2023/06/28 by Jérémie Guilhot, Guilhot, Jérémie, Cédric Lecouvey +3
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2306.16105
openalex publication_date 2023/06/28 · openalex created_date 2023/06/30 · openalex updated_date 2026/07/28
Let \mathfrakg be an untwisted affine Lie algebra with associated Weyl group Wa. To any level 0 weight γ we associate a weighted graph Γγ that encodes the orbit of γ under the action Wa. We show that the graph Γγ encodes the periodic orientation of certain subsets of alcoves in Wa 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 Γγ 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 Γγ yields the key ingredients to study a large class of central random walks on alcoves.