Cellular Bases for Algebras with a Jones Basic Construction

Abstract

We construct analogues for the Brauer, BMW, partition, and Jones-TemperleyLieb algebras of the Murphy basis of the Hecke algebra of the symmetric group. The bases are cellular bases indexed by paths on branching diagrams, and compatible with restriction of cell modules. The Jucys-Murphy elements for each class of algebras act by triangular matrices on the Murphy basis.

Diagram Algebras, Dominance Triangularity and Skew Cell Modules

“…Since F t is a minimal idempotent, it follows that G t = F t , which proves part (4). Part (5) follows from parts (2), (3), and (4).…”

“…Following [4], we observe that algebras fitting into our framework possess analogues of both the Murphy and seminormal bases of the Hecke algebras of the symmetric groups. We prove that the transition matrix between these bases is dominance unitriangular.…”

“…For the remainder of the paper, we shall let R be an integral domain with field of fractions F. In this section, we shall define diagram algebras and recall the construction of their Murphy bases, following [4]. We first recall the definition of a cellular algebra, as in [9].…”

“…(D5) The sequence of algebras (A r ) r 0 is restriction-coherent. As discussed in [4,Section 3], under the assumptions (D1)-(D5) above, there exists a well defined multiplicity-free branching diagram A associated with the sequence (A r ) r 0 . The branching diagram is an infinite, graded, directed graph with vertices A r at level k and edges determined as follows.…”

“…(2) The towers of Tempreley-Lieb algebras, partition algebras, Brauer algebras, and Birman-Murakami-Wenzl algebras [4].…”