Combinatorial Theory

Let \(\mathfrak{g}\) be an untwisted affine Lie algebra with associated Weyl group \(W_a\) and let \(W\) be the Weyl group of the corresponding simple Lie algebra. To any level-0 weight \(\gamma\) we associate a rooted weighted graph \(\Gamma_\gamma\) that encodes the orbit of \(\gamma\) under the action \(W_a\). From a combinatorial point of view, the graph \(\Gamma_{\gamma}\) is the weak version of the quantum Bruhat graph for the strong Bruhat order on either \(W\) or one of its parabolic quotients. 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 the graph \(\Gamma_{\gamma}\) is positively multiplicative: there exists, in some sense, a maximal family of commuting adjacency matrices which also commute with the one of \(\Gamma_{\gamma}\). This yields the key ingredients to study a large class of random walks which can be either seen as random paths on alcoves or as interacting particle systems.

Mathematics Subject Classifications: 05E05, 05E10, 22E47, 57T15

Keywords: Root systems, affine Grassmannians, homology ring, oriented graphs, alcove walks