相关论文: Rigidity of two-dimensional Coxeter groups

A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some…

Given a Coxeter system (W,S) equipped with an involutive automorphism T, the set of twisted identities is i(T) = {T(w)^{-1}w : w \in W}. We point out how i(T) shows up in several contexts and prove that if there is no s \in S such that…

Reflection length and codimension of fixed point spaces induce partial orders on a complex reflection group. While these partial orders are of independent combinatorial interest, our investigation is motivated by a connection between the…

Buan, Iyama, Reiten and Smith proved that the superpotential of a quiver corresponding to an element of Coxeter group is rigid. In this paper, we extend this result to the Berenstein-Fomin-Zelevinsky quivers corresponding to double Bruhat…

A 2-dimensional point-line framework is a collection of points and lines in the plane which are linked by pairwise constraints that fix some angles between pairs of lines and also some point-line and point-point distances. It is rigid if…

In 1991, Dajczer and Rodriguez proved in [10] that a complete minimal real Kahler submanifold of codimension 2, if with complex dimension > 2, would be either holomorphic, or a cylinder, or complex ruled. In this article, we generalize…

A topological commutative ring is said to be rigid when for every set $X$, the topological dual of the $X$-fold topological product of the ring is isomorphic to the free module over $X$. Examples are fields with a ring topology, discrete…

The double Cayley Grassmannian is a unique smooth equivariant completion with Picard number one of the 14-dimensional exceptional complex Lie group $G_2$, and it parametrizes eight-dimensional isotropic subalgebras of the complexified…

For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy…

Given a metric space X, one defines its Wasserstein space W2(X) as a set of sufficiently decaying probability measures on X endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of W2(X)…

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and proper…

For any Coxeter group W, we define a filtration of H^*(W;ZW) by W-submodules and then compute the associated graded terms. More generally, if U is a CW complex on which W acts as a reflection group we compute the associated graded terms for…

We consider generalisations of the elliptic Calogero--Moser systems associated to complex crystallographic groups in accordance to [1]. In our previous work [2], we proposed these systems as candidates for Seiberg--Witten integrable systems…

A (closed) dynamical system is a notion of how things can be, together with a notion of how they may change given how they are. The idea and mathematics of closed dynamical systems has proven incredibly useful in those sciences that can…

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…

In this note, we characterize affine and non-affine Coxeter systems among all Coxeter systems in terms of the structure of their reflection orders. For an infinite irreducible system $(W,S)$, we show that affineness can be characterized in…

A hypertree, or $\mathbb{Q}$-acyclic complex, is a higher-dimensional analogue of a tree. We study random $2$-dimensional hypertrees according to the determinantal measure suggested by Lyons. We are especially interested in their…

We show a rigid connection between two classical objects: the R.Thompson group F and the Dunce hat. A diagram group of a directed 2-complex contains a copy of F if and only if the complex contains a copy of the Dunce hat.

This paper introduces new structures called conic frameworks and their rigidity. They are composed by agents and a set of directed constraints between pairs of agents. When the structure cannot be flexed while preserving the constraints, it…

Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…