K-theoretic Kostant partition function

The course will introduce the deformation of the classical Kostant partition function which is the basic object of the representation theory. We relate this deformation with the combinatorial objects — Ehrhart polynomials of certain lattice polytopes.

Lecturer

Date

10th September ~ 10th December, 2024

Location

Weekday | Time | Venue | Online | ID | Password |
---|---|---|---|---|---|

Tuesday,Thursday | 13:30 - 15:05 | A3-1a-204 | Zoom 41a | 812 4478 6077 | BIMSA |

Prerequisite

Basic knowledge of representation theory and graph theory will be useful.

Syllabus

Part 1: Kostant partition function (type A) and representation theory.

1. Basic facts on irreducible representations (irreps), weight spaces, Weyl theorem.

2. Kostant partition function: basic properties and connections with representation theory

3. Kostant multiplicity, and triple multiplicity theorem.

Part 2: Graphs and lattice polytopes.

1. Lattices, graphs, and lattice polytopes associated with a graph.

2. Volume of lattice polytope, Ehrhart polynomial associated with a lattice polytope.

3. Examples: permutahedron, associahedron, flow polytopes.

4. Definition and basic properties of flow polytopes, zonotope.

Part 3: 3-term relations algebras.

1. Fomin–Kirillov (FK) algebras and some of their basic properties, Dunkl elements, Bruhat representations.

2. Some open problems connected with FK algebras.

Part 4: Some topics of algebraic combinatorics.

1. Definitions and properties of Catalan numbers, Fuss–Catalan numbers, Schröder numbers and q-polynomials.

2. Higher genus analogues of the Fuss–Catalan numbers.

Part 5: Reduced polynomials associated with abelian 3-term relations algebras and K-theoretic analogue of Kostant partition function.

1. Definition of the reduced polynomial corresponding to an ordered product of elements of $3T_n$ algebra.

2. $\beta$-Coxeter and $\beta$-Kostant polynomials, basic properties of these polynomials, piecewise polynomiality.

3. Connection with Kostant partition function ($\beta = 0$), $\beta$ Fuss–Katalan polynomials.

4. Main result: For a finite loopless graph $G$, $\beta$-reduced polynomial $R_\beta(G)$ coinsides with the Hodge polynomial $H(G, \beta)$. Moreover, $H(G, \beta)$ is $\beta$-Kostant function $K^{(\beta)}_G(\gamma)$ for some point $\gamma \in G$ depending on graph $G$.

1. Basic facts on irreducible representations (irreps), weight spaces, Weyl theorem.

2. Kostant partition function: basic properties and connections with representation theory

3. Kostant multiplicity, and triple multiplicity theorem.

Part 2: Graphs and lattice polytopes.

1. Lattices, graphs, and lattice polytopes associated with a graph.

2. Volume of lattice polytope, Ehrhart polynomial associated with a lattice polytope.

3. Examples: permutahedron, associahedron, flow polytopes.

4. Definition and basic properties of flow polytopes, zonotope.

Part 3: 3-term relations algebras.

1. Fomin–Kirillov (FK) algebras and some of their basic properties, Dunkl elements, Bruhat representations.

2. Some open problems connected with FK algebras.

Part 4: Some topics of algebraic combinatorics.

1. Definitions and properties of Catalan numbers, Fuss–Catalan numbers, Schröder numbers and q-polynomials.

2. Higher genus analogues of the Fuss–Catalan numbers.

Part 5: Reduced polynomials associated with abelian 3-term relations algebras and K-theoretic analogue of Kostant partition function.

1. Definition of the reduced polynomial corresponding to an ordered product of elements of $3T_n$ algebra.

2. $\beta$-Coxeter and $\beta$-Kostant polynomials, basic properties of these polynomials, piecewise polynomiality.

3. Connection with Kostant partition function ($\beta = 0$), $\beta$ Fuss–Katalan polynomials.

4. Main result: For a finite loopless graph $G$, $\beta$-reduced polynomial $R_\beta(G)$ coinsides with the Hodge polynomial $H(G, \beta)$. Moreover, $H(G, \beta)$ is $\beta$-Kostant function $K^{(\beta)}_G(\gamma)$ for some point $\gamma \in G$ depending on graph $G$.

Reference

[1] B. Kostant. A formula for the multiplicity of weight.

[2] A.N. Kirillov. On some quadratic algebras I 1/2.

[3] M. Yip. Fuss–Catalan variation of caracol flow polytope.

[4] C. Mészáros and A.H. Morales. Volumes and Ehrhart polynomials of flow polytopes.

[5] R. Stanley and A. Postnikov. Acyclic flow polytope and Kostant partition functions.

[2] A.N. Kirillov. On some quadratic algebras I 1/2.

[3] M. Yip. Fuss–Catalan variation of caracol flow polytope.

[4] C. Mészáros and A.H. Morales. Volumes and Ehrhart polynomials of flow polytopes.

[5] R. Stanley and A. Postnikov. Acyclic flow polytope and Kostant partition functions.

Audience

Advanced Undergraduate
, Graduate
, Postdoc

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.