Jadyn V. Breland | Searcharxiv

arXiv2025

Brauer pairs for splendid Rickard equivalences

We define the notion of a Brauer pair of a chain complex, extending the notion of a Brauer pair of a $p$-permutation module introduced by Boltje and Perepelitsky. In fact, the Brauer pairs of a splendid Rickard equivalence $C$ coincide with the set of Brauer pairs of the corresponding $p$-permutation equivalence $Λ(C)$ induced by $C$. As a result, we derive structural results for splendid Rickard equivalences that correspond to known structural properties for $p$-permutation equivalences. In particular, we show splendid Rickard equivalences induce local splendid Rickard equivalences between normalizer block algebras as well as centralizer block algebras.

math.RT↗

arXiv2024

Braid graphs in simply-laced triangle-free Coxeter systems are median

Any two reduced expressions for the same Coxeter group element are related by a sequence of commutation and braid moves. Two reduced expressions are said to be braid equivalent if they are related via a sequence of braid moves. Braid equivalence is an equivalence relation and the corresponding equivalence classes are called braid classes. Each braid class can be encoded in terms of a braid graph in a natural way. In a recent paper, Awik et al.~proved that when a Coxeter system is simply laced and triangle free (i.e., the corresponding Coxeter graph has no three-cycles), the braid graph for a reduced expression is a partial cube (i.e., isometric to a subgraph of a hypercube). In this paper, we will provide an alternate proof of this fact, as well as determine the minimal dimension hypercube into which a braid graph can be isometrically embedded, which addresses an open question posed by Awik et al. For our main result, we prove that braid graphs in simply-laced triangle-free Coxeter systems are median, which is a strengthening of previous results.

math.CO↗