Representations of the marked Brauer algebra

Abstract

The marked Brauer algebra is a generalization of the diagrammatic Brauer algebra which diagrammatizes Moon's centralizer algebra for the type \mathfrak{p} Lie superalgebra. We prove that the marked Brauer algebra is a standard based algebra in the sense of Du-Rui and determine the circumstances under which the algebra is quasi-hereditary. In particular, we observe that the category of modules for the marked Brauer algebra is a highest weight category. Standard modules are constructed and the simple modules are classified using the general framework of standard based algebras. We describe the induction and restriction of standard modules and study weights of these modules which arise from the action of a collection of Jucys-Murphy type elements in the marked Brauer algebra. Finally, we introduce arc diagrams for the marked Brauer algebra, which provide a combinatorial criterion for determining decomposition multiplicities of standard modules.