arXiv subjects

Christian Gaetz

Publications and source records attributed to Christian Gaetz.

At least 37 records · Page 2Linked to original sources

arXiv2024

Repeatable patterns and the maximum multiplicity of a generator in a reduced word

We study the maximum multiplicity $\mathcal{M}(k,n)$ of a simple transposition $s_k=(k \: k+1)$ in a reduced word for the longest permutation $w_0=n \: n-1 \: \cdots \: 2 \: 1$, a problem closely related to much previous work on sorting networks and on the "$k$-set" problem. After reinterpreting the problem in terms of monotone weakly separated paths, we show that, for fixed $k$ and sufficiently large $n$, the optimal density is realized by paths which are periodic in a precise sense, so that \[ \mathcal{M}(k,n)=c_k n + p_k(n) \] for a periodic function $p_k$ and constant $c_k$. In fact we show that $c_k$ is always rational, and compute several bounds and exact values for this quantity with "repeatable patterns", which we introduce.

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arXiv2024

Positivity of permutation pattern character polynomials

Let $N_σ(π)$ denote the number of occurrences of a permutation pattern $σ\in S_k$ in a permutation $π\in S_n$. Gaetz and Ryba (2021) showed using partition algebras that the $d$-th moment $M_{σ,d,n}(π)$ of $N_σ$ on the conjugacy class of $π$ is given by a polynomial in $n,m_1,\dots,m_{dk}$, where $m_i$ denotes the number of $i$-cycles of $π$. They also showed that the coefficient $\langle χ^{λ[n]}, M_{σ,d,n}\rangle$ agrees with a polynomial $a_{σ,d}^λ(n)$ in $n$. This work is motivated by the conjecture that when $σ=\text{id}_k$ is the identity permutation, all of these coefficients are nonnegative. We directly compute closed forms for the polynomials $a_{\text{id}_k}^λ(n)$ in the cases $λ=(1),(1,1),$ and $(2)$, and use this to verify the positivity conjecture for those cases by showing that the polynomials are real-rooted with all roots less than $k$. We also study the case $a_σ^{(1)}(n)$, for which we give a formula for the polynomials and their leading coefficients.

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arXiv2023

Balance constants for Coxeter groups

The $1/3$-$2/3$ Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant (a quantity determined by the linear extensions) of any non-total order is at least $1/3$. By reinterpreting balance constants of posets in terms of convex subsets of the symmetric group, we extend the study of balance constants to convex subsets $C$ of any Coxeter group. Remarkably, we conjecture that the lower bound of $1/3$ still applies in any finite Weyl group, with new and interesting equality cases appearing. We generalize several of the main results towards the $1/3$-$2/3$ Conjecture to this new setting: we prove our conjecture when $C$ is a weak order interval below a fully commutative element in any acyclic Coxeter group (an generalization of the case of width-two posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce generalized semiorders for which we resolve the conjecture. We hope this new perspective may shed light on the proper level of generality in which to consider the $1/3$-$2/3$ Conjecture, and therefore on which methods are likely to be successful in resolving it.

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arXiv2023

One-skeleton posets of Bruhat interval polytopes

Introduced by Kodama and Williams, Bruhat interval polytopes are generalized permutohedra closely connected to the study of torus orbit closures and total positivity in Schubert varieties. We show that the 1-skeleton posets of these polytopes are lattices and classify when the polytopes are simple, thereby resolving open problems and conjectures of Fraser, of Lee--Masuda, and of Lee--Masuda--Park. In particular, we classify when generic torus orbit closures in Schubert varieties are smooth.

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arXiv2023

Products of reflections in smooth Bruhat intervals

A permutation is called smooth if the corresponding Schubert variety is smooth. Gilboa and Lapid prove that in the symmetric group, multiplying the reflections below a smooth element $w$ in Bruhat order in a compatible order yields back the element $w$. We strengthen this result by showing that such a product in fact determines a saturated chain $e \to w$ in Bruhat order, and that this property characterizes smooth elements.

