The partition algebra revisited
The partition algebra introduced by P.~Martin is a generalization of both the Temperley-Lieb algebra and the Brauer algebra. In the reviewer's paper [in Compos. Math. 119, No. 1, 99-109 (1999; Zbl 0939.16006)] it is proved that the partition algebra is also cellular in the sense of \textit{J.~J.~Graham} and \textit{G.~I.~Lehrer} [Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)]. Given a natural number \(n\) and an element \(Q\) from some fixed field \(K\), the partition algebra \(P_n(Q)\) over the field \(K\) can be defined, and the cell modules are parametrized by integer partitions \(\lambda\) with \(|\lambda|\leq n\). When \(K\) is the field of complex numbers and \(Q\neq 0\), \textit{P.~Martin} [in J. Algebra 183, No. 2, 319-358 (1996; Zbl 0863.20009)] gives a condition for which there is a non-zero homomorphism between two cell modules \(\Delta_n(\lambda)\) and \(\Delta_n(\mu)\). In the present paper, the authors provide a more combinatorial view of Martin's result. Moreover, the authors not only extend Martin's results to include the case \(Q=0\), but also close a gap in the paper of Martin. This is a very powerful result.
scientific article; zbMATH DE number 1525274