Homology rings of affine grassmannians and positively multiplicative graphs

TL;DR

This paper introduces a new graph-based approach to understanding the homology rings of affine Grassmannians, revealing positive multiplicativity and enabling computation of structure constants through elementary linear algebra.

Contribution

It constructs weighted graphs from affine Lie algebra data that encode alcove orientations and demonstrates their positive multiplicativity, facilitating algebraic computations.

Findings

Graphs encode orbit and alcove orientation information.

Homology ring quotients are positively multiplicative.

Structure constants can be computed via linear algebra on graphs.

Abstract

Let $\mathfrak{g}$ be an untwisted affine Lie algebra with associated Weyl group $W_a$. To any level 0 weight $\gamma$ we associate a weighted graph $\Gamma_\gamma$ that encodes the orbit of $\gamma$ under the action $W_a$. We show that the graph $\Gamma_\gamma$ encodes the periodic orientation of certain subsets of alcoves in $W_a$ and therefore can be interpreted as an automaton determining the reduced expressions in these subsets. Then, by using some relevant quotients of the homology ring of affine Grassmannians, we show that $\Gamma_\gamma$ is positively multiplicative. This allows us in particular to compute the structure constants of the homology rings using elementary linear algebra on multiplicative graphs. In another direction, the positivity of $\Gamma_\gamma$ yields the key ingredients to study a large class of central random walks on alcoves.