Algebraic combinatorics and quantum groups / edited by Naihuan Jing.

Format:

Book

Conference/Event

Contributor:

Jing, Naihuan.

Conference Name:

CBMS Conference on Algebraic Combinatorics (2001 : North Carolina State University)

Language:

English

Subjects (All):

Combinatorial analysis--Congresses.

Combinatorial analysis.

Quantum groups--Congresses.

Quantum groups.

Algebra--Congresses.

Algebra.

Physical Description:

1 online resource (171 p.)

Place of Publication:

River Edge, NJ : World Scientific, c2003.

Language Note:

English

Summary:

Algebraic combinatorics has evolved into one of the most active areas of mathematics during the last several decades. Its recent developments have become more interactive with not only its traditional field representation theory but also algebraic geometry, harmonic analysis and mathematical physics.This book presents articles from some of the key contributors in the area. It covers Hecke algebras, Hall algebras, the Macdonald polynomial and its deviations, and their relations with other fields.

Contents:

Preface; Contents; UNO'S CONJECTURE ON REPRESENTATION TYPES OF HECKE ALGEBRAS; 1. INTRODUCTION; 2. REDUCTION TO HECKE ALGEBRAS ASSOCIATED TO IRREDUCIBLE WEYL GROUPS; 3. TYPE B AND TYPE D; 4. APPENDIX; REFERENCES; QUIVER VARIETIES, AFFINE LIE ALGEBRAS, ALGEBRAS OF BPS STATES, AND SEMICANONICAL BASIS; 1. INTRODUCTION; 2. PRELIMINARIES; 2.1. Quivers.; 2.2. Convolution product.; 2.3. Geometric realization of the enveloping algebra.; 2.4. Lie algebra based on the Euler cocycle: classical ADE case.; 2.5. Lie algebra based on the Euler cocycle: affine ADE case.

3. STABILITY AND SIMPLE LIE ALGEBRAS OF TYPE ADE3.1. Stability after King and Rudakov.; 3.1.1. Stability after King.; 3.1.2. Stability after Rudakov.; 3.2. Stability Lemma.; 3.3. Stable construction.; 4. STABILITY, AFFINE LIE ALGEBRAS OF TYPE ADE, AND A CONJECTURAL CONSTRUCTION OF THE ALGEBRAS OF BPS STATES; 4.1. Physics.; 4.2. The conjecture.; 5. REMARKS ON SEMICANONICAL BASIS FOR SIMPLE LIE ALGEBRAS; 5.1. Semicanonical basis.; 5.2. Open problem.; 5.3. Irreducible components arising from orientations.; 5.4. Conjugacy classes of the Weyl group.; 5.5. Example: D4.; 5.6. Example: D5.

5.7. More open questions.REFERENCES; DIVIDED DIFFERENCES OF TYPE D AND THE GRASSMANNIAN OF COMPLEX STRUCTURES; 1. INTRODUCTION; 2. PRELIMINARIES, NOTATION, AND CONVENTIONS; 3. COMBINATORICS OF DIVIDED DIFFERENCES OF TYPE D; 4. A GROUP-THEORETIC APPROACH TO SCHUBERT CALCULUS FOR CSn; 5. SCHUBERT CYCLES OP COMPLEX STRUCTURES ON R2n; REFERENCES; TABLEAUX STATISTICS FOR TWO PART MACDONALD POLYNOMIALS; 1. INTRODUCTION; 2. DEFINITIONS; 3. ALGEBRAIC SIDE; 4. TABLEAUX SIDE; 4.1. Definition and background.; 4.2. Tableaux operators.; 5. A STATISTIC FOR MACDONALD POLYNOMIALS IN 2 PARTS; REFERENCES

2.1. Symmetric functions, partitions and columns strict tableaux.

Notes:

Papers from the "... CBMS conference "Algebraic Combinatorics" held at North Carolina State University during June 4-8, 2001." -- preface.

Includes bibliographical references.

ISBN:

9786611928124

9781281928122

1281928127

9789812775405

9812775404