Split Grothendieck rings of rooted trees and skew shapes via monoid representations

We study commutative ring structures on the integral span of rooted trees and $n$-dimensional skew shapes. The multiplication in these rings arises from the smash product operation on monoid representations in pointed sets. We interpret these as Grothendieck rings of indecomposable monoid representations over \fun - the "field" of one element. We also study the base-change homomorphism from \mt-modules to $k[t]$-modules for a field $k$ containing all roots of unity, and interpret the result in terms of Jordan decompositions of adjacency matrices of certain graphs.Comment: arXiv admin note: text overlap with arXiv:1706.0390

Hopf Algebras in General and in Combinatorial Physics: a practical introduction

This tutorial is intended to give an accessible introduction to Hopf algebras. The mathematical context is that of representation theory, and we also illustrate the structures with examples taken from combinatorics and quantum physics, showing that in this latter case the axioms of Hopf algebra arise naturally. The text contains many exercises, some taken from physics, aimed at expanding and exemplifying the concepts introduced

Excision for deformation K-theory of free products

Associated to a discrete group $G$, one has the topological category of finite dimensional (unitary) $G$-representations and (unitary) isomorphisms. Block sums provide this category with a permutative structure, and the associated $K$-theory spectrum is Carlsson's deformation $K$-theory of G. The goal of this paper is to examine the behavior of this functor on free products. Our main theorem shows the square of spectra associated to $G*H$ (considered as an amalgamated product over the trivial group) is homotopy cartesian. The proof uses a general result regarding group completions of homotopy commutative topological monoids, which may be of some independent interest.Comment: 32 pages, 1 figure. Final version: The title has changed, and the paper has been substantially revised to improve clarit

The topological Atiyah-Segal map

Associated to each finite dimensional linear representation of a group $G$, there is a vector bundle over the classifying space $BG$. We introduce a framework for studying this construction in the context of infinite discrete groups, taking into account the topology of representation spaces. This involves studying the homotopy group completion of the topological monoid formed by all unitary (or general linear) representations of $G$, under the monoid operation given by block sum. In order to work effectively with this object, we prove a general result showing that for certain homotopy commutative topological monoids $M$, the homotopy groups of $\Omega BM$ can be described explicitly in terms of unbased homotopy classes of maps from spheres into $M$. Several applications are developed. We relate our constructions to the Novikov conjecture; we show that the space of flat unitary connections over the 3-dimensional Heisenberg manifold has extremely large homotopy groups; and for groups that satisfy Kazhdan's property (T) and admit a finite classifying space, we show that the reduced $K$-theory class associated to a spherical family of finite dimensional unitary representations is always torsion.Comment: 57 pages. Comments welcome

Multi-Colour Braid-Monoid Algebras

We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the critical dilute A-D-E models which were recently introduced by Warnaar, Nienhuis, and Seaton as well as by Roche. These and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure. It is conjectured that the Yang-Baxterization of general multi-colour braid-monoid algebras will lead to the construction of further solvable lattice models.Comment: 32 page