Scott Akin MS Thesis
On April 29, 2026, Scott Akin successfully defended his MS thesis! The title of his thesis is An isomorphism theorem for quotient commutation graphs and here is the abstract:
Any two reduced expressions for an element of a Coxeter group are related by a sequence of commutation and braid moves. Two reduced expressions are called braid equivalent if they are related by a sequence of only braid moves. Braid equivalence is an equivalence relation, and the corresponding equivalence classes are called braid classes, which are encoded in braid graphs. Similarly, reduced expressions related by only commutation moves form commutation classes. The relationships between commutation classes are encoded in a graph called the quotient commutation graph, where vertices are commutation classes and edges represent braid moves between representatives of adjacent classes.
In this thesis, we investigate the structural relationship between braid graphs and quotient commutation graphs in simply-laced Coxeter systems. We introduce the transversal property: a condition where a single braid class contains exactly one representative from every commutation class of a given element. We prove an isomorphism theorem stating that for certain Coxeter systems, the quotient commutation graph is isomorphic to the braid graph of one of its reduced expressions if and only if the element possesses the transversal property. This work extends the theoretical framework established in recent research on reduced expressions and the combinatorial properties of simply-laced Coxeter systems. We conclude by providing several families of elements that exhibit the transversal property and hence satisfy the isomorphism.
He’s one of the good guys. Watch out, world—here he comes!
I’ve been blessed with the opportunity to work with some amazing students throughout my career, and this year has been no exception. Mentoring students on mathematics research projects is unquestionably my favorite part of my job. But it is mostly about the relationships. I am full of gratitude.