Christian Gaetz

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


5. Differential posets and restriction in critical groups. with Ayush Agarwal (2017).
This research was the result of my work with Ayush, whom I mentored through MIT's PRIMES program. Our paper shows that group homomorphisms induced maps between critical groups of group representations, and that, in the abelian group case, this induced map agrees with a map between graph critical groups of a related pair of Cayley graphs. It also relates certain critical groups to words in the up and down operators when the group is an element of a differential tower of groups, such as the tower of symmetric groups. This allows the critical group of the generalized permutation representation of the wreath product of an abelian group and the symmetric group to be computed exactly. It also allows for an exact enumeration of the number of factors in critical groups which result from repeatedly applying induction and restriction to the trivial representation in a differential tower of groups.
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.