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R. M. Green

Publications and source records attributed to R. M. Green.

At least 19 recordsLinked to original sources
arXiv2026

Perfect matchings, Fano planes, and orthogonal bases of type $E_8$

We use perfect matchings and labelled Fano planes to construct and study the $2025$ orthogonal bases of positive roots in the $E_8$ root system. The set of these bases forms a highly structured, Bruhat-like graded poset $(\Omega, \leq_Q)$ whose rank function can be computed from the cardinalities of so-called generalized Rothe diagrams. We give combinatorial characterizations of these diagrams in terms of matchings and Fano planes, and we explain how to compute the ranks of the elements of $\Omega$ using suitable combinatorial statistics such as the weights of perfect matchings. We establish simple formulas for the rank generating functions of $\Omega$ and of its 50 congruence classes under a natural order congruence relation. Our derivation of the generating functions contains some intermediate results on general perfect matchings and labelled Fano planes that can be stated without mentioning root systems and may be of independent interest.

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arXiv2026

Branching rules of minuscule representations via a new partial order

We introduce a new partial order on the set of all antichains of a fixed size in any poset. When applied to minuscule posets, these partial orders give rise to distributive lattices that appear in the branching rules for minuscule representations of complex simple Lie algebras.

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arXiv2025

A partial order on the 240 packings of PG(3,2)

It has long been known that the most symmetrical solutions of Kirkman's Schoolgirl Problem can be constructed from the $240$ packings of the projective space $PG(3, 2)$, but it seems to have escaped notice that these packings have the structure of a partially ordered set. In this paper, we construct a shellable Bruhat-like graded partial order on the packings of $PG(3, 2)$ that refines the partial order on the product of four chains $[8]\times[5]\times[3]\times[2]$ and defines a Lehmer code on the packings. The partial order exists because the packings of $PG(3, 2)$ form a quasiparabolic set (in the sense of Rains--Vazirani) that is in bijective correspondence with a certain collection of maximal orthogonal subsets of the $E_8$ root system. The $E_8$ construction also induces transitive actions of the Weyl groups of type $D_n$ on the packings for $5 \leq n \leq 8$, and these actions are faithful for $n < 8$. It is possible to define both the signed permutation action and the partial order using the combinatorics of labelled Fano planes.

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arXiv2025

Generalized Rothe diagrams for orthogonal roots

Let $U$ be a set of positive roots of type $ADE$, and let $\Omega_U$ be the set of all maximum-cardinality orthogonal subsets of $U$. We associate a generalized Rothe diagram to each element $R\in \Omega_U$ as a broad, root-theoretic generalization of the traditional Rothe diagrams of permutations, and we use the generalized Rothe diagrams to define a $q$-polynomial in $U$ that we call the generalized quantum Hafnian of $U$. We study a large number of examples where these constructions recover a variety of widely studied algebraic and combinatorial objects. One of our motivating examples involves a certain set $U$ of $k^2$ roots in type $D_{2k}$, where the elements of $\Omega_U$ can be identified with permutations in $S_k$, the generalized Rothe diagrams are the traditional Rothe diagrams of permutations, and the generalized quantum Hafnian is the $q$-permanent. In another example, the generalized quantum Hafnian gives a non-recursive method to compute the 45 terms of a well-known invariant cubic polynomial of type $E_6$. More generally, all our examples in types $A$ and $D$ are closely related to perfect matchings and rook configurations, and our examples in type $E$ have applications to labelled Fano planes, del Pezzo surfaces, and minuscule representations. Each of our examples also gives rise to a matroid, and many of our examples have an associated equal-rank simply-laced symmetric pair.

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arXiv2025

Orthogonal roots, Macdonald representations, and quasiparabolic sets

Let $W$ be a simply laced Weyl group of finite type and rank $n$. If $W$ has type $E_7$, $E_8$, or $D_n$ for $n$ even, then the root system of $W$ has subsystems of type $nA_1$. This gives rise to an irreducible Macdonald representation of $W$ spanned by $n$-roots, which are products of $n$ orthogonal roots in the symmetric algebra of the reflection representation. We prove that in these cases, the set of all maximal sets of orthogonal positive roots has the structure of a quasiparabolic set in the sense of Rains--Vazirani. The quasiparabolic structure can be described in terms of certain quadruples of orthogonal positive roots which we call crossings, nestings, and alignments. This leads to nonnesting and noncrossing bases for the Macdonald representation, as well as some highly structured partially ordered sets. We use the $8$-roots in type $E_8$ to give a concise description of a graph that is known to be non-isomorphic but quantum isomorphic to the orthogonality graph of the $E_8$ root system.

