Combinatorics of Exterior Algebra, Graded Multiplicities and Generalized Exponents of Small Representations

Sabino di Trani (University of Florence)

14-Jan-2021, 13:00-14:00 (3 years ago)

Abstract: Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$, and consider the exterior algebra $\wedge\mathfrak{g}$ as $\mathfrak{g}$-representations. In the talk we will give an overview of some conjectures and of many elegant results proved in the past century about the irreducible decomposition of $\wedge\mathfrak{g}$. We will focus on a Conjecture due by Reeder that generalizes the classical result about invariants in $\wedge\mathfrak{g}$ to a special class of representations, called "small". Reeder conjectured that it is possible to compute the graded multiplicities in $\wedge\mathfrak{g}$ of this special class of representations reducing to a "Weyl group representation" problem. We will give an idea of the strategy we used to prove the conjecture in the classical case, introducing the most relevant instruments we used and we will outline the difficulties we faced with. Finally, we will expose how our formulae can be rearranged involving the Generalized Exponents of small representations, obtaining a generalization of some classical formulae for graded multiplicities in the adjoint and little adjoint case.

group theoryrings and algebras

Audience: researchers in the topic


Al@Bicocca take-away

Organizer: Claudio Quadrelli*
*contact for this listing

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