Mathematical Springtime in Rome 2023

Monday, May 22nd, 15.00, room 1B1 (block RM002)

Mirko Primc, University of Zagreb

Groebner basis approach to combinatorial bases of standard modules for affine Lie algebras

 Abstract:

I'll describe the "Groebner basis approach" used in the joint works with A. Meurman on combinatorial bases of standard modules for A_1^(1) and on A_2^(1), and comment on the advantages and drawbacks of this approach in general. If time permits, I'll comment on some technical differences that appear in I. Siladić's work on A_2^(2).

Wednesday, May 24th, 15.00, room 1B1

Tomislav Šikić,  University of Zagreb

Combinatorial relations among relations for standard $C^{(1)}_n$-modules of level 2,3,…?

Abstract:

In the first part of the talk, it will be presented the construction of combinatorial bases of basic modules for affine symplectic Lie algebras $C^{(1)}_n$ (Journal of Mathematical Physics [P\v S 2016]). This construction is some kind generalization of A. Meurman and M. Primc's results (Memoirs of AMS [MP 1999]). 

The rest of the talk will be devoted to the construction of combinatorial bases of standard $C^{(1)}_n$-modules. Special accent of this talk will be devoted to the combinatorial parametrization of leading terms of defining relations for all standard modules for affine Lie algebra of type $C^{(1)}_n$. This parametrization is the base of a conjecture on the standard modules $L(k\Lambda_0)$ where $n\geq 2$ and $k\geq 2$ (The Ramanujan Journal  [P\v S 2019]). The numerical evidence for some characteristic examples which supports our conjecture will be given.

At the end of talk, will be presented recent result for level $2$ standard $C^{(1)}_2$-module [P\v S 2023 (arXiv:2301.11222)] and expectations for larger levels in the case $n=2$. 

 This talk is based on joint work with Mirko Primc.

Friday, May 26th, 11.00, room 1B1

Jehanne Dousse, University of Geneva,

Title: Integer partitions and characters of affine Lie algebras

A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. In the 1980's, Lepowsky and Wilson established a connection between the Rogers-Ramanujan partition identities and characters of affine Lie algebras. Since then, a nice interplay between combinatorics and representation theory has led to the discovery of new results in both fields. After presenting the history of these interactions, we will introduce grounded partitions, a generalisation of partitions which is well-suited for a connection with representation theory, and show how they can be used to prove partition identities and character formulas.