Critical points of master functions and flag varieties

TL;DR

This paper studies the critical points of master functions linked to Kac-Moody algebras, introduces a generating process for these points, and conjectures a deep geometric connection to flag varieties, proving it for certain algebras.

Contribution

It introduces a new generating procedure for critical points and conjectures their isomorphism to flag varieties, proving this for specific Kac-Moody algebras.

Findings

Population of critical points is isomorphic to flag variety for certain algebras.

Populations correspond to intersection points of Schubert cycles in Grassmannians.

Proved the conjecture for $sl_{N+1}$, $so_{2N+1}$, and $sp_{2N}$ algebras.

Abstract

We consider critical points of master functions associated with integral dominant weights of Kac-Moody algebras and introduce a generating procedure constructing new critical points starting from a given one. The set of all critical points constructed from a given one is called a population. We formulate a conjecture that a population is isomorphic to the flag variety of the Langlands dual Kac-Moody algebra and prove the conjecture for algebras $sl_{N+1}, so_{2N+1}$, and $sp_{2N}$. We show that populations associated with a collection of integral dominant $sl_{N+1}$-weights are in one to one correspondence with intersection points of suitable Schubert cycles in a Grassmannian variety.