1 4 Ju n 20 02 Ramified Partition Algebras
نویسنده
چکیده
For each natural number n, poset T , and |T |–tuple of scalars Q, we introduce the ramified partition algebra P (T) n (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 = 2 := ({1, 2}, ≤). We show that P (2) n (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 P (2) n (Q), and construct standard modules with this index set. We show that there are unboundedly many choices of Q such that P (2) n (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 spe-cializations of P (2) n (Q), and show how to use these to build clock model transfer matrices [24] in arbitrary physical dimensions.
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