Research

Modular Representation Theory

• Brauer pairs for splendid Rickard equivalences

  with S.K. Miller

  Journal of Algebra, Volume 691, Pages 694-729, 2026

  Journal arXiv:2312.10258

  Abstract

  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. We prove that the Brauer pairs of a splendid Rickard equivalence \(C\) coincide with the set of Brauer pairs of the corresponding \(p\)-permutation equivalence \(\Lambda(C)\) 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.
• Splendid relatively stable equivalences for blocks of finite groups

  J.V. Breland

  In preparation

  Summary

  This work characterizes splendid relatively stable equivalences in terms of their local pieces, generalizing Rouquier's work on splendid stable equivalences. It establishes a framework to detect relative stable equivalences by checking derived equivalences on specific local levels determined by the Brauer pair poset.

Algebraic Combinatorics

• Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes

  with F. Awik, Q. Cadman, and D.C. Ernst

  European Journal of Combinatorics, Vol 118, 103931, 2024

  Journal arXiv:2104.12318

  Abstract

  We study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid graph into a box product of the braid graphs for each link factor. When the Coxeter graph has no three-cycles, we use the decomposition to prove that braid graphs are partial cubes, i.e., can be isometrically embedded into a hypercube. For a special class of links, called Fibonacci links, we prove that the corresponding braid graphs are Fibonacci cubes.
  
• Braid graphs in simply-laced triangle-free Coxeter systems are median

  with J. Barnes, D.C. Ernst, and R. Perry

  Submitted (2024)

  arXiv:2408.16839

  Abstract

  In this paper, we will provide an alternate proof that braid graphs in simply-laced triangle-free Coxeter systems are partial cubes, as well as determine the minimal dimension hypercube into which a braid graph can be isometrically embedded. 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.
  
• Braid graphs in simply-laced triangle-free Coxeter systems and ribbon polyomino lattices

  with J.K. Howard

  In preparation