Related papers: Stack-Sorting for Coxeter Groups

This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to…

We define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice…

(Abridged abstract) For a finite real reflection group W and a W-orbit O of flats in its reflection arrangement---or equivalently a conjugacy class of its parabolic subgroups---we introduce a statistic on elements of W. We then study the…

The sorting indices $\text{sor}_B$ and $\text{sor}_D$ on the Coxeter groups of type $B$ and $D$ respectively are defined by Petersen and it is proved that $(\text{inv}_B, \text{rlmin})$ and $(\text{sor}_B, \ell'_B)$ have the same joint…

We classify all homogeneous odd (i.e., parity-reversing) Rota--Baxter operators of weight zero on the modified Witt-type Lie superalgebra $W = \langle L_m, G_n \rangle_{m,n\in\Z}$. Our classification shows that nontrivial such operators are…

We define the notion of braided Coxeter category, which is informally a tensor category carrying compatible, commuting actions of a generalised braid group B_W and Artin's braid groups B_n on the tensor powers of its objects. The data which…

We prove that certain families of Coxeter groups and inclusions $W_1\hookrightarrow W_2\hookrightarrow...$ satisfy homological stability, meaning that in each degree the homology $H_\ast(BW_n)$ is eventually independent of $n$. This gives a…

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the characters of a finite Coxeter group $W$ afforded by the homogeneous components of its Orlik-Solomon algebra as sums of characters induced from linear…

Solomon's descent algebra is used to define a family of signed measures M(W,x) for a finite Coxeter group W and non-zero x. The measures corresponding to W of types A and B are known to arise from the theory of card shuffling and to be…

We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new…

We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\cE = \ZZ[e^{2 \pi i/3}]$: there are only four such lattices,…

We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the $2$-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show…

Coxeter groups are equipped with a partial order known as the weak order, such that $u \leq v$ if the inversions of $u$ are a subset of the inversions of $v$. In finite Coxeter groups, weak order is a complete lattice, but in infinite…

Let $\mathrm{Mac}(W)$ be the MacNeille completion of the Bruhat order of a Coxeter group $W$. We introduce an action of the $0$-Hecke monoid of type $W$ on $\mathrm{Mac}(W)$, which allows us to define a weak order and a descent set…

One possible way to obtain the quasicrystallographic structures is the projections of the higher dimensional lattices into 2D or 3D subspaces. In this work we introduce a general technique applicable to any higher dimensional lattice. We…

In this paper, we give a new class of rigid Coxeter groups. Let $(W,S)$ be a Coxeter system. Suppose that (0) for each $s,t\in S$ such that $m(s,t)$ is even, $m(s,t)=2$, (1) for each $s\neq t\in S$ such that $m(s,t)$ is odd, $\{s,t\}$ is a…

In this paper, we introduce the dotted pattern-avoiding map $s_{\dot{\tau}}$, which avoids the dotted pattern $\dot{\tau}$ instead of descents as West's stack-sorting map $s$ does. We also extend the pattern-avoiding machine, which is…

We study the restriction of the absolute order on a Coxeter group $W$ to an interval $[1,w]_T$, where $w\in W$ is an involution. We characterize and classify those involutions $w$ for which $[1,w]_T$ is a lattice, using the notion of…

Let $s$ be West's deterministic stack-sorting map. A well-known result (West) is that any length $n$ permutation can be sorted with $n-1$ iterations of $s.$ In 2020, Defant introduced the notion of highly-sorted permutations -- permutations…

We introduce a technique of projection onto the Coxeter plane of an arbitrary higher dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by the simple roots of the Coxeter graph I2 (h) where h is the…