Tonal partition algebras: fundamental and geometrical aspects of representation theory

Abstract

For we define tonal partition algebra over . We construct modules for over , and hence over any integral domain containing (such as ), that pass to a complete set of irreducible modules over the field of fractions. We show that is semisimple there. That is, we construct for the tonal partition algebras a modular system in the sense of Brauer. Using a “geometrical” index set for the Δ-modules, we give an order with respect to which the decomposition matrix over (with ) is upper-unitriangular. We establish several crucial properties of the Δ-modules. These include a tower property, with respect to n, in the sense of Green and Cox-Martin-Parker-Xi; contravariant forms with respect to a natural involutive antiautomorphism; a highest weight category property; and branching rules.