Christian Gaetz

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

Look for my research on the arXiv, on Google scholar, or on MathSciNet.

Papers

7. Path counting and rank gaps in differential posets. with Praveen Venkataramana (2018).
We study the gaps between consecutive rank sizes in r-differential posets by introducing a projection operator whose matrix entries can be expressed in terms of the number of certain paths in the Hasse diagram. We stengthen Miller's result that these rank gaps are strictly positive, which resolved a longstanding conjecture of Stanley, by showing that they are at least 2r. We also obtain stronger bounds in the case that the poset has many substructures called threads.
arXiv
6. Dual graded graphs and Bratteli diagrams of towers of groups. (2018).
An r-dual tower of groups, introduced by Miller and Reiner, is a nested sequence of finite groups, like the symmetric groups, whose Bratteli diagram forms an r-dual graded graph. In this paper I prove that when r is one or prime, wreath products of a fixed group with the symmetric groups are the only r-dual tower of groups, and conjecture that this is the case for general values of r. This implies that these wreath products are the only groups for which one can define an analog of the Robinson-Schensted bijection in terms of a growth rule in a dual graded graph.
arXiv
5. Differential posets and restriction in critical groups. with Ayush Agarwal (2017).
We show 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 the (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.
arXiv
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.
arXiv
journal
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.
pdf
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.
pdf
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.
arXiv
journal