From Fall 2021 until Spring 2022 I worked with JMU student Colby Sherwood.






Colby was interested in the longstanding problem of determining the critical group of the n-cube graph. Since the Laplacian matrix of n-cube graph is a map that respects the action of the symmetric group, we can understand the critical group by computing the dimensions of certain modules for the symmetric group.
We warmed ourselves up to this challenge by learning a lot about the modular representation theory of S_n due to Gordon James, and applying it to determine the rank of the incidence matrices describing the intersection of 2-subsets with n-subsets in a set of size one. See our paper describing this result here.

Colby wrote an excellent Honors thesis about this open problem, and won the JMU Phi Beta Kappa best honors thesis award (out of submissions by more that 100 departments at JMU). In his thesis he recasts the n-cube critical group problem in the language of representation theory, and manages to recover some known results such as the 2-rank of the Laplacian and also the number of 2s appearing in the Laplacian's Smith normal form (only the 2-primary component of the critical group is unknown). You can read his excellent thesis here.

Colby gave several talks about this work at JMU and other conferences.
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