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arXiv2023

Positivity properties for spherical functions of maximal Young subgroups

Let $S_k \times S_{n-k}$ be a maximal Young subgroup of the symmetric group $S_n$. We introduce a basis ${\mathcal B}_{n,k}$ for the coset space $S_n/S_k \times S_{n-k}$ that is naturally parametrized by the set of standard Young tableaux with $n$ boxes, at most two rows, and at most $k$ boxes in the second row. The basis ${\mathcal B}_{n,k}$ has positivity properties that resemble those of a root system, and there is a composition series of the coset space in which each term is spanned by the basis elements that it contains. We prove that the spherical functions of the associated Gelfand pair are nonnegative linear combinations of the ${\mathcal B}_{n,k}$.

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arXiv2023

Kazhdan--Lusztig cells of $\mathbf{a}$-value 2 in $\mathbf{a}(2)$-finite Coxeter systems

A Coxeter group is said to be \emph{$\mathbf{a}(2)$-finite} if it has finitely many elements of $\mathbf{a}$-value 2 in the sense of Lusztig. In this paper, we give explicit combinatorial descriptions of the left, right, and two-sided Kazhdan--Lusztig cells of $\mathbf{a}$-value 2 in an irreducible $\mathbf{a}(2)$-finite Coxeter group. In particular, we introduce elements we call \emph{stubs} to parameterize the one-sided cells and we characterize the one-sided cells via both star operations and weak Bruhat orders. We also compute the cardinalities of all the one-sided and two-sided cells.

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arXiv2023

2-roots for simply laced Weyl groups

We introduce and study "2-roots", which are symmetrized tensor products of orthogonal roots of Kac--Moody algebras. We concentrate on the case where $W$ is the Weyl group of a simply laced Y-shaped Dynkin diagram $Y_{a,b,c}$ having $n$ vertices and with three branches of arbitrary finite lengths $a$, $b$ and $c$; special cases of this include types $D_n$, $E_n$ (for arbitrary $n \geq 6$), and affine $E_6$, $E_7$ and $E_8$. We show that a natural codimension-$1$ submodule $M$ of the symmetric square of the reflection representation of $W$ has a remarkable canonical basis $\mathcal{B}$ that consists of 2-roots. We prove that, with respect to $\mathcal{B}$, every element of $W$ is represented by a column sign-coherent matrix in the sense of cluster algebras. If $W$ is a finite simply laced Weyl group, each $W$-orbit of 2-roots has a highest element, analogous to the highest root, and we calculate these elements explicitly. We prove that if $W$ is not of affine type, the module $M$ is completely reducible in characteristic zero and each of its nontrivial direct summands is spanned by a $W$-orbit of 2-roots.

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arXiv2023

On Representations of Affine Temperley--Lieb Algebras

We study the finite-dimensional simple modules, over an algebraically closed field, of the affine Temperley--Lieb algebra corresponding to the affine Weyl group of type $A$. These turn out to be closely related to the simple modules for a certain $q$-analogue of the annular algebra of V.F.R. Jones.

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arXiv2022

The nil Temperley--Lieb algebra of type affine C

We introduce a type affine $C$ analogue of the nil Temperley--Lieb algebra, in terms of generators and relations. We show that this algebra $T(n)$, which is a quotient of the positive part of a Kac--Moody algebra of type $D_{n+1}^{(2)}$, has an easily described faithful representation as an algebra of creation and annihilation operators on particle configurations, reminiscent of the open TASEP model in statistical physics. The centre of $T(n)$ consists of polynomials in a certain element $Q$, and $T(n)$ is a free module of finite rank over its centre. We show how to localize $T(n)$ by adjoining an inverse of $Q$, and prove that the resulting algebra is a full matrix ring over a ring of Laurent polynomials over a field. Although $T(n)$ has wild representation type, over an algebraically closed field we can classify all the finite dimensional indecomposable representations of $T(n)$ in which $Q$ acts invertibly.

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arXiv2015

A Classification of the Veldkamp Lines of the Near Hexagon L_3 times GQ(2, 2)

Using a standard technique sometimes (inaccurately) known as Burnside's Lemma, it is shown that the Veldkamp space of the near hexagon L_3 times GQ(2, 2) features 156 different types of lines. We also give an explicit description of each type of a line by listing the types of the three geometric hyperplanes it consists of and describing the properties of its core set, that is the subset of points of L_3 times GQ(2, 2) shared by the three geometric hyperplanes in question.

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arXiv2014

Relations among complementary and supplementary pairings of Saalschutzian 4F3(1) series

We investigate sums $K(\vec{x})$ and $L(\vec{x})$ of pairs of (suitably normalized) Saalschützian ${}_4F_3(1)$ hypergeometric series, and develop a theory of relations among these $K$ and $L$ functions. The function $L(\vec{x})$ has been studied extensively in the literature, and has been shown to satisfy a number of two-term and three-term relations with respect to the variable $\vec{x}$. More recent works have framed these relations in terms of Coxeter group actions on $\vec{x}$, and have developed a similar theory of two-term and three-term relations for $K(\vec{x})$. In this article, we derive "mixed" three-term relations, wherein any one of the $L$ (respectively, $K$) functions arising in the above context may be expressed as a linear combination of two of the above $K$ (respectively, $L$) functions. We show that, under the appropriate Coxeter group action, the resulting set of three-term relations (mixed and otherwise) among $K$ and $L$ functions partitions into eighteen orbits. We provide an explicit example of a relation from each orbit. We further classify the eighteen orbits into five types, with each type uniquely determined by the distances (under a certain natural metric) between the $K$ and $L$ functions in the relation. We show that the type of a relation dictates the complexity (in terms of both number of summands and number of factors in each summand) of the coefficients of the $K$ and $L$ functions therein.

