Christian Gaetz
Home  Research
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 rdifferential
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 rdual tower of groups, introduced by Miller and Reiner, is a nested
sequence of finite groups, like the symmetric groups, whose Bratteli diagram
forms an rdual 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
rdual 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 RobinsonSchensted 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) 9199.
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 McKayCartan 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 McKayCartan 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 extendedShi
and nestedIsh arrangements.

pdf

1. KKnuth 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 KKnuth 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 KKnuth equivalent.

arXiv
journal

