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When is the Chromatic Quasisymmetric Function Symmetric?
We investigate the problem of when a chromatic quasisymmetric function (CQF) $X_G(x;q)$ of a graph $G$ is in fact symmetric. We first prove the remarkable fact that if a product of two quasisymmetric functions $f$ and $g$ in countably infinitely many variables is symmetric, then in fact $f$ and $g$ must be symmetric. This allows the problem to be reduced to the case of connected graphs. We then show that any labeled graph having more than one source or sink has a nonsymmetric CQF. As a corollary, we find that all trees other than a directed path have a nonsymmetric CQF. We also show that a family of graphs we call ``mixed mountain graphs'' always have symmetric CQF.
Copyright (c) 2025 Algebraic Combinatorics
Decomposition numbers for weight 3 blocks of Iwahori–Hecke algebras of type B
Let B be a weight-3 block of an Iwahori–Hecke algebra of type B over any field. We develop the combinatorics of B to prove that the decomposition numbers for B are all 0 or 1.
Iwahori-Hecke algebra; Specht module; decomposition number; bipartition.
More on the corner-vector construction for spherical designs
This paper explores a full generalization of the classical corner-vector method for constructing weighted spherical designs, which we call the generalized corner-vector method. First we establish a uniform upper bound for the degree of designs obtained from the proposed method. Our proof is a hybrid argument that employs techniques in analysis and combinatorics, especially a famous result by Xu(1998) on the interrelation between spherical designs and simplical designs, and the cross-ratio comparison method for Hilbert identities introduced by Nozaki and Sawa(2013). We extensively study conditions for the existence of designs obtained from our method, and present many curious examples of degree 7 through 13, some of which are, to our surprise, characterized in terms of integral lattices.
Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone
A numerical semigroup is a cofinite subset of $\mathbb Z_{\ge 0}$ containing $0$ and closed under addition. Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in the Kunz cone $\mathcal C_m \subseteq \mathbb R^{m-1}$, and the face of $\mathcal C_m$ containing that integer point determines certain algebraic properties of $S$. In this paper, we introduce the Kunz fan, a pure, polyhedral cone complex comprised of a faithful projection of certain faces of $\mathcal C_m$. We characterize several aspects of the Kunz fan in terms of the combinatorics of Kunz nilsemigroups, which are known to index the faces of $\mathcal C_m$, and our results culminate in a method of ``walking'' the face lattice of the Kunz cone in a manner analogous to that of a Gr\"obner walk. We apply our results in several contexts, including a wealth of computational data obtained from the aforementioned ``walks'' and a proof of a recent conjecture concerning which numerical semigroups achieve the highest minimal presentation cardinality when one fixes the smallest positive element and the number of generators.
A conjectural basis for the $(1,2)$-bosonic-fermionic coinvariant ring
We give the first conjectural construction of a monomial basis for the coinvariant ring $R_n^{(1,2)}$, for the symmetric group $\mathfrak{S}_n$ acting on one set of bosonic (commuting) and two sets of fermionic (anticommuting) variables. Our construction interpolates between the modified Motzkin path basis for $R_n^{(0,2)}$ of Kim--Rhoades (2022) and the super-Artin basis for $R_n^{(1,1)}$ conjectured by Sagan--Swanson (2024) and proven by Angarone et al. (2024). We prove that our proposed basis has cardinality $2^{n-1}n!$, aligning with a conjecture of Zabrocki (2020) on the dimension of $R_n^{(1,2)}$, and show how it gives a combinatorial expression for the Hilbert series. We also conjecture a Frobenius series for $R_n^{(1,2)}$. We show that these proposed Hilbert and Frobenius series are equivalent to conjectures of Iraci, Nadeau, and Vanden Wyngaerd (2023) on $R_n^{(1,2)}$ in terms of segmented Smirnov words, by exhibiting a weight-preserving bijection between our proposed basis and their segmented permutations. We extend some of their results on the sign character to hook characters, and give a formula for the $m_\mu$ coefficients of the conjectural Frobenius series. Finally, we conjecture a monomial basis for the analogous ring in type $B_n$, and show that it has cardinality $4^nn!$.
A type $B$ analog of the Ish arrangement
The Shi arrangement due to Shi (1986) and the Ish arrangement due to Armstrong (2013) are deformations of the type $A$ Coxeter arrangement that share many common properties. Motivated by a question of Armstrong and Rhoades since 2012 to seek for Ish arrangements of other types, in this paper we introduce an Ish arrangement of type $B$. We study this Ish arrangement through various aspects similar to as known in type $A$ with a main emphasis on freeness and supersolvability. Our method is based on the concept of $\psi$-digraphic arrangements recently introduced due to Abe and the authors with a type $B$ extension.
