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

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.