A presentation for the symplectic blob algebra

TL;DR

This paper introduces a presentation for the symplectic blob algebra, providing a new algebraic description that is proven to be isomorphic to the original diagram-based algebra, enhancing understanding of its structure.

Contribution

The paper presents a new algebraic presentation for the symplectic blob algebra and proves its isomorphism to the diagram-based algebra, offering a novel perspective on its structure.

Findings

The algebraic presentation is linearly growing in generators and relations.

The new presentation is proven to be isomorphic to the original diagram-based algebra.

Provides insights into the structure and rank of the symplectic blob algebra.

Abstract

The symplectic blob algebra $b_n$ ($n \in \mathbb{N}$) is a finite dimensional algebra defined by a multiplication rule on a basis of certain diagrams. The rank $r(n)$ of $b_n$ is not known in general, but $r(n)/n$ grows unboundedly with $n$. For each $b_n$ we define an algebra by presentation, such that the number of generators and relations grows linearly with $n$. We prove that these algebras are isomorphic.