Web invariants for flamingo Specht modules
Webs yield an especially important realization of certain Specht modules, irreducible representations of symmetric groups, as they provide a pictorial basis with a convenient diagrammatic calculus. In recent work, the last three authors associated polynomials to noncrossing partitions without singleton blocks, so that the corresponding polynomials form a web basis of the pennant Specht module $S^{(d,d,1^{n-2d})}$. These polynomials were interpreted as global sections of a line bundle on a $2$-step partial flag variety. Here, we both simplify and extend this construction. On the one hand, we show that these polynomials can alternatively be situated in the homogeneous coordinate ring of a Grassmannian, instead of a $2$-step partial flag variety, and can be realized as tensor invariants of classical (but highly nonplanar) tensor diagrams. On the other hand, we extend these ideas from the pennant Specht module $S^{(d,d,1^{n-2d})}$ to more general flamingo Specht modules $S^{(d^r,1^{n-rd})}$. In the hook case $r=1$, we obtain a spanning set that can be restricted to a basis in various ways. In the case $r>2$, we obtain a basis of a well-behaved subspace of $S^{(d^r,1^{n-rd})}$, but not of the entire module.
Tridendriform algebras on hypergraph polytopes
We extend the works of Loday-Ronco and Burgunder-Ronco on the tridendriform decomposition of the shuffle product on the faces of associahedra and permutohedra, to other families of hypergraph polytopes (or nestohedra), including simplices, hypercubes and some new families. We also extend the shuffle product to take more than two arguments, and define accordingly a new algebraic structure, that we call polydendriform, from which the original tridendriform equations can be crisply synthesized.
HIVE-TYPE POLYTOPES FOR QUIVER SEMI-INVARIANTS AND THE MEMBERSHIP PROBLEM FOR QUIVER MOMENT CONES
Let $\mathcal{Q}$ be a bipartite quiver with vertex set $\mathcal{Q}_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $\beta=(\beta(x))_{x \in \mathcal{Q}_0}$ be a dimension vector of $\mathcal{Q}$ with positive integer coordinates. Let $\mathcal{Q}_{\beta}$ be the flag-extension of $\mathcal{Q}$ obtained by attaching a flag $\mathcal{F}(x)$ of length $\beta(x)-1$ at every vertex $x$ of $\mathcal{Q}$, and let $\widetilde{\beta}$ be the extension of $\beta$ to $\mathcal{Q}_{\beta}$ that takes values $1, \ldots, \beta(x)$ along the vertices of the flag $\F(x)$ for every vertex $x$ of $\mathcal{Q}$. For an integral weight $\widetilde{\sigma}$ of $\mathcal{Q}_{\beta}$, let $K_{\widetilde{\sigma}}$ be the dimension of the space of semi-invariants of weight $\widetilde{\sigma}$ on the representation space of $\widetilde{\beta}$-dimensional complex representations of $\mathcal{Q}_{\beta}$. We show that $K_{\widetilde{\sigma}}$ can be expressed as the number of lattice points of a polytope obtained by gluing together two Knutson-Tao hive polytopes. Furthermore, this polytopal description together with Derksen-Weyman's Saturation Theorem for quiver semi-invariants allows us to use Tardos' algorithm to solve the membership problem for the moment cone associated to $(\mathcal{Q},\beta)$ in strongly polynomial time.
Hive polytopes, Littlewood-Richardson coefficients, moment cones, strongly polynomial time, quiver exceptional sequences, quiver semi-invariants.
Projective dimension of weakly chordal graphic arrangements
A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices.
Crystals for set-valued decomposition tableaux
We describe two crystal structures on set-valued decomposition tableaux. These provide the first examples of interesting ``$K$-theoretic'' crystals on shifted tableaux. Our first crystal is modeled on a similar construction of Monical, Pechenik, and Scrimshaw for semistandard (unshifted) set-valued tableaux. Our second crystal is adapted from the ``square root'' operators introduced by Yu on the same set. Neither of our shifted crystals is normal, but we conjecture that our second construction is connected with a unique highest weight element. These results lead to partial progress on a conjectural formula of Cho--Ikeda for $K$-theoretic Schur $P$-functions. We also study a new category of ``square root crystals'' that includes our second construction and Yu's set-valued tableau crystals as examples. We observe that Buch's formula for the coefficients expanding products of symmetric Grothendieck functions has a simple description in terms of the tensor product for this category.