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arXiv2022

On automorphisms of undirected Bruhat graphs

The (directed) Bruhat graph $\hatΓ(u,v)$ has the elements of the Bruhat interval $[u,v]$ as vertices, with directed edges given by multiplication by a reflection. Famously, $\hatΓ(e,v)$ is regular if and only if the Schubert variety $X_v$ is smooth, and this condition on $v$ is characterized by pattern avoidance. In this work, we classify when the undirected Bruhat graph $Γ(e,v)$ is vertex-transitive; surprisingly this class of permutations is also characterized by pattern avoidance and sits nicely between the classes of smooth permutations and self-dual permutations. This leads us to a general investigation of automorphisms of $Γ(u,v)$ in the course of which we show that special matchings, which originally appeared in the theory Kazhdan--Lusztig polynomials, can be characterized as certain $Γ(u,v)$-automorphisms which are conjecturally sufficient to generate the orbit of $e$ under $Aut(Γ(e,v))$.

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arXiv2022

The hull metric on Coxeter groups

We reinterpret an inequality, due originally to Sidorenko, for linear extensions of posets in terms of convex subsets of the symmetric group $\mathfrak{S}_n$. We conjecture that the analogous inequalities hold in arbitrary (not-necessarily-finite) Coxeter groups $W$, and prove this for the hyperoctahedral groups $B_n$ and all right-angled Coxeter groups. Our proof for $B_n$ (and new proof for $\mathfrak{S}_n$) use a combinatorial insertion map closely related to the well-studied promotion operator on linear extensions; this map may be of independent interest. We also note that the inequalities in question can be interpreted as a triangle inequalities, so that convex hulls can be used to define a new invariant metric on $W$ whenever our conjecture holds. Geometric properties of this metric are an interesting direction for future research.

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arXiv2021

Curious cyclic sieving on increasing tableaux

We prove a cyclic sieving result for the set of $3 \times k$ packed increasing tableaux with maximum entry $m :=3+k$ under K-promotion. The "curiosity" is that the sieving polynomial arises from the $q$-hook formula for standard tableaux of "toothbrush shape" $(2^3, 1^{k-2})$ with $m+1$ boxes, whereas K-promotion here only has order $m$.

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arXiv2021

Diameters of graphs of reduced words and rank-two root subsystems

We study the diameter of the graph $G(w)$ of reduced words of an element $w$ in a Coxeter group $W$ whose edges correspond to applications of the Coxeter relations. We resolve conjectures of Reiner--Roichman and Dahlberg--Kim by proving a tight lower bound on this diameter when $W=S_n$ is the symmetric group and by characterizing the equality cases. We also give partial results in other classical types which illustrate the limits of current techniques.

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arXiv2021

Spherical Schubert varieties and pattern avoidance

A normal variety $X$ is called $H$-spherical for the action of the complex reductive group $H$ if it contains a dense orbit of some Borel subgroup of $H$. We resolve a conjecture of Hodges--Yong by showing that their spherical permutations are characterized by permutation pattern avoidance. Together with results of Gao--Hodges--Yong this implies that the sphericality of a Schubert variety $X_w$ with respect to the largest possible Levi subgroup is characterized by this same pattern avoidance condition.

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arXiv2021

Minimal elements for the limit weak order on affine Weyl groups

The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide with the infinite fully commutative reduced words and with infinite powers of Coxeter elements. We answer several open problems raised there by classifying minimal elements in all affine types and relating these elements to the classes of fully commutative and Coxeter elements. Interestingly, the infinite fully commutative elements correspond to the minuscule and cominuscule nodes of the Dynkin diagram, while the infinite Coxeter elements correspond to a single node, which we call the heavy node, in all affine types other than type $A$.

