Ramified partition algebras

For each natural number n, poset T, and |T|–tuple of scalars Q, we introduce the ramified partition algebra (Q), which is a physically motivated and natural generalization of the partition algebra [24, 25] (the partition algebra coincides with case |T|=1). For fixed n and T these algebras, like the partition algebra, have a basis independent of Q. We investigate their representation theory in case T \({{:=(\{1,2\},\leq)}}\). We show that (Q) is quasi–hereditary over field k when Q 1 Q 2 is invertible in k and k is such that certain finite group algebras over k are semisimple (e.g. when k is algebraically closed, characteristic zero). Under these conditions we determine an index set for simple modules of (Q), and construct standard modules with this index set. We show that there are unboundedly many choices of Q such that (Q) is not semisimple for sufficiently large n, but that it is generically semisimple for all n. We construct tensor space representations of certain non–semisimple specializations of (Q), and show how to use these to build clock model transfer matrices [24] in arbitrary physical dimensions.