Recent developments in representation theory : Maurice Auslander Distinguished Lectures and International Conference, May 1-6, 2014 : Woods Hole Oceanographic Institute, Woods Hole, MA / Alex Martsinkovsky, Gordana Todorov, Kiyoshi Igusa, editors.

Format:

Book

Conference/Event

Contributor:

Martsinkovsky, A. (Alex), editor.

Todorov, G. (Gordana), editor.

Igusa, Kiyoshi, editor.

Conference Name:

Maurice Auslander Distinguished Lectures and International Conference (2014 : Woods Hole, Mass.)

Series:

Contemporary mathematics (American Mathematical Society) ; 673.

Contemporary Mathematics, 1098-3627 ; 673

Language:

English

Subjects (All):

Associative rings--Congresses.

Associative rings.

Representations of rings (Algebra)--Congresses.

Representations of rings (Algebra).

Physical Description:

1 online resource (258 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2016.

Summary:

This volume contains selected expository lectures delivered at the Maurice Auslander Distinguished Lectures and International Conference, held May 1-6, 2014, at the Woods Hole Oceanographic Institute, Woods Hole, MA. Several significant developments of the last decade in representation theory of finite-dimensional algebras are related to combinatorics. Three of the five lectures in this volume deal, respectively, with the Catalan combinatorics, the combinatorics of Gelfand-Zetlin polytopes, and the combinatorics of tilting modules. The remaining papers present history and recent advances in the study of left orders in left Artinian rings and a survey on invariant theory of Artin-Schelter regular algebras.

Contents:

Cover

Title page

Contents

Preface

Orders in Artinian rings, Goldie's Theorem and the largest left quotient ring of a ring

1. Introduction

Part 1. New criteria for a ring to have a semisimple left quotient ring

2. Four new criteria for a ring to have a semisimple left quotient ring

Part 2. Left Orders in Left Artinian Rings

3. Old criteria for a ring to have a left Artinian left quotient ring

4. Necessary and sufficient conditions for a ring to have a left Artinian left quotient ring

5. A criterion via associated graded ring

6. Criteria similar to Robson's Criterion

7. A left quotient ring of a factor ring

Part 3. The Largest Left Quotient Ring of a Ring

8. The largest denominator sets and the largest left quotient ring of a ring

9. The maximal left quotient rings of a ring

10. Examples

Acknowledgements

References

Invariant theory of Artin-Schelter regular algebras: a survey

0. Introduction

1. Artin-Schelter regular subrings of invariants

2. Artin-Schelter Gorenstein subrings of invariants

3. Complete intersection subrings of invariants

4. Related research directions

The Catalan combinatorics of the hereditary artin algebras

Introduction

Outline

Notes

1. Numbers

1.1. The setting

1.2. Dynkin functions

1.3. The exponents

1.4. The height partition

1.5. Inductive determination of the exponents

Notes to Chapter 1

2. Tilting Theory

2.1. Linearity of tilting torsion pairs

2.2. Exceptional antichains and normal partial tilting modules

2.3. The category ( )

2.4. The Ingalls-Thomas bijections

2.5. Perpendicular pairs and exceptional sequences

2.6. Torsion pairs and perpendicular pairs

2.7. Partial orderings on the set of antichains

Notes to Chapter 2

3. The Poset \A(\moΛ) of Exceptional Antichains.

3.1. The poset \A(\moΛ): Definition and first properties

3.2. The poset \A(\moΛ): Is it a lattice?

3.3. The poset \A(\moΛ): Intervals

3.4. The poset \A(\moΛ): Automorphisms and anti-automorphisms

3.5. The poset \A(\moΛ): Maximal chains, complete exceptional sequences

3.6. The braid group operation on \E(\moΛ) and on \M(\A(\moΛ))

3.7. Generalized non-crossing partitions

3.8. The braid group operation on \F( , )

Notes to Chapter 3

4. The Hereditary Artin Algebra Λ_{ }

4.1. The lattice \NC( ) of non-crossing partitions of an -element set

4.2. The categorification of \NC( )

4.3. Perpendicular pairs and the Kreweras complement

4.4. Non-crossing partitions and binary trees

4.5. The ( +1)ⁿ⁻¹-problems: Maximal chains of non-crossing partitions, parking functions, labeled trees

Notes to Chapter 4

5. Appendix

5.1. What is Catalan combinatorics? A first answer

5.2. What is Catalan combinatorics? A second answer

Grassmannians, flag varieties, and Gelfand-Zetlin polytopes

2. Grassmannians

3. Flag varieties

4. Toric varieties

5. An approach to Schubert calculus via Khovanskii-Pukhlikov rings

On the combinatorics of the set of tilting modules

1. The partial order ( _{Λ},\le) of tilting modules

2. The quiver of tilting modules

3. The simplicial complex of tilting modules

4. Acknowledgments

Back Cover.

Notes:

Includes bibliographical references at the end of each chapters.

Description based on print version record.

ISBN:

1-4704-3530-6