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arXiv2020

Self-dual intervals in the Bruhat order

Björner-Ekedahl prove that general intervals $[e,w]$ in Bruhat order are "top-heavy", with at least as many elements in the $i$-th corank as the $i$-th rank. Well-known results of Carrell and of Lakshmibai-Sandhya give the equality case: $[e,w]$ is rank-symmetric if and only if the permutation $w$ avoids the patterns $3412$ and $4231$ and these are exactly those $w$ such that the Schubert variety $X_w$ is smooth. In this paper we study the finer structure of rank-symmetric intervals $[e,w]$, beyond their rank functions. In particular, we show that these intervals are still "top-heavy" if one counts cover relations between different ranks. The equality case in this setting occurs when $[e,w]$ is self-dual as a poset; we characterize these $w$ by pattern avoidance and in several other ways.

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arXiv2020

Separable elements and splittings of Weyl groups

We continue the study of separable elements in finite Weyl groups. These elements generalize the well-studied class of separable permutations. We show that the multiplication map $W/U \times U \to W$ is a length-additive bijection, or splitting, of the Weyl group $W$ when $U$ is an order ideal in right weak order generated by a separable element; this generalizes a result for the symmetric group, answering 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 in right weak order generated by a separable element, thereby classifying those generalized quotients which induce splittings of the symmetric group, resolving a problem of Björner and Wachs from 1988. We also prove that this map is always surjective when $U$ is an order ideal in right weak order. Interpreting these sets of permutations as linear extensions of 2-dimensional posets gives the first direct combinatorial proof of an inequality due originally to Sidorenko in 1991, answering an open problem Morales, Pak, and Panova. We also prove a new $q$-analog of Sidorenko's formula. 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 all dimensions 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.

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arXiv2020

Padded Schubert polynomials and weighted enumeration of Bruhat chains

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 ${n \choose 2}!$ 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.

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arXiv2020

On the Sperner property for the absolute order on complex reflection groups

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 $D_n$, for which this property is conjectural. The Sperner property had previously been established for the noncrossing partition lattice $NC_W$, a certain maximal interval in the absolute order, but not for the entire poset, except in the case of the symmetric group. We also show that neither the codimension order nor the prefix order has the Sperner property for general complex reflection groups.

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arXiv2020

The Sperner property for $132$-avoiding intervals in the weak order

A well-known result of Stanley from 1980 implies that the weak order on a maximal parabolic quotient of the symmetric group $S_n$ has the Sperner property; this same property was recently established for the weak order on all of $S_n$ by Gaetz and Gao, resolving a long-open problem. In this paper we interpolate between these results by showing that the weak order on any parabolic quotient of $S_n$ (and more generally on any $132$-avoiding interval) has the Sperner property. This result is proven by exhibiting an action of $\mathfrak{sl}_2$ respecting the weak order on these intervals. As a corollary we obtain a new formula for principal specializations of Schubert polynomials. Our formula can be seen as a strong Bruhat order analogue of Macdonald's reduced word formula. This proof technique and formula generalize work of Hamaker, Pechenik, Speyer, and Weigandt and Gaetz and Gao.

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arXiv2020

Stable characters from permutation patterns

For a fixed permutation $σ\in S_k$, let $N_σ$ denote the function which counts occurrences of $σ$ as a pattern in permutations from $S_n$. We study the expected value (and $d$-th moments) of $N_σ$ on conjugacy classes of $S_n$ and prove that the irreducible character support of these class functions stabilizes as $n$ grows. This says that there is a single polynomial in the variables $n, m_1, \ldots, m_{dk}$ which computes these moments on any conjugacy class (of cycle type $1^{m_1}2^{m_2}\cdots$) of any symmetric group. This result generalizes results of Hultman and of Gill, who proved the cases $(d,k)=(1,2)$ and $(1,3)$ using ad hoc methods. Our proof is, to our knowledge, the first application of partition algebras to the study of permutation patterns.

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arXiv2020

On $q$-analogs of descent and peak polynomials

Descent polynomials and peak polynomials, which enumerate permutations with given descent and peak sets respectively, have recently received considerable attention. We give several formulas for $q$-analogs of these 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.

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