHN01 UBR01 Volltext |
| Summary: | 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 |
| Physical Description: | 1 Online-Ressource (xv, 232 Seiten) |
| ISBN: | 9780511470899 |
| DOI: | 10.1017/CBO9780511470899 |
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| 505 | 8 | |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 | 8 | |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 | 8 | |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 | ||
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Record in the Search Index
| _version_ | 1804176884720730112 |
|---|---|
| any_adam_object | |
| author | Martin, Stuart 1964- |
| author_facet | Martin, Stuart 1964- |
| author_role | aut |
| author_sort | Martin, Stuart 1964- |
| author_variant | s m sm |
| building | Verbundindex |
| bvnumber | BV043942193 |
| classification_rvk | SK 260 |
| collection | ZDB-20-CBO |
| contents | 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 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 A.7 Simple modules for reductive groups -- A.8 General linear group schemes |
| ctrlnum | (ZDB-20-CBO)CR9780511470899 (OCoLC)849890746 (DE-599)BVBBV043942193 |
| dewey-full | 512/.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-search | 512/.55 |
| dewey-sort | 3512 255 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1017/CBO9780511470899 |
| format | Electronic eBook |
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| id | DE-604.BV043942193 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:39:17Z |
| institution | BVB |
| isbn | 9780511470899 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029351162 |
| oclc_num | 849890746 |
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| owner_facet | DE-12 DE-92 DE-355 DE-BY-UBR |
| physical | 1 Online-Ressource (xv, 232 Seiten) |
| psigel | ZDB-20-CBO ZDB-20-CBO BSB_PDA_CBO ZDB-20-CBO FHN_PDA_CBO ZDB-20-CBO UBR Einzelkauf (Lückenergänzung CUP Serien 2018) |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Cambridge University Press |
| record_format | marc |
| series2 | Cambridge tracts in mathematics |
| spelling | Martin, Stuart 1964- Verfasser aut Schur algebras and representation theory Stuart Martin Schur Algebras & Representation Theory Cambridge Cambridge University Press 1993 1 Online-Ressource (xv, 232 Seiten) txt rdacontent c rdamedia cr rdacarrier Cambridge tracts in mathematics 112 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 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 A.7 Simple modules for reductive groups -- A.8 General linear group schemes 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 Representations of groups Representations of algebras Schur-Algebra (DE-588)4180242-1 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Schur-Algebra (DE-588)4180242-1 s Darstellungstheorie (DE-588)4148816-7 s DE-604 Erscheint auch als Druck-Ausgabe 978-0-521-41591-0 Erscheint auch als Druck-Ausgabe 978-0-521-10046-5 https://doi.org/10.1017/CBO9780511470899 Verlag URL des Erstveröffentlichers Volltext |
| spellingShingle | Martin, Stuart 1964- Schur algebras and representation theory 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 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 A.7 Simple modules for reductive groups -- A.8 General linear group schemes Representations of groups Representations of algebras Schur-Algebra (DE-588)4180242-1 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| subject_GND | (DE-588)4180242-1 (DE-588)4148816-7 |
| title | Schur algebras and representation theory |
| title_alt | Schur Algebras & Representation Theory |
| title_auth | Schur algebras and representation theory |
| title_exact_search | Schur algebras and representation theory |
| title_full | Schur algebras and representation theory Stuart Martin |
| title_fullStr | Schur algebras and representation theory Stuart Martin |
| title_full_unstemmed | Schur algebras and representation theory Stuart Martin |
| title_short | Schur algebras and representation theory |
| title_sort | schur algebras and representation theory |
| topic | Representations of groups Representations of algebras Schur-Algebra (DE-588)4180242-1 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| topic_facet | Representations of groups Representations of algebras Schur-Algebra Darstellungstheorie |
| url | https://doi.org/10.1017/CBO9780511470899 |
| work_keys_str_mv | AT martinstuart schuralgebrasandrepresentationtheory AT martinstuart schuralgebrasrepresentationtheory |