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.
讲师
日期
2024年09月10日 至 12月10日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周二,周四 | 13:30 - 15:05 | A3-1a-204 | Zoom 41a | 812 4478 6077 | BIMSA |
修课要求
Basic knowledge of representation theory and graph theory will be useful.
课程大纲
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 $G$.
参考资料
[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.
听众
Advanced Undergraduate
, Graduate
, 博士后
视频公开
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笔记公开
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语言
英文
讲师介绍
Anatol Kirillov的研究领域是可积系统、表示论、特殊函数、代数组合学和代数几何。他在过去的20年里，在日本不同的大学担任教授。2022年，他加入BIMSA任研究员。