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John Shareshian

Publications and source records attributed to John Shareshian.

At least 19 recordsLinked to original sources
arXiv2026

Sprout Symmetric Functions: Part 1

A \emph{sprout sequence} is a sequence $\frakr=(R_0=1,R_1,R_2,\dots)$ of symmetric functions in the variables $\bmx=(x_1,x_2,\dots)$ over a field $K$ generated from a power series $F(t)=1+a_1t+a_2t^2+\cdots$ by the rule $\sum_{n\geq 0}R_nt^n = \prod_{i\geq 1} F(x_it)$. The power series $F(t)$ is called the \emph{seed} of $\frakr$. This concept originated in the work of Littlewood and Richardson (though not with the name ``sprout sequence''), and numerous examples of sprout sequences have appeared in the literature. They are related to chromatic Tutte polynomials of complete graphs and complete hypergraphs, binomial posets, upper homogeneous (upho) posets, topological genera, etc. We first develop the basic theory of sprout sequences and then look at the special case $F(t)=\sec(\sqrt{t})$. We give five characterizations of sprout sequences and consider the expansion of sprout symmetric functions in terms of well-known symmetric function bases. The Schur positivity, elementary symmetric function positivity, and complete homogeneous symmetric function positivity of $R_n$ for all $n$ are completely characterized using the Edrei-Thoma theorem from the theory of total positivity. The seed $F(t)=\sec(\sqrt{t})$ is especially interesting. The expansion of $R_n$ in the power sum or monomial basis is related to alternating permutations. The Schur function expansion is related to standard Young skew tableaux. The expansion in terms of the complete symmetric functions has nonnegative integer coefficients, but we don't know a combinatorial interpretation. Finally we give a formula for $R_n$ as a sum of chromatic symmetric functions of interval orders.

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arXiv2025

Posets of decompositions in spherical buildings

We propose definitions of the common bases complex, the poset of decompositions, and the poset of partial decompositions for arbitrary spherical buildings. We show that the poset of decompositions is Cohen-Macaulay, and that the poset of partial decompositions is spherical and homotopy equivalent to the common bases complex. To prove these results, we rely on the concepts of opposition, Levi spheres, and convexity in buildings. In particular, our results extend the already known constructions for the linear case (vector spaces) to arbitrary buildings. As a byproduct, we see that the poset of ordered partial decompositions carries the square of the Steinberg representation.

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arXiv2024

Gamma vectors of partitioned permutohedra

We determine that $\gamma$-vectors of partitioned permutohedra, thereby generalizing a result of Foata and Sch\"utzenberger. Our result is closely related to a result of Athanasiadis on the representation of the symmetric group on the cohomology of the permutohedral variety. We explain how to derive Athanasiadis' result from ours and vice versa.

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arXiv2023

$K$-Orbit closures and Hessenberg varieties

This article explores the relationship between Hessenberg varieties associated with semisimple operators with two eigenvalues and orbit closures of a spherical subgroup of the general linear group. We establish the specific conditions under which these semisimple Hessenberg varieties are irreducible. We determine the dimension of each irreducible Hessenberg variety under consideration and show that the number of such varieties is a Catalan number. We then apply a theorem of Brion to compute a polynomial representative for the cohomology class of each such variety. Additionally, we calculate the intersections of a standard (Schubert) hyperplane section of the flag variety with each of our Hessenberg varieties and prove this intersection possess a cohomological multiplicity-free property.

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arXiv2023

Ramanujan sums and rectangular power sums

For a fixed nonnegative integer $u$ and positive integer $n$, we investigate the symmetric function \[\sum_{d|n} \left(c_d(\tfrac{n}{d})\right)^u p_d^{\tfrac{n}{d}},\] where $p_n$ denotes the $n$th power sum symmetric function, and $c_d(r)$ is a Ramanujan sum, equal to the sum of the $r$th powers of all the primitive $d$th roots of unity. We establish the Schur positivity of these functions for $u=0$ and $u=1$, showing that, in each case, the associated representation of the symmetric group $\mathfrak{S}_n$ decomposes into a sum of Foulkes representations, that is, representations induced from the irreducibles of the cyclic subgroup generated by the long cycle. We also conjecture Schur positivity for the case $u= 2$.

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arXiv2022

Hessenberg varieties of codimension one in the flag variety

We study geometric and topological properties of Hessenberg varieties of codimension one in the type A flag variety. Our main results: (1) give a formula for the Poincar\'e polynomial, (2) characterize when these varieties are irreducible, and (3) show that all are reduced schemes. We prove that the singular locus of any nilpotent codimension one Hessenberg variety is also a Hessenberg variety. A key tool in our analysis is a new result applying to all (type A) Hessenberg varieties without any restriction on codimension, which states that their Poincar\'e polynomials can be computed by counting the points in the corresponding variety defined over a finite field. The results below were originally motivated by work of the authors in [arXiv:2107.07929] studying the precise relationship between Hessenberg and Schubert varieties, and we obtain a corollary extending the results from that paper to all codimension one (type A) Schubert varieties.

