arXiv subjects

Harrison Gimenez

Publications and source records attributed to Harrison Gimenez.

3 recordsLinked to original sources

arXiv2026

On the Weak Right Order of a Right-Angled Coxeter System

Let $ (W,S)$ be a Coxeter system, and let $ w\in W$. Let $ [1,w] := \{ x\in W \mid x \leq_{R} w \} $ where $ \leq_{R}$ denotes the weak right order of $ (W,S)$. The element $ w$ is said to have the \emph{ancestor property} if there is a unique non-trivial involution of maximal length in the set $ [1,w]$. The ancestor property was first defined by Hart and Rowley in \cite{hart2025noteinvolutionprefixescoxeter} where they conjectured that all non-identity elements in a finite Coxeter system have the ancestor property. In an arbitrary Coxeter system $(W,S)$, we show that the ancestor property holds for any non-identity fully commutative element (see \cite{stembridge1996fully} for the definition of a fully commutative element). In particular, since any element of a right-angled Coxeter system is fully commutative, we show that the ancestor property holds for all non-identity elements of a right-angled Coxeter system. Lastly, we also provide an axiomatization of right-angled Coxeter systems as reflection systems with a reflection cocycle that obeys a certain property called the \emph{meet intersection condition}.

math.GR↗

arXiv2026

Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids

We extend the basic theory of the groupoids introduced by Brink and Howlett in their study of normalizers of parabolic subgroups of Coxeter groups, by studying both their abstract root systems and root systems realized in real vector spaces. Such root systems have some properties formally analogous to those of root systems of Borcherds-Kac-Moody Lie algebras; in particular, some contain imaginary simple roots. Further, positive roots correspond to certain reflection subgroups. We also extend the most basic properties of the Tits cone and imaginary cone of Coxeter groups to corresponding cones defined for Brink-Howlett groupoids. The results linearize the study of certain classes of reflection subgroups of Coxeter groups in a similar way as root systems of Coxeter groups linearize the study of reflections.

math.GR↗

arXiv2025

Coxeter systems, left inversion sets, and higher dimensional cubes

Let $ (W,S)$ be a Coxeter system. We investigate the equation $ w(\Phi_{x}) = \Phi_{y}$ where $ w,x,y\in W$ and $ \Phi_{x}$, $\Phi_{y}$ denote the left inversion sets of $ x$ and $ y$. We then define a commutative square diagram called a Coxeter square which describes the relationship between 4 non-identity elements of the Coxeter group $ W$ and the equation $ w(\Phi_{x}) = \Phi_{y}$. Coxeter squares were first introduced by Dyer, Wang in \cite{dyer2011groupoids2} and \cite{dyer2019characterization}. Coxeter squares can be \textquotedblleft glued" together by compatible edges to form commutative diagrams in the shape of higher dimensional cubes called Coxeter $n$-cubes, which were first defined by Dyer in Example 12.5 of \cite{dyer2011groupoids2}. When $ |W| < \infty$ and $ |S| = n$, we show that Coxeter $n$-cubes must exist within $ (W,S)$. We then prove results about Coxeter $n$-cubes in the $A_{n}$ Coxeter system. We establish an explicit bijection between Coxeter $n$-cubes (modulo orientation) in $ A_{n}$ and binary trees with $n+1$ leaves. We also show that an element $x$ of $ A_{n}$ appears as the edge of some Coxeter $n$-cube if and only if $ x$ is a bigrassmannian permutation.

math.GR↗