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arXiv2012

On the Cyclically Fully Commutative Elements of Coxeter Groups

Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in W. We say that w is "cyclically fully commutative" (CFC) if every cyclic shift of any reduced expression for w is fully commutative (i.e., avoids long braid relations). These generalize Coxeter elements in that their reduced expressions can be described combinatorially by acyclic directed graphs, and cyclically shifting corresponds to source-to-sink conversions. In this paper, we explore the combinatorics of the CFC elements and enumerate them in all Coxeter groups. Additionally, we characterize precisely which CFC elements have the property that powers of them remain fully commutative, via the presence of a simple combinatorial feature called a "band." This allows us to give necessary and sufficient conditions for a CFC element w to be "logarithmic," that is, l(w^k) = k l(w) for all k > 0, for a large class of Coxeter groups that includes all affine Weyl groups and simply-laced Coxeter groups. Finally, we give a simple non-CFC element that fails to be logarithmic under these conditions.

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arXiv2011

Morse matchings on polytopes

We show how to construct homology bases for certain CW complexes in terms of discrete Morse theory and cellular homology. We apply this technique to study certain subcomplexes of the half cube polytope studied in previous works. This involves constructing explicit complete acyclic Morse matchings on the face lattice of the half cube; this procedure may be of independent interest for other highly symmetric polytopes.

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arXiv2010

Homology representations arising from the half cube, II

In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes of the $n$-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least $k$, and we proved that the homology of such a subcomplex is concentrated in degree $k-1$. This homology group supports a natural action of the Coxeter group $W(D_n)$ of type $D$. In this paper, we explicitly determine the characters (over ${\Bbb C}$) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group $S_n$ by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of $\sym_n$ agree (over ${\Bbb C}$) with the representations of $\sym_n$ on the $(k-2)$-nd homology of the complement of the $k$-equal real hyperplane arrangement.

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arXiv2008

Homology representations arising from the half cube

We construct a CW decomposition $C_n$ of the $n$-dimensional half cube in a manner compatible with its structure as a polytope. For each $3 \leq k \leq n$, the complex $C_n$ has a subcomplex $C_{n, k}$, which coincides with the clique complex of the half cube graph if $k = 4$. The homology of $C_{n, k}$ is concentrated in degree $k-1$ and furthermore, the $(k-1)$-st Betti number of $C_{n, k}$ is equal to the $(k-2)$-nd Betti number of the complement of the $k$-equal real hyperplane arrangement. These Betti numbers, which also appear in theoretical computer science, numerical analysis and engineering, are the coefficients of a certain Pascal-like triangle (Sloane's sequence A119258). The Coxeter groups of type $D_n$ act naturally on the complexes $C_{n, k}$, and thus on the associated homology groups.

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arXiv2008

Coxeter group actions on 4F3(1) hypergeometric series

We investigate a certain linear combination $K(\vec{x})=K(a;b,c,d;e,f,g)$ of two Saalschutzian hypergeometric series of type ${_4}F_3(1)$. We first show that $K(a;b,c,d;e,f,g)$ is invariant under the action of a certain matrix group $G_K$, isomorphic to the symmetric group $S_6$, acting on the affine hyperplane $V=\{(a,b,c,d,e,f,g)\in\Bbb C^7\colon e+f+g-a-b-c-d=1\}$. We further develop an algebra of three-term relations for $K(a;b,c,d;e,f,g)$. We show that, for any three elements $μ_1,μ_2,μ_3$ of a certain matrix group $M_K$, isomorphic to the Coxeter group $W(D_6)$ (of order 23040), and containing the above group $G_K$, there is a relation among $K(μ_1\vec{x})$, $K(μ_2\vec{x})$, and $K(μ_3\vec{x})$, provided no two of the $μ_j$'s are in the same right coset of $G_K$ in $M_K$. The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in $a,b,c,d,e,f,g$. The set of $({|M_K|/|G_K|\atop 3})=({32\atop 3})=4960$ resulting three-term relations may further be partitioned into five subsets, according to the Hamming type of the triple $(μ_1,μ_2,μ_3) $ in question. This Hamming type is defined in terms of Hamming distance between the $μ_j$'s, which in turn is defined in terms of the expression of the $μ_j$'s as words in the Coxeter group generators. Each three-term relation of a given Hamming type may be transformed into any other of the same type by a change of variable. An explicit example of each of the five types of three-term relations is provided.

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