A Coxeter group is an abstract group generated by a finite number of reflections. These reflections are subject to commutation and braid relations, which reflect (pun intended!) the geometric fact that the product of two reflections is a rotation. Any group element can be represented as a product of reflections, called a word, in many different ways. If the number of reflections in a word is minimal, we call it reduced. The relationships among the reduced words for a fixed element can be encoded into a graph: the vertices are the words, and the edges are the commutation and braid relations linking them. In 1964, Japanese mathematician Hideya Matsumoto proved that this graph is always connected: any two reduced words for a given element differ only by a sequence of these simple relations. The Matsumoto graph beautifully encodes the combinatorics of a Coxeter group element, but its global structure can be wildly complex. To make sense of it, we instead study its local structure by focusing on smaller pieces—called braid graphs—which cover the entire graph. In this talk, I will discuss what we know (and don't know!) about these graphs, including a surprising appearance of the Fibonacci numbers and a hidden connection to geometric group theory: braid graphs are gluings of hypercubes in a very precise sense. Note: No prior knowledge is required to enjoy the core ideas of the talk.

The poset of Brauer pairs for a finite group algebra $FG$ over a field $F$ of positive characteristic $p$ is a refinement of the poset of $p$-subgroups of $G$ which encodes information about the representation theory of the $p$-local subgroups of $G$. Brauer pairs have been associated to various objects of interest, such as blocks, indecomposable $p$-permutation $FG$-modules, and $p$-permutation equivalences. In each of these cases, one obtains a downward closed subposet of Brauer pairs which is stable under the action of $G$ and satisfies a "Sylow" theorem. In this talk, I will explain how Brauer pairs can be associated to a bounded chain complex of $p$-permutation modules. Then, I will describe the structure of the set of Brauer pairs for a splendid Rickard complex between two block algebras. This talk is based on joint work with Sam K. Miller.

The strongest version of Broué's abelian defect group conjecture predicts that a block with abelian defect group is splendidly Rickard equivalent to its Brauer correspondent. Such an equivalence is realized by a bounded complex of $p$-permutation bimodules with twisted diagonal vertices inducing an equivalence of bounded homotopy categories. In order to study the structure of these equivalences, we introduce the poset of Brauer pairs for a chain complex of $p$-permutation modules. We show that, under ideal conditions, this poset controls the vertices of the constituents of the complex. We determine the Brauer pair poset for a splendid Rickard equivalence, establishing control of vertices by a single conjugacy class of maximal elements which encodes the isomorphism of defect groups and fusion systems.

The strongest version of Broué's abelian defect group conjecture predicts that a block with abelian defect group is splendidly Rickard equivalent to its Brauer correspondent. Such an equivalence is realized by a bounded complex of $p$-permutation bimodules with twisted diagonal vertices inducing an equivalence of bounded homotopy categories. In order to study the structure of these equivalences, we introduce the poset of Brauer pairs for a chain complex of $p$-permutation modules. We show that, under ideal conditions, this poset controls the vertices of the constituents of the complex. We determine the Brauer pair poset for a splendid Rickard equivalence, establishing control of vertices by a single conjugacy class of maximal elements which encodes the isomorphism of defect groups and fusion systems.