We show that for any permutation w, the increasing flip graph on acyclic pipe dreams with exiting permutation w is a lattice quotient of the interval [e,w] of the weak order. We then discuss conjectural generalizations of this result to acyclic facets of subword complexes on arbitrary finite Coxeter groups.
Centraliser algebras of monomial representations and applications in combinatorics
Centraliser algebras of monomial representations of finite groups may be constructed and studied using methods similar to those employed in the study of permutation groups. Guided by results of D. G. Higman and others, we give an explicit construction for a basis of the centraliser algebra of a monomial representation. The character table of this algebra is then constructed via character sums over double cosets. We locate the theory of group-developed and cocyclic-developed Hadamard matrices within this framework. We apply Grobner bases to produce a new classification of highly symmetric complex Hadamard matrices. MSC: 05B20, 20B25.
Tropical Fock-Goncharov coordinates \\ for $\operatorname{SL}(3)$-webs on surfaces II: naturality
In \cite{DouglasArxiv20}, we constructed nonnegative integer coordinates $\Phi_\mathscr{T}(\mathscr{W}_{3, \widehat{S}}) \subset \mathbb{Z}_{\geq 0}^N$ for the collection $\mathscr{W}_{3, \widehat{S}}$ of reduced $\mathrm{SL}_3$-webs on a finite-type punctured surface $\widehat{S}$, depending on an ideal triangulation $\mathscr{T}$ of $\widehat{S}$. We show that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $\mathscr{T}^\prime$ is chosen, then the coordinate change map relating $\Phi_\mathscr{T}(\mathscr{W}_{3, \widehat{S}})$ to $\Phi_{\mathscr{T}^\prime}(\mathscr{W}_{3, \widehat{S}})$ is a tropical $\mathcal{A}$-coordinate cluster transformation. We can therefore view the webs $\mathscr{W}_{3, \widehat{S}}$ as a concrete topological model for the Fock-Goncharov-Shen positive integer tropical points~$\mathcal{A}_{\mathrm{PGL}_3, \widehat{S}}^+(\mathbb{Z}^t)$.
Root polytopes, tropical types, and toric edge ideals
We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data, analogous to the covectors of an oriented matroid. By work of Develin-Sturmfels and Fink-Rincon, these `tropical complexes' are dual to (regular) subdivisions of root polytopes, which in turn are in bijection with mixed subdivisions of certain generalized permutohedra. Extending previous work with Joswig-Sanyal, we show how a natural monomial labeling of these complexes describes polynomial relations (syzygies) among `type ideals' which arise naturally from the combinatorial data of the arrangement. In particular, we show that the cotype ideal is Alexander dual to a corresponding initial ideal of the lattice ideal of the underlying root polytope. This leads to novel ways of studying algebraic properties of various monomial and toric ideals, as well as relating them to combinatorial and geometric properties. In particular, our methods of studying the dimension of the tropical complex leads to new formulas for homological invariants of toric edge ideals of bipartite graphs, which have been extensively studied in the commutative algebra community.
We investigate the ``stratified Ehrhart ring theory'' for periodic graphs, which gives an algorithm for determining the growth sequences of periodic graphs. The growth sequence $(s_{\Gamma, x_0, i})_{i \ge 0}$ is defined for a graph $\Gamma$ and its fixed vertex $x_0$, where $s_{\Gamma, x_0, i}$ is defined as the number of vertices of $\Gamma$ at distance $i$ from $x_0$. Although the sequences $(s_{\Gamma, x_0, i})_{i \ge 0}$ for periodic graphs are known to be of quasi-polynomial type, their determination had not been established, even in dimension two. Our theory and algorithm can be applied to arbitrary periodic graphs of any dimension. As an application of the algorithm, we determine the growth sequences in several new examples.
