Murphy bases of generalized Temperley-Lieb algebras
The author considers the action of the Hecke algebra \({\mathcal H}_n\) of the symmetric group \({\mathcal S}_n\) on the tensor power \(V^{\otimes n}\), where \(V\) is a free \(R\)-module of rank \(m\). The paper is concerned with the quotient of \({\mathcal H}_n\) which acts faithfully on the tensor space. When \(m=2\), this quotient is the Temperley-Lieb algebra, and in general, a fixed value of \(m\) will produce a ``generalized Temperley-Lieb algebra''. The main point of the paper is that, for any \(m\), the quotients of the Hecke algebra occurring are compatible with Murphy's basis for \({\mathcal H}_n\). As the author hints, the theory can also be developed from the starting point of the Kazhdan-Lusztig basis \(\{C_w':w\in W\}\). The paper concludes by comparing the image of the Murphy basis to Kauffman's diagram basis in the case \(m=2\), showing that the change of basis matrix is triangular with powers of \(q\) on the diagonal. This can be regarded as a statement about the relationship between the Murphy basis and the Kazhdan-Lusztig basis, as the latter is known to be compatible with Kauffman's basis.
scientific article; zbMATH DE number 1309371