Schubert calculus and symmetric polynomials
The main goal of the course is to look on some classical problems in the symmetric polynomials, algebraic combinatorics and algebraic topology from the point of view of the hidden Yang-Baxter structures, which origins from mathematical physics. We will start from the refreshment of the classical items in combinatorics and algebraic geometry. The second part of the course is devoted to application of the hidden Yang-Baxter structure in the classical problems of Schubert and Grothendieck calculus. The last part will contain the modern generalizations of these ideas, which I am developing now.
Lecturer
Date
19th September ~ 19th December, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Friday | 13:30 - 15:15 | A3-1a-205 | - | - | - |
Prerequisite
The knowledge of linear algebra and basic algebraic structures (groups, rings, polynomial rings) is required. The familiarity with the basic facts about cohomologies and K-theory will be quite useful in the second part of the course.
Syllabus
Part 1. Preliminaries.
1. Symmetric group, definition and basic properties
2. Ring of polynomials and action of symmetric group on it; ring of symmetric functions;
3. Basic symmetric functions
4. Elementary, complete and power sums;
5. Schur functions, definition and basic properties;
6. Jacobi-Trudy identities, Cauchy identities;
7. Diagrams, tableaux and basic properties; Euler and Euler-Bethe identities; diagrams and tableaux;
8. Tableaux formula for Schur polynomials; Kostka polynomials
9. Flag varieties. Definition and basic properties: Schubert cells, cohomology ring, Schubert classes and Schubert polynomials.
10. Divided differences and Lascaux-Schützenberger Schubert polynomials.
Part 2. Yang-Baxter relations and Schubert and Grothendieck polynomials.
11. Nil-Coxeter, Id-Coxeter, and Plactic algebras. Definitions and basic properties.
12. Yang-Baxter relations, symmetric polynomials and non-commutative Cauchy kernel.
13. Schubert expression, divided differences and Schubert polynomials.
14. Id-Coxeter algebra and Grothendieck polynomials.
15. Some combinatorial rites of Schubert and Grothendieck polynomials: compatible sequences, reduced decompositions, principal specialization formulas for Schubert and Grothendieck polynomials.
16. Plactic relations, monoid and algebras. Plactic Cauchy kernel, plactic double and Stanley polynomials. ASM and TSPP matrices.
Part 3. Non-commutative approach to (small) quantum Schubert and Grothendieck polynomials and Fomin-Kirillov algebras.
17. Graphs, quadratic Algebras and Dunkl elements. Definitions and basic properties.
18. Fomin-Kirillov algebras. Definition and basic properties of FK algebras for complete digraphs.
19. Relations between Dunkl element and a commutative subalgebra generated by Dunkl elements.
20. Nil-Fomin-Kirillov algebra and cohomology ring of type A flag varieties.
21. FK algebras and small quantum cohomology of type A flag varieties.
22. Multiplicative Dunkl elements and K-theory of type A flag varieties.
23. The case of Multipartite graphs and parabolic flag varieties ( e.g. Grassmannians,…).
24. Elliptic representations of FK algebras and commutative subalgebras generated by Dunkl elements.
1. Symmetric group, definition and basic properties
2. Ring of polynomials and action of symmetric group on it; ring of symmetric functions;
3. Basic symmetric functions
4. Elementary, complete and power sums;
5. Schur functions, definition and basic properties;
6. Jacobi-Trudy identities, Cauchy identities;
7. Diagrams, tableaux and basic properties; Euler and Euler-Bethe identities; diagrams and tableaux;
8. Tableaux formula for Schur polynomials; Kostka polynomials
9. Flag varieties. Definition and basic properties: Schubert cells, cohomology ring, Schubert classes and Schubert polynomials.
10. Divided differences and Lascaux-Schützenberger Schubert polynomials.
Part 2. Yang-Baxter relations and Schubert and Grothendieck polynomials.
11. Nil-Coxeter, Id-Coxeter, and Plactic algebras. Definitions and basic properties.
12. Yang-Baxter relations, symmetric polynomials and non-commutative Cauchy kernel.
13. Schubert expression, divided differences and Schubert polynomials.
14. Id-Coxeter algebra and Grothendieck polynomials.
15. Some combinatorial rites of Schubert and Grothendieck polynomials: compatible sequences, reduced decompositions, principal specialization formulas for Schubert and Grothendieck polynomials.
16. Plactic relations, monoid and algebras. Plactic Cauchy kernel, plactic double and Stanley polynomials. ASM and TSPP matrices.
Part 3. Non-commutative approach to (small) quantum Schubert and Grothendieck polynomials and Fomin-Kirillov algebras.
17. Graphs, quadratic Algebras and Dunkl elements. Definitions and basic properties.
18. Fomin-Kirillov algebras. Definition and basic properties of FK algebras for complete digraphs.
19. Relations between Dunkl element and a commutative subalgebra generated by Dunkl elements.
20. Nil-Fomin-Kirillov algebra and cohomology ring of type A flag varieties.
21. FK algebras and small quantum cohomology of type A flag varieties.
22. Multiplicative Dunkl elements and K-theory of type A flag varieties.
23. The case of Multipartite graphs and parabolic flag varieties ( e.g. Grassmannians,…).
24. Elliptic representations of FK algebras and commutative subalgebras generated by Dunkl elements.
Reference
[1] I.G. Macdonald, Symmetric functions and Hall polynomials (1995）
[2] I.G. Macdonald, Notes on Schubert polynomials (1991)
[3] L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci (2001)
[4] R.P. Stanley, Enumerative combinatorics, vol.2 (2001)
[5] S.I. Fomin, A.N. Kirillov, Quadratic algebras, Dunkl Elements and Schubert Calculus (1999)
[6] S. Fomin, A.N. Kirillov, Yang-Baxter equation, symmetric functions and Grothendieck polynomials (hep-th/9306005)
[7] A.N. Kirillov, On some quadratic algebras 1+1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and Reduced Polynomials (arXiv:1501.07337)
[2] I.G. Macdonald, Notes on Schubert polynomials (1991)
[3] L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci (2001)
[4] R.P. Stanley, Enumerative combinatorics, vol.2 (2001)
[5] S.I. Fomin, A.N. Kirillov, Quadratic algebras, Dunkl Elements and Schubert Calculus (1999)
[6] S. Fomin, A.N. Kirillov, Yang-Baxter equation, symmetric functions and Grothendieck polynomials (hep-th/9306005)
[7] A.N. Kirillov, On some quadratic algebras 1+1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and Reduced Polynomials (arXiv:1501.07337)
Video Public
Yes
Notes Public
Yes
Lecturer Intro
Anatol Kirillov is a researcher in the area of integrable systems, representation theory, special functions, algebraic combinatorics, and algebraic geometry. He worked as a professor in different universities in Japan for the last 20 years. In 2022 he joined BIMSA as a research fellow.