periodic graphs, growth sequence, growth series, Ehrhart theory, convex geometry
Integer Diagonal Forms for Subset Intersection Relations
For integers $0 \leq \ell \leq k_{r} \leq k_{c} \leq n$, we give a description for the Smith group of the incidence matrix with rows (columns) indexed by the size $k_r$ ($k_c$, respectively) subsets of an $n$-element set, where incidence means intersection in a set of size $\ell$. This generalizes work of Wilson and Bier from the 1990s which dealt only with the case where incidence meant inclusion. Our approach also describes the Smith group of any matrix in the $\Z$-linear span of these matrices so includes all integer matrices in the Bose-Mesner algebra of the Johnson association scheme: for example, the association matrices themselves as well as the Laplacian, signless Laplacian, Seidel adjacency matrix, etc. of the associated graphs. In particular, we describe the critical (also known as sandpile) groups of these graphs. The complexity of our formula grows with the $k$ parameters, but is independent of $n$ and $\ell$, which often leads to an efficient algorithm for computing these groups. We illustrate our techniques to give diagonal forms of matrices attached to the Kneser and Johnson graphs for subsets of size $3$, whose invariants have never before been described, and recover results from a variety of papers in the literature in a unified way.
Polytopal realizations of non-crystallographic associahedra
We use the folding technique to show that generalized associahedra for non-simply-laced root systems (including non-crystallographic ones) can be obtained as sections of simply-laced generalized associahedra constructed by Bazier-Matte, Chapelier-Laget, Douville, Mousavand, Thomas and Yıldırım.
Three results related to the half-plane property of matroids
We settle three problems from the literature on stable and real zero polynomials and their connection to matroid theory. We disprove the weak real zero amalgamation conjecture by Schweighofer and the second author. We disprove a conjecture by Brändén and D'León by finding a relaxation of a matroid with the weak half-plane property that does not have the weak half-plane property itself. Finally, we prove that every quaternionic unimodular matroid has the half-plane property which was conjectured by Pendavingh and van Zwam.
Cluster Monomials in Graph Laurent Phenomenon Algebras
Laurent Phenomenon algebras, first introduced by Lam and Pylyavskyy, are a generalization of cluster algebras that still possess many salient features of cluster algebras. Graph Laurent Phenomenon algebras, defined by Lam and Pylyavskyy, are a subclass of Laurent Phenomenon algebras whose structure is given by the data of a directed graph. Our main result is that cluster monomials of a graph Laurent Phenomenon algebra form a linear basis, analogously to a result of Caldero and Keller for cluster algebras.
Explicit forms in lower degrees of rank 2 cluster scattering diagrams
In this paper, we study wall elements of rank 2 cluster scattering diagrams based on dilogarithm elements. We derive two major results. First, we give a method to calculate wall elements in lower degrees. By this method, we may see the explicit forms of wall elements including the Badlands, which is the complement of G-fan. In this paper, we write one up to 7 degrees. Also, by using this method, we derive some walls independent of their degrees. Second, we find a certain admissible form of them. In the proof of these facts, we introduce a matrix action on a structure group, which we call a similarity transformation, and we argue the relation between this action and ordered products.
Combinatorial Göttsche-Schroeter invariants in any genus
Göttsche-Schroeter invariants are a genus 0 extension of Block-Göttsche invariants. They interpolate between Welschinger invariants involving pairs of complex conjugated points and genus 0 descendant Gromov-Witten invariants. They can be computed by a floor diagram algorithm. In this paper, we show that this floor diagrams recipe actually leads to some invariants in any genus. This generalizes Göttsche-Schroter invariant in higher genus in a combinatorial way. We then prove some polynomiality result and establish a link with invariants defined by Shustin and Sinichkin. We provide many examples. In particular, we conjecture that these combinatorial invariants satisfy the Abramovich-Bertram formula.
Pattern bounds for principal specializations of $\beta$-Grothendieck polynomials
There has been recent interest in lower bounds for the principal specializations of Schubert polynomials $\nu_w := \mathfrak{S}_w(1,\dots,1)$. We prove a conjecture of Yibo Gao in the setting of $1243$-avoiding permutations that gives a lower bound for $\nu_w$ in terms of the permutation patterns contained in $w$. We extended this result to principal specializations of $\beta$-Grothendieck polynomials $\nu^{(\beta)}_w := \mathfrak{G}^{(\beta)}_w(1,\dots,1)$ by restricting to the class of vexillary $1243$-avoiding permutations. Our methods are bijective, offering a combinatorial interpretation of the coefficients $c_w$ and $c^{(\beta)}_w$ appearing in these conjectures.
