Christian Gaetz

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My research interests include algebraic combinatorics, representation theory, graph theory, and connections between these areas and computation.


4. Critical groups of group representations. Linear Algebra and its Applications 508 (2016) 91-99.
This paper computes the order of the critical group of a faithful representation of a finite group and gives some restrictions on its subgroup structure. It also computes the exact critical group for the reflection representation of the symmetric group and for the regular representation of any finite group.
3. Critical groups of McKay-Cartan matrices. 2016.
This is my undergraduate honors thesis from the University of Minnesota; my thesis advisor was Vic Reiner. This thesis gives a longer exposition of the results from (4) and also includes an additional theorem which identifies a subset of the superstable configurations of a McKay-Cartan matrix, answering a question of Benkart, Klivans, and Reiner.
2. Extensions of Shi/Ish duality. (with Michelle Bodnar, Nitin Prasad, and Bjorn Wehlin). 2015.
This is our report from the Summer 2015 CURE math research program at UCSD, where we were advised by Brendon Rhoades. We generalized a bijection between the regions of the Shi and Ish hyperplane arrangements to a bijection between the regions of the extended-Shi and nested-Ish arrangements.
1. K-Knuth equivalence for increasing tableaux. (with M. Mastrianni, R. Patrias, H. Peck, C. Robichaux, D. Schwein, and Ka Yu Tam). Elec. J. of Combin., Vol. 23(1) (2016).
This research was conducted at the Summer 2014 University of Minnesota math REU. This paper studies the K-Knuth equivalence relations on words and increasing tableaux. We give several new families of so called "unique rectification targets" and describe an algorithm to determine if two words are K-Knuth equivalent.