Birational rowmotion is a discrete dynamical system on the set of all positive real-valued functions finite poset, which birational lift combinatorial order ideals. It known that for minuscule poset has equal to Coxeter number, and exhibits file homomesy phenomenon refined ideal cardinality statistics. In this paper we generalize these results setting. Moreover, as generalization promotion prod...

Motivated by the Coxeter complex associated to a Coxeter system (W, S), we introduce a simplicial regular cell complex ∆(G, S) with a G-action associated to any pair (G, S) where G is a group and S is a finite set of generators for G which is minimal with respect to inclusion. We examine the topology of ∆(G, S), and in particular the representations of G on its homology groups. We look closely ...

Coxeter groups have presentations 〈S : (st)st∀s, t ∈ S〉 where for all s, t ∈ S, mst ∈ {1, 2, . . . ,∞}, mst = mts and mst = 1 if and only if s = t. A fundamental question in the theory of Coxeter groups is: Given two such “Coxeter” presentations, do they present the same group? There are two known ways to change a Coxeter presentation, generally referred to as twisting and simplex exchange. 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 ∈ S such that m(s, t) is even, m(s, t) = 2, (1) for each s 6= t ∈ S such that m(s, t) is odd, {s, t} is a maximal spherical subset of S, (2) there does not exist a three-points subset {s, t, u} ⊂ S such that m(s, t) and m(t, u) are odd, and (3) for each s 6= t ∈ S such that...

The isomorphism problem for finitely generated Coxeter groups is the problem of deciding if two finite Coxeter matrices define isomorphic Coxeter groups. Coxeter [3] solved this problem for finite irreducible Coxeter groups. Recently there has been considerable interest and activity on the isomorphism problem for arbitrary finitely generated Coxeter groups. In this paper, we determine some stro...

For a Coxeter group (W,S), a permutation of the set S is called a Coxeter word and the group element represented by the product is called a Coxeter element. Moving the first letter to the end of the word is called a rotation and two Coxeter elements are rotation equivalent if their words can be transformed into each other through a sequence of rotations and legal commutations. We prove that Cox...

In mathematics, a group is the set of symmetries of an object. Coxeter groups are a broad and natural class of groups that are related to reflectional symmetries. Each Coxeter group is determined by a diagram, called a labeled graph, that encodes algebraic information about the group. In general, two different labeled graphs can give rise to the same group. It is therefore natural to ask: are t...

We show that the vertex barycenter of generalized associahedra and permutahedra coincide for any finite Coxeter system.

The aim is to apply string-rewriting methods to compute left Kan extensions, or, equivalently, induced actions of monoids, categories, groups or groupoids. This allows rewriting methods to be applied to a greater range of situations and examples than before. The data for the rewriting is called a Kan extension presentation. The paper has its origins in earlier work by Carmody and Walters who ga...

Let (W,S) be a Coxeter system and let s ∈ S. We call s a right-angled generator of (W,S) if st = ts or st has infinite order for each t ∈ S. We call s an intrinsic reflection of W if s ∈ RW for all Coxeter generating sets R of W . We give necessary and sufficient conditions for a right-angled generator s ∈ S of (W,S) to be an intrinsic reflection of W .