Year: 2023
Type: article
Abstract: For l,n∈N we define tonal partition algebra Pnl over Z[δ]. We construct modules {Δμ¯}μ¯ for Pnl over Z[δ], and hence over any integral domain containing Z[δ] (such as C[δ]), that pass to a complete set of irreducible modules over the field of fractions. We show that Pnl 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 C (with δ∈C×) 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.
Source: Communications in Algebra
Authors: Chwas Ahmed, Paul Martin, Volodymyr Mazorchuk
Institutions: University of Leeds, University of Sulaimani, Uppsala University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2993539115