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 q-analogs of descent and peak polynomials. (with Yibo Gao)

We give several formulas for q-analogs of descent and peak polynomials which refine the enumeration by the length of the permutations. In the case of q-descent polynomials we prove that the coefficients in one basis are strongly q-log concave, and conjecture this property in another basis. For peaks, we prove that the q-peak polynomial is palindromic in q, resolving a conjecture of Diaz-Lopez, 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 length-additive 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 2-dimensional 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 well-studied class of separable permutations. We prove that the upper and lower order ideals in weak Bruhat order generated by a separable element are rank-symmetric and rank-unimodal, 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 n-choose-2 factorial for both the code weights and the Chevalley weights. We also define weights which give a one-parameter 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_2-action and the Sperner property for the weak order. (with Yibo Gao) Proc. Amer. Math. Soc. 148 (2020).

We construct a simple combinatorially-defined 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 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 journal
6. Dual graded graphs and Bratteli diagrams of towers of groups. Electron. J. Combin. 26 (1) (2019).

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 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 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) Electron. J. Combin. 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