Affine diagram categories, algebras and monoids

TL;DR

This paper introduces affine versions of classical diagram algebras, explores their algebraic structures and representation theory, and proves a conjecture on the asymptotic growth of indecomposable summands in tensor powers.

Contribution

It presents new affine diagram algebras, provides their generators and relations, and proves a conjecture on asymptotic growth in monoid representation tensor powers.

Findings

Affine diagram algebras are characterized and presented with generators and relations.

Representation theory of these algebras is developed and analyzed.

A conjecture on the asymptotic growth of indecomposable summands is proved.

Abstract

We introduce and study several affine (=annular in this paper) versions of the classical diagram algebras such as Temperley-Lieb, partition, Brauer, Motzkin, rook Brauer, rook, planar partition, and planar rook algebras. We give generators and relation presentation for them and their associated categories, study their representation theory, and the asymptotic behavior of tensor products of their representations in the monoid case. Under a mild hypothesis, we also prove a previous conjecture concerning the asymptotic growth of the number of indecomposable summands in the tensor powers of representations for finite monoids.