Tonal partition algebras: fundamental and geometrical aspects of representation theory

For (Formula presented.) we define tonal partition algebra (Formula presented.) over (Formula presented.). We construct modules (Formula presented.) for (Formula presented.) over (Formula presented.), and hence over any integral domain containing (Formula presented.) (such as (Formula presented.)), that pass to a complete set of irreducible modules over the field of fractions. We show that (Formula presented.) 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 (Formula presented.) (with (Formula presented.)) 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.