CU Algebraic Lie Theory Seminar

CU Algebraic Lie Theory Seminar
Roots, weights, and distributive lattices Richard Green, CU
January 18, 2011 Math 220 2-3pm	Abstract Let $W$ be a finite Weyl group, and let $\alpha_p$ be a simple root of $W$ corresponding to a minuscule weight of the associated Lie algebra. The set of positive roots, $R_p$, of $W$ in which $\alpha_p$ appears with nonzero coefficient is equipped with a natural partial order. Remarkably, this partial order makes $R_p$ into a distributive lattice. Also remarkably, the lattice $J(R_p)$ of ideals of $R_p$ is isomorphic to the lattice of weights of the minuscule representation. There are signs that this result is folklore, but I have never seen it stated. I will discuss the strangeness of this result, speculate on possible extensions of it, and lament the lack (to my knowledge) of a conceptual proof.

Roots, weights, and distributive lattices

Richard Green, CU

January 18, 2011

Math 220

2-3pm

Abstract

Let $W$ be a finite Weyl group, and let $\alpha_p$ be a simple root of $W$ corresponding to a minuscule weight of the associated Lie algebra. The set of positive roots, $R_p$, of $W$ in which $\alpha_p$ appears with nonzero coefficient is equipped with a natural partial order. Remarkably, this partial order makes $R_p$ into a distributive lattice. Also remarkably, the lattice $J(R_p)$ of ideals of $R_p$ is isomorphic to the lattice of weights of the minuscule representation. There are signs that this result is folklore, but I have never seen it stated. I will discuss the strangeness of this result, speculate on possible extensions of it, and lament the lack (to my knowledge) of a conceptual proof.