A finiteness theorem for W-graphs

A W-graph for a Coxeter group W is a combinatorial structure that encodes a module for the group algebra of W, or more generally, a module for the associated Iwahori–Hecke algebra. Of special interest are the W-graphs that encode the action of the Hecke algebra on its Kazhdan–Lusztig basis, as well as the action on individual cells. In previous work, we isolated a few basic features common to the W-graphs in Kazhdan–Lusztig theory and used these to define the class of “admissible” W-graphs. The main result of this paper resolves one of the basic question about admissible W-graphs: there are only finitely many admissible W-cells (i.e., strongly connected admissible W-graphs) for each finite Coxeter group W. Ultimately, the finiteness depends only on the fact that admissible W-graphs have nonnegative integer edge weights. Indeed, we formulate a much more general finiteness theorem for “cells” in finite-dimensional algebras which in turn is fundamentally a finiteness theorem for nonnegative integer matrices satisfying a polynomial identity.     

A factorization theorem for matrices

It is shown that a nonscalar invertible square matrix can be written as a product of two square matrices with prescribed eigenvalues subject only to the obvious determinant condition. As corollaries, we give short proofs of some known results such as Ballantine's characterization of products of four or five positive definite matrices, the commutator theorem of Shoda-Thompson for fields with sufficiently many elements and other results.     

A covering theorem for quasi-groups

It is shown that if Q is a quasi-group of order n and k is moderately large, there exists a subset A of Q of size k such that if t is the least number of left translates of A needed to cover Q, then $\text{t >}\text{c(n}\text{log}\text{n)}\text{k}$.     

A decomposition theorem for neutrices

Neutrices are convex additive subgroups of the nonstandard space R^k, most of them are external sets. Because of the convexity and the invariance under some translations and multiplications, external neutrices are models for orders of magnitude. One dimensional neutrices have been applied to asymptotics, singular perturbations, and statistics. This paper shows that in R^k, with standard k, every neutrix is the direct sum of k neutrices of R. These components may be chosen to be orthogonal.     

A decomposition theorem for G-groups

Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations. Moreover, we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.     

A Ramsey theorem for multiposets

In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result from the 1980s claims that the class of all finite posets with a linear extension has the Ramsey property. In 2010 Sokić proved that the class of all finite structures consisting of several linear orders has the Ramsey property. This was followed by a 2017 result of Solecki and Zhao that the class of all finite posets with several linear extensions has the Ramsey property.Using the categorical reinterpretation of the Ramsey property in this paper we prove a common generalization of all these results. We consider multiposets to be structures consisting of several partial orders and several linear orders. We allow partial orders to extend each other in an arbitrary but fixed way, and require that every partial order is extended by at least one of the linear orders. We then show that the class of all finite multiposets conforming to a fixed template has the Ramsey property.     

A symmetry theorem for condensers

We prove the radial symmetry of the solution of degenerate quasilinear capacity problems, when constant overdetermined Neumann data are assigned.     

A pullback theorem for cofibrations

In this note we prove a pullback theorem for cofibrations, which extends a well known theorem of Strøm [5]. It also implies the pullback theorem of Heath [4] for locally equiconnected spaces. In addition, we comment on the dual problem of attaching fibrations.     

A partition theorem for ordinals

In this note we show that α →^trans(α, m)² for all m < ω and every α of the form ω^β, where α →^trans(α, m)² is a weakening of the usual partition property obtained by considering only partitions whose first member is a transitive relation.     

A Brody theorem for orbifolds

We study the Kobayashi pseudodistance for orbifolds, proving an orbifold version of Brody’s theorem and classifying which one-dimensional orbifolds are hyperbolic.     

A delay theorem for pointlikes

This paper shows that, for a pseudovariety V of monoids, V * D has decidable pointlikes if and only if V does. In the process, we develop a theory of pointlike sets for categories and a generalization of the Derived Category Theorem to understand how pointlike sets behave with respect to the semidirect product. This paper is intended to be the first of two papers concerning algorithmic problems for semidirect products of pseudovarieties.