Decomposition matrices and blocks for the symplectic blob algebra over the complex field

TL;DR

This paper analyzes the symplectic blob algebra over the complex field, determining its block structure, Gram determinants, and conditions for semisimplicity, providing new insights into its representation theory.

Contribution

It explicitly finds the blocks, Gram determinants, and decomposition numbers for the symplectic blob algebra over complex numbers, extending understanding of its module structure.

Findings

The algebra is semisimple unless certain parameters are integral or roots of unity.

Blocks are classified for all parameter specializations over complex numbers.

Decomposition numbers are computed in many generic cases.

Abstract

The symplectic blob algebra is a physically motivated quotient of the Hecke algebra $H(\tilde{C}_n)$ with a diagram calculus. We find the blocks for the symplectic blob algebra for all specialisations of its parameters over the complex numbers. We determine Gram determinants for the cell modules with respect to a canonical contravariant form. We show in particular that the algebra is semisimple over the complex numbers unless at least one of the "quantisation" parameters, or the sum or difference of two of these parameters is integral, or the bulk parameter $q$ is a root of unity. We find decomposition numbers in many of the $q$-generic cases.