Ramified partition algebras |
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Authors: | PP Martin A Elgamal |
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Institution: | (1) Mathematics Department, City University, Northampton Square, London, EC1V 0HB, UK |
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Abstract: | For each natural number n, poset T, and |T|–tuple of scalars Q, we introduce the ramified partition algebra P
n
(T)
(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 ${{T=\underline{{2}}:=({1,2},\leq)}}$. We show that ${{P_n^{(\underline{{2}})$ (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 ${{P_n^{(\underline{{2}})$ (Q), and construct standard modules with this index set. We show that there are unboundedly many choices of Q such that ${{P_n^{(\underline{{2}})$ (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 ${{P_n^{(\underline{{2}})$ (Q), and show how to use these to build clock model transfer matrices 24] in arbitrary physical dimensions.
Sadly Ahmed died before this work was completed. His memory lives on. |
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Keywords: | |
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