Christian Gaetz
Home  Research
I have quite wide research interests, centered around the areas of algebraic, enumerative, and geometric
combinatorics and representation theory.
Look for my research on
the arXiv, on Google scholar,
or on MathSciNet.
Papers
14. On qanalogs of descent and peak polynomials.
(with Yibo Gao)
We give several formulas for qanalogs of descent and peak polynomials which refine the enumeration
by the length of the permutations. In the case of qdescent polynomials we prove that the coefficients
in one basis are strongly qlog concave, and conjecture this property in another basis. For peaks,
we prove that the qpeak polynomial is palindromic in q, resolving a conjecture of DiazLopez,
Harris, and Insko.

arXiv

13. Separable elements and splittings of Weyl groups.
(with Yibo Gao)
We show that the multiplication map from W/U x U to W is a lengthadditive bijection, or splitting,
of the Weyl group W when U is an order ideal in weak order generated by a separable element;
this answers an open problem of Wei.
For a generalized quotient of the symmetric group, we show that this multiplication map is a bijection
if and only if U is an order ideal generated by a separable element, thereby classifying those
generalized quotients which induce splittings of the symmetric group, resolving a problem of Bjorner
and Wachs from 1988. We also prove that this map is always surjective when U is an order ideal in
weak order. Interpreting these sets of permutations as linear extensions of 2dimensional posets
answers an open problem Morales, Pak, and Panova. All of these results are conjectured to extend
to arbitrary finite Weyl groups.
Finally, we show that separable elements in W are in bijection with the faces of several copies of
the graph associahedron of the Dynkin diagram of W. This correspondence associates to each separable
element w a certain nested set; we give product formulas for the rank generating functions of the
principal upper and lower order ideals generated by w in terms of these nested sets, generalizing
several known formulas.

arXiv

12. Separable elements in Weyl groups.
(with Yibo Gao) Adv. Appl. Math. 113 (2020).
We define the notion of a separable element in a finite Weyl group, generalizing the wellstudied
class of separable permutations. We prove that the upper and lower order ideals in weak Bruhat
order generated by a separable element are ranksymmetric and rankunimodal, and that the product
of their rank generating functions gives that of the whole group, answering an open problem of Fan
Wei. We also prove that separable elements are characterized by pattern avoidance in the sense of
Billey and Postnikov.

arXiv
journal

11. Padded Schubert polynomials and weighted enumeration of Bruhat chains.
(with Yibo Gao)
We prove a common generalization of the fact that the weighted number of maximal chains in the
strong Bruhat order on the symmetric group is nchoose2 factorial for both the code weights and the Chevalley
weights. We also define weights which give a oneparameter family of strong order analogues of
Macdonald's reduced word identity for Schubert polynomials.

arXiv

10. On the Sperner property for the absolute order on complex reflection groups.
(with Yibo Gao)
Two partial orders on a reflection group, the codimension order and the prefix order, are together
called the absolute order when they agree. We show that in this case the absolute order on a
complex reflection group has the strong Sperner property, except possibly for the Coxeter group of
type D, for which this property is conjectural. We also show that neither the codimension order
nor the prefix order has the Sperner property for general complex reflection groups.

arXiv

9. A combinatorial duality between the weak and strong Bruhat orders.
(with Yibo Gao) J. Combin. Theory Ser. A 171 (2020).
In recent work, the authors used an order lowering operator D, introduced by Stanley, to prove the
strong Sperner property for the weak Bruhat order on the symmetric group. Hamaker, Pechenik,
Speyer, and Weigandt interpreted this operator as a differential operator on Schubert polynomials.
In this paper we study a raising operator for the strong Bruhat order, which is in many ways dual
to D. We prove a Schubert identity dual to that of Hamaker et al. and derive formulas for counting
weighted paths in the Hasse diagrams of the strong order which agree with path counting formulas
for the weak order. We also show that powers of these operators have the same Smith normal forms, which we
describe explicitly, answering a question of Stanley.

arXiv
journal

8. A combinatorial sl_2action and the Sperner property for the weak order.
(with Yibo Gao) Proc. Amer. Math. Soc. 148 (2020).
We construct a simple combinatoriallydefined representation of sl_2 which respects the order structure
of the weak order on the symmetric group. This is used to prove that the weak order has the strong
Sperner property, and this therefore a Peck poset, solving a problem raised by Bjorner (1984); a
positive answer to this question had been conjectured by Stanley (2017).

arXiv
journal

7. Path counting and rank gaps in differential posets.
(with Praveen Venkataramana) Order (2019).
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
journal

6. Dual graded graphs and Bratteli diagrams of towers of groups.
Electron. J. Combin. 26 (1) (2019).
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
journal

5. Differential posets and restriction in critical groups.
(with Ayush Agarwal) Algebraic Combin. 2 (6) (2019).
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.

arXiv
journal

4. Critical groups of group representations.
Linear Algebra Appl. 508 (2016).
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) Electron. J. Combin. 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