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arXiv2021

Which Schubert Varieties are Hessenberg Varieties?

After proving that every Schubert variety in the full flag variety of a complex reductive group $G$ is a general Hessenberg variety, we show that not all such Schubert varieties are adjoint Hessenberg varieties. In fact, in types A and C, we provide pattern avoidance criteria implying that the proportion of Schubert varieties that are adjoint Hessenberg varieties approaches zero as the rank of $G$ increases. We show also that in type A, some Schubert varieties are not isomorphic to any adjoint Hessenberg variety.

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arXiv2021

Toric orbifolds associated with partitioned weight polytopes in classical types

Given a root system $Φ$ of type $A_n$, $B_n$, $C_n$, or $D_n$ in Euclidean space $E$, let $W$ be the associated Weyl group. For a point $p \in E$ not orthogonal to any of the roots in $Φ$, we consider the $W$-permutohedron $P_W$, which is the convex hull of the $W$-orbit of $p$. The representation of $W$ on the rational cohomology ring $H^\ast(X_Φ)$ of the toric variety $X_Φ$ associated to (the normal fan to) $P_W$ has been studied by various authors. Let $\{s_1,\ldots,s_n\}$ be a complete set of simple reflections in $W$. For $K \subseteq [n]$, let $W_K$ be the standard parabolic subgroup of $W$ generated by $\{s_k:k \in K\}$. We show that the fixed subring $H^\ast(X_Φ)^{W_K}$ is isomorphic to the cohomology ring of the toric variety $X_Φ(K)$ associated to a polytope obtained by intersecting $P_W$ with half-spaces bounded by reflecting hyperplanes for the given generators of $W_K$. By a result of Balibanu--Crooks, the cohomology rings $H^\ast(X_Φ(K))$ are isomorphic with cohomology rings of certain regular Hessenberg varieties.

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arXiv2018

Gamma-positivity of variations of Eulerian polynomials

An identity of Chung, Graham and Knuth involving binomial coefficients and Eulerian numbers motivates our study of a class of polynomials that we call binomial-Eulerian polynomials. These polynomials share several properties with the Eulerian polynomials. For one thing, they are $h$-polynomials of simplicial polytopes, which gives a geometric interpretation of the fact that they are palindromic and unimodal. A formula of Foata and Schützenberger shows that the Eulerian polynomials have a stronger property, namely $γ$-positivity, and a formula of Postnikov, Reiner and Williams does the same for the binomial-Eulerian polynomials. We obtain $q$-analogs of both the Foata-Schützenberger formula and an alternative to the Postnikov-Reiner-Williams formula, and we show that these $q$-analogs are specializations of analogous symmetric function identities. Algebro-geometric interpretations of these symmetric function analogs are presented.

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arXiv2016

Chromatic quasisymmetric functions

We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's expansion in Gessel's basis of fundamental quasisymmetric functions. We present a conjectural refinement of Stanley's power sum basis expansion, which we prove in special cases. We describe connections between the chromatic quasisymmetric function and both the $q$-Eulerian polynomials introduced in our earlier work and, conjecturally, representations of symmetric groups on cohomology of regular semisimple Hessenberg varieties, which have been studied by Tymoczko and others. We discuss an approach, using the results and conjectures herein, to the $e$-positivity conjecture of Stanley and Stembridge for incomparability graphs of $(3+1)$-free posets.

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arXiv2015

On the lattice of subracks of the rack of a finite group

In this paper we initiate the study of racks from the combined perspective of combinatorics and finite group theory. A rack R is a set with a self-distributive binary operation. We study the combinatorics of the partially ordered set {\cal R}(R) of all subracks of R with inclusion as the order relation. Groups G with the conjugation operation provide an important class of racks. For the case R = G we show that -> the order complex of {\cal R}(R) has the homotopy type of a sphere, -> the isomorphism type of {\cal R}(R) determines if G is abelian, nilpotent, supersolvable, solvable or simple, -> {\cal R}(R) is graded if and only if G is abelian, G = S_3, G = D_8 or G = Q_8. In addition, we provide some examples of subracks R of a group G for which {\cal R}(R) relates to well studied combinatorial structures. In particular, the examples show that the order complex of {\cal R}(R) for general R is more complicated than in the case R = G.

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