Ramified partition algebras.
Classical partition algebras are towers of finite-dimensional unital algebras over a ring. These algebras first arose in the context of transfer matrix algebras in Statistical Mechanics. In the paper under review the authors introduce a generalization of partition algebras, which they call `ramified partition algebras'. A ramified partition algebra \(P_n^{(T)}(\mathbf Q)\) is associated with a natural number \(n\), a poset \(T\), and a \(|T|\)-tuple of scalars \(\mathbf Q\) (the classical partition algebra is obtained in the case \(|T|=1\)). For fixed \(n\) and \(T\) the algebras \(P_n^{(T)}(\mathbf Q)\) have a basis independent of \(\mathbf Q\). In the present paper the authors study the representation theory of \(P_n^{(T)}(\mathbf Q)\) in the case \(T=\underline 2:=(\{1,2\},\leq)\). It is shown that the algebra \(P_n^{(\underline 2)}(\mathbf Q)\) is quasi-hereditary (over a field \(\mathbf k\)) provided that \(Q_1Q_2\) is invertible in \(\mathbf k\) and \(\mathbf k\) is such that certain finite group algebras over \(\mathbf k\) are semisimple. In the latter case the authors present a classification of all simple \(P_n^{(\underline 2)}(\mathbf Q)\)-modules and an explicit construction of standard modules determining the quasi-hereditary structure. The algebra \(P_n^{(\underline 2)}(\mathbf Q)\) turns out to be generically semisimple for all \(n\), however, it is shown that there are infinitely many choices of \(\mathbf Q\) such that \(P_n^{(\underline 2)}(\mathbf Q)\) is not semisimple for sufficiently large \(n\). For certain non-semisimple \(P_n^{(\underline 2)}(\mathbf Q)\) some tensor space representations are constructed and used to build clock model transfer matrices in arbitrary physical dimension. Several open problems on the representation theory of \(P_n^{(\underline 2)}(\mathbf Q)\) are formulated.
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