Thrall's problem asks for the Schur decomposition of the higher Lie modules $\mathcal{L}_\lambda$, which are defined using the free Lie algebra and decompose the tensor algebra as a general linear group module. Although special cases have been solved, Thrall's problem remains open in general. We generalize Thrall's problem to the free Lie superalgebra, and prove extensions of three known results in this setting: Brandt's formula, Klyachko's identification of the Schur--Weyl dual of $\mathcal{L}_n$, and Kr{\'a}skiewicz--Weyman's formula for the Schur decomposition of $\mathcal{L}_n$. The latter involves a new version of the major index on super tableaux, which we show corresponds to a $q,t$-hook formula of Macdonald.
Degrees of $P$-Grothendieck polynomials and regularity of Pfaffian varieties
We prove a formula for the degrees of Ikeda and Naruse's $P$-Grothendieck polynomials using combinatorics of shifted tableaux. We show this formula can be used in conjunction with results of Hamaker, Marberg, and Pawlowski to obtain an upper bound on the Castelnuovo--Mumford regularity of certain Pfaffian varieties known as vexillary skew-symmetric matrix Schubert varieties. Similar combinatorics additionally yields a new formula for the degree of Grassmannian Grothendieck polynomials and the regularity of Grassmannian matrix Schubert varieties, complementing a 2021 formula of Rajchgot, Ren, Robichaux, St. Dizier, and Weigandt.
Pfaffian variety, P-Grothendieck polynomial, regularity, skew-symmetric matrix Schubert variety
Triangulations of cosmological polytopes
A cosmological polytope is defined for a given Feynman diagram, and its canonical form may be used to compute the contribution of the Feynman diagram to the wavefunction of certain cosmological models. Given a subdivision of a polytope, its canonical form is obtained as a sum of the canonical forms of the facets of the subdivision. In this paper, we identify such formulas for the canonical form via algebraic techniques. It is shown that the toric ideal of every cosmological polytope admits a Gröbner basis with a squarefree initial ideal, yielding a regular unimodular triangulation of the polytope. In specific instances, including trees and cycles, we recover graphical characterizations of the facets of such triangulations that may be used to compute the desired canonical form. For paths and cycles, these characterizations admit simple enumeration. Hence, we obtain formulas for the normalized volume of these polytopes, extending previous observations of Kühne and Monin.
Weights on homogeneous coherent configurations
D.~G.~Higman generalized a coherent configuration and defined a weight. In this article, we will modify the definition and investigate weights on coherent configurations. If our weights are on a thin homogeneous coherent configuration, that is essentially a finite group, then there is a natural correspondence between the set of equivalence classes of weights and $2$-cohomology group of the group. We also give a construction of weights as a generalization of Higman's method using monomial representations of finite groups.
homogeneous coherent configuration, weight, association scheme, factor set, cohomology
Multigraded strong Lefschetz property for balanced simplicial complexes
Generalizing the strong Lefschetz property for an $\mathbb{N}$-graded algebra, we introduce the multigraded strong Lefschetz property for an $\mathbb{N}^m$-graded algebra. We show that, for $\bm{a} \in \mathbb{N}^m_+$, the generic $\mathbb{N}^m$-graded Artinian reduction of the Stanley-Reisner ring of an $\bm{a}$-balanced homology sphere over a field of characteristic $2$ satisfies the multigraded strong Lefschetz property. A corollary is the inequality $h_{\bm{b}} \leq h_{\bm{c}}$ for $\bm{b} \leq \bm{c} \leq \bm{a}-\bm{b}$ among the flag $h$-numbers of an $\bm{a}$-balanced simplicial sphere. This can be seen as a common generalization of the unimodality of the $h$-vector of a simplicial sphere by Adiprasito and the balanced generalized lower bound inequality by Juhnke-Kubitzke and Murai. We further generalize these results to $\bm{a}$-balanced homology manifolds and $\bm{a}$-balanced simplicial cycles over a field of characteristic $2$.
Lefschetz property, Stanley-Reisner ring, balancedness, multigraded algebra, unimodality
On the automorphism group of a putative Conway 99-graph
Conway 99-graph, strongly regular graph, automorphism group, orbit partition, orbit valencies.
Intersection cohomology of type-A toric varieties
Type-A toric varieties may be obtained as GIT quotients with respect to a torus action with weights corresponding to roots of the group SL(k) for some k>1. These varieties appear in various important applications, in particular, as normal cones to strata in moduli spaces of vector bundles. In this paper, we describe the intersection Betti numbers of these varieties, and those of some associated projective varieties. We present an elegant combinatorial model for these numbers, and, using the work of Hausel and Sturmfels, we show that the relevant intersection cohomology groups are endowed with a canonical product structure.