Fixed points of involutive automorphisms of the Bruhat order

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 genera…

Q-Systems, Factorization Dynamics, and the Twist Automorphism

We provide a concrete realization of the cluster algebras associated with Q-systems as amalgamations of cluster structures on double Bruhat cells in simple algebraic groups. For nonsimply-laced groups, this provides a cluster-algebraic formulation of Q-systems of twisted type. It also yields a uniform proof of the discrete integrability of these Q-systems by identifying them with the dynamics of factorization mappings on quotients of double Bruhat cells. On the double Bruhat cell itself, we find these dynamics are closely related to those of the Fomin-Zelevinsky twist map. This leads to an exp…

Fixed points of zircon automorphisms

A zircon is a poset in which every principal order ideal is finite and equipped with a so-called special matching. We prove that the subposet induced by the fixed points of any automorphism of a zircon is itself a zircon. This provides a natural context in which to view recent results on Bruhat orders on twisted involutions in Coxeter groups.

pircon 自同构的不动元素

Fixed elements of pircon automorphisms

本文证明 pircon 自同构的不动元素仍构成 pircon，并将其应用于对称群，证明无不动点带符号对合在超八面体群 Bruhat 序对偶下形成 EL 可壳的 pircon。

Coxeter群的固定点陪集中最小长度元素与Bruhat序

Elements of minimal length and Bruhat order on fixed point cosets of Coxeter groups

研究Coxeter群中固定点陪集上的Bruhat序，刻画最小长度元素并证明其与序极小性等价。

A word property for twisted involutions in Coxeter groups

Given an involutive automorphism $θ$ of a Coxeter system $(W,S)$, let $\mathfrak{I}(θ) \subseteq W$ denote the set of twisted involutions. We provide a minimal set of moves that can be added to the braid moves, in order to connect all reduced $\underline{S}$-expressions (also known as admissible sequences, reduced $I_θ$-expressions, or involution words) for any given $w \in \mathfrak{I}(θ)$. This can be viewed as an analogue of the well-known word property for Coxeter groups. It improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types…