北京雁栖湖应用数学研究院

2026年03月03日 至 05月29日

Part I. Refreshment
Lectures 1-6
ˆ Lie groups and Lie algebras.
ˆ Main examples: symmetric and general linear groups; definitions and basic properties.
ˆ (Overview) Representation theory of symmetric and general linear groups: Schur functions, Cauchy identities, Schur-Weyl duality, and Howe duality.

Part II. Combinatorics
Lectures 7-15
ˆ Compositions (strong and weak).
ˆ Diagrams of partitions; Young tableaux (strong or strict semi-simple; column/rowstrict tableaux).
ˆ Symmetric functions and (semi-simple) Young tableaux.
ˆ Symmetric functions and characters of finite-dimensional irreducible representations of the symmetric and general linear groups.
ˆ Dimension formulas for characters.
ˆ Kostant partition functions; Kostka numbers and polynomials.
ˆ Weight multiplicities, Littlewood-Richardson rule, and Robinson-Schensted correspondence.

Part III. Fermionic Formulas for Parabolic Kostka Polynomials
Lectures 18-21
ˆ Overview of the Bethe Ansatz (BA) and Bethe Ansatz equations (BAE) for the (generalized) Heisenberg spin chain.
ˆ Combinatorial analysis of BAE: string conjectures and vacancy numbers.
ˆ Rigged configurations; special matrices and riggings.
ˆ Initial data: \((\lambda, R)\), where \(\lambda \vdash n\) is a partition, and \(R = \{(\mu_a^{n_a})\}_{a \geq 1}\) is a set of rectangular-shape partitions such that \(\sum_{a \geq 1} \mu_a n_a = n\).
ˆ Special matrices \(\{m_{ij} \in \mathbb{Z} \mid 1 \leq i, j \leq n\}\) (note that, in general, \(m_{ij}\) may be negative integers).
ˆ Framing (or riggings) of special matrices, given by explicit formulas using the set of rectangles \(R\).
ˆ Charge of a framing configuration and the polynomial associated with a given framed matrix \(\{m_{ij}\}\).
ˆ Parabolic Kostka polynomials \(K_{\lambda,R}(q)\), where \(\lambda \vdash n\), \(R = \{\mu_a^{n_a}\}_{a \geq 1}\), and \(\mu_1 \geq \mu_2 \geq \dots\).
ˆ **Main Theorem**: The parabolic Kostka polynomial \(K_{\lambda,R}(q)\) is equal to the sum of charged framed (or rigged) special matrices of type \((\lambda, R)\).

Part IV. Rigged Configuration Bijection
Bijection between framed special matrices and the corresponding Littlewood-Richardson tableaux.

[1] Berenstein, A. D., and Kirillov, A. N. (2016). Cactus group and Gelfand-Tsetlin group. Res. Inst. Math. Sci. preprint RIMS-1858.
[2] Macdonald, I. G. (1998). Symmetric functions and Hall polynomials. Oxford University Press.
[3] Kirillov, A. N. (2001). Combinatorics of Young tableaux and configurations. Proceedings of the St. Petersburg Mathematical Society Volume VII, 203, 17-98.
[4] Kirillov, A. N. (2015). Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials: Polynomiality, Unimodality and Logconcavity. arXiv preprint arXiv:1505.01542.
[5] Kirillov, A. N., et al. (2016). On some quadratic algebras I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 12, 002.
[6] Kirillov, A. N. (2021). Rigged Configurations and Unimodality. In Representation Theory, Mathematical Physics, and Integrable Systems: In Honor of Nicolai Reshetikhin (pp. 453-496). Springer.

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