Ramified partition algebras
Ramified partition algebras
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发表时间：
2002
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通讯作者：
A. Elgamal
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作者：
P. Martin;A. Elgamal
摘要。对于标量Q的每个自然数n，偏置T和|T| -元组，我们引入分支划分代数(Q)，它是划分代数的物理动机和自然推广[24,25]（划分代数与情况|T|=1一致）。对于固定的n和T，这些代数和划分代数一样，具有独立于q的基。我们研究了它们在T${{:=({1,2},leq)}}$的表示理论。我们证明了当Q1Q2在k中可逆且k使得k上的有限群代数是半单的（例如当k是代数闭的，特征为零）时，(Q)在域k上是拟遗传的。在这些条件下，我们确定了(Q)的简单模的索引集，并用该索引集构造了标准模。我们证明了Q有无限多的选择，使得(Q)对足够大的n不是半简单的，而是对所有n都是一般半简单的。我们构造了(Q)的某些非半简单专门化的张量空间表示，并展示了如何使用这些来构建任意物理维度的时钟模型转移矩阵[24]。
Abstract.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 Q1Q2 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.