Symmetric Groups, Symmetric Functions, and Grothendieck Polynomials
The main goal of these lectures is to describe some different realizations of the symmetric group (such as Bruhat, Hurwitz, Star, …), as well as specific properties of these realizations (e.g., Hilbert polynomials, certain factorizations of its elements, …).
Next, I suppose to consider in more detail the Bruhat realization (Young tableaux, symmetric functions, Schur polynomials, Cauchy identities, Jacobi–Trudy formulae, RSK, …). At the end of this part, I plan to explain and draw the audience's attention to some realizations of the braid group.
In the second part of these lectures, I will refresh some basic definitions and results concerning flag varieties (of type A) used in this part. Next, I introduce the so-called nil-Coxeter and 0-Hecke algebras and define the Cauchy kernels in these algebras. The Schubert (Grothendieck) polynomials are the polynomials that can be obtained from the decomposition of the corresponding Cauchy kernels by the standard basis in nil-Coxeter, or 0-Hecke algebras.
Next, I suppose to consider in more detail the Bruhat realization (Young tableaux, symmetric functions, Schur polynomials, Cauchy identities, Jacobi–Trudy formulae, RSK, …). At the end of this part, I plan to explain and draw the audience's attention to some realizations of the braid group.
In the second part of these lectures, I will refresh some basic definitions and results concerning flag varieties (of type A) used in this part. Next, I introduce the so-called nil-Coxeter and 0-Hecke algebras and define the Cauchy kernels in these algebras. The Schubert (Grothendieck) polynomials are the polynomials that can be obtained from the decomposition of the corresponding Cauchy kernels by the standard basis in nil-Coxeter, or 0-Hecke algebras.
Lecturer
Date
18th February ~ 13th May, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 13:30 - 15:05 | A3-1a-204 | Zoom 46a | 864 0037 8637 | 1 |
Prerequisite
The knowledge of linear algebra (vector spaces, matrices, determinants, …) and basic algebraic structures (groups, algebras, rings, polynomial rings, …). Basic facts about algebraic varieties, (for example, flag varieties) and about cohomologies and K-theory will be used in the second part of the course about Grothendieck polynomials.
Syllabus
1. Symmetric group: definition and properties (Bruhat realization).
2. Hurwitz and Star realizations of symmetric groups.
3. Bruhat realization: the ring of symmetric polynomials; monomial and complete symmetric polynomials, power sums.
4. Relations between monomial, complete symmetric, and elementary symmetric polynomials.
5. Schur functions/polynomials and some of their combinatorial properties.
6. Young tableaux, standard and semi-standard, column strict tableaux.
7. Cauchy identities and their combinatorial interpretation.
8. Robinson–Schensted–Knuth (RSK) correspondence.
9. Jacobi–Trudy identities.
10. Braid group; Birman–Ko–Lee realization of the braid group.
11. Flag varieties: definition, Bruhat decomposition. Schubert cells and classical Schubert polynomials.
12. Divided difference operators and some relations between the latter.
13. Lascoux–Schützenberger’s Schubert (and Grothendieck) polynomials.
14. Key (K-Key) polynomials. Generalization to 5-parameter family of polynomials. Some conjectures.
15. Nil-Coxeter, 0-Hecke, and Plactic algebras. Definitions.
16. Quantum Yang–Baxter relations and symmetric polynomials.
17. Non-commutative Cauchy kernels and Schubert, and
18. Grothendieck polynomials.
19. Some combinatorial properties of (non-commutative) Cauchy kernel.
20. Combinatorial formulas for Schubert, and
21. Grothendieck polynomials.
22. Principal specialization of Schubert polynomials (S. Fomin, R. Stanley).
23. Main specialization of Grothendieck polynomials.
24. Plactic algebra and Grothendieck polynomials (MacNeil completion).
2. Hurwitz and Star realizations of symmetric groups.
3. Bruhat realization: the ring of symmetric polynomials; monomial and complete symmetric polynomials, power sums.
4. Relations between monomial, complete symmetric, and elementary symmetric polynomials.
5. Schur functions/polynomials and some of their combinatorial properties.
6. Young tableaux, standard and semi-standard, column strict tableaux.
7. Cauchy identities and their combinatorial interpretation.
8. Robinson–Schensted–Knuth (RSK) correspondence.
9. Jacobi–Trudy identities.
10. Braid group; Birman–Ko–Lee realization of the braid group.
11. Flag varieties: definition, Bruhat decomposition. Schubert cells and classical Schubert polynomials.
12. Divided difference operators and some relations between the latter.
13. Lascoux–Schützenberger’s Schubert (and Grothendieck) polynomials.
14. Key (K-Key) polynomials. Generalization to 5-parameter family of polynomials. Some conjectures.
15. Nil-Coxeter, 0-Hecke, and Plactic algebras. Definitions.
16. Quantum Yang–Baxter relations and symmetric polynomials.
17. Non-commutative Cauchy kernels and Schubert, and
18. Grothendieck polynomials.
19. Some combinatorial properties of (non-commutative) Cauchy kernel.
20. Combinatorial formulas for Schubert, and
21. Grothendieck polynomials.
22. Principal specialization of Schubert polynomials (S. Fomin, R. Stanley).
23. Main specialization of Grothendieck polynomials.
24. Plactic algebra and Grothendieck polynomials (MacNeil completion).
Audience
Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
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.