Related papers: Regular cell complexes in total positivity

For strongly connected, pure $n$-dimensional regular CW-complexes, we show that {\it evenness} (each $(n{-}1)$-cell is contained in an even number of $n$-cells) is equivalent to generalizations of both cycle decomposition and…

We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces,…

We study the relationship between two sets of coordinates on a double Bruhat cell, the cluster variables introduced by Berenstein, Fomin, and Zelevinsky and the $\CX$-coordinates defined by the coweight parametrization of Fock and…

The main purpose of this paper is to introduce a new smooth version of a CW complex named a fat CW complex, and to show that it includes all closed manifolds, because existing smooth versions of CW complexes (e.g. [Iwa22]) do not have such…

We introduce the notion of smooth cell complexes and its subclass consisting of gathered cell complexes within the category of diffeological spaces (cf. Definitions 1 and 3). It is shown that the following hold. (1) With respect to the…

We describe CW decompositions of complex Lagrangian Grassmannians, that contain as subcomplexes, CW decompositions of real Lagrangian Grassmannians by Schubert-Arnol'd cells. The degrees of attaching maps are explicitly computed in terms of…

The purpose of this article is to shed new light on the combinatorial structure of Kazhdan-Lusztig cells in infinite Coxeter groups $W$. Our main focus is the set $\D$ of distinguished involutions in $W$, which was introduced by Lusztig in…

We establish a loop space decomposition for certain $CW$-complexes with a single top cell in the presence of a spherical pair, thereby generalizing several known decompositions of Poincar\'{e} duality complexes in which a loop of a product…

Zaslavsky conjectures that the bounded complex of a simple hyperplane arrangement is homeomorphic to a ball. We prove this conjecture for the more general uniform affine oriented matroids.

We use the natural homeomorphism between a regular CW-complex X and its face poset P_X to establish a canonical isomorphism between the cellular chain complex of X and the result of applying the poset construction of arXiv:0806.4532v1 to…

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…

This article introduces a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed rectangular region. We call this the branched…

Lusztig varieties are subvarieties in flag manifolds $G/B$ associated to an element $w$ in the Weyl group $W$ and an element $x$ in $G$, introduced in Lusztig's papers on character sheaves. We study the geometry of these varieties when $x$…

Let C be a one- or two-sided Kazhdan--Lusztig cell in a Coxeter group (W,S), and let Reduced(C) denote the set of reduced expressions of all w in C, regarded as a language over the alphabet S. Casselman has conjectured that Reduced(C) is…

We define the totally nonnegative matroid Schubert variety $\mathcal Y_V$ of a linear subspace $V \subset \mathbb R^n$. We show that $\mathcal Y_V$ is a regular CW complex homeomorphic to a closed ball, with strata indexed by pairs of…

Let $\cH$ be the one-parameter Hecke algebra associated to a finite Weyl group $W$, defined over a ground ring in which ``bad'' primes for $W$ are invertible. Using deep properties of the Kazhdan--Lusztig basis of $\cH$ and Lusztig's…

We explicitly describe a structure of a regular cell complex $K(L)$ on the moduli space $M(L)$ of a planar polygonal linkage $L$. The combinatorics is very much related (but not equal) to the combinatorics of the permutahedron. In…

We show that a finite regular cell complex with the intersection property is a Cohen-Macaulay space iff the top enriched cohomology module is the only nonvanishing one. We prove a comprehensive generalization of Balinski's theorem on convex…

Let $G$ be any connected and simply connected complex semisimple Lie group, equipped with a standard holomorphic multiplicative Poisson structure. We show that the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalized minors on…

We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings.…