On the Semisimplicity of the Brauer and Birman-Murakami-Wenzl algebras
Friday 13th August, 12:05-12:55pm,
The Birman-Murakami-Wenzl (or B-M-W) algebras arise as centraliser
algebras for the action of the Drinfeld-Jimbo quantum groups of type
B, C and D on their respective natural representations. The B-M-W
algebras are known to be cellular in the sense of Graham and
Lehrer. We give an explicit construction of new cellular bases for the
B-M-W algebras, bases which are indexed by paths in the Bratteli
diagram associated with the B-M-W algebras and with respect to which
the Jucys-Murphy elements in the B-M-W algebra act upper triangularly;
we also give explicit formulae, in terms of paths in the Bratteli
diagram associated with the B-M-W algebras, for the generalised
eigenvalues arising from the action of the Jucys-Murphy operators on
the cell (or Specht) modules of the B-M-W algebras.
Our construction is motivated by the Specht module theory for the
representations of the Iwahori-Hecke algebra of the symmetric
group. The Jucys-Murphy elements act upper triangularly with respect
to the Murphy bases for the Specht modules of the Iwahori-Hecke
algebra of the symmetric group, allowing one to distinguish between
the Specht modules and to state explicitly in terms of standard
tableaux, the necessary and sufficient conditions for the
Iwahori-Hecke algebra of the symmetric group to be semisimple.
The Jucys-Murphy operators play a somewhat weaker role in the
representation theory of the B-M-W algebras. Here, the action of the
Jucys-Murphy operators allow us to state explicitly, in terms of paths
in the associated Bratteli diagrams, sufficient but not necessary
conditions for the B-M-W algebras to be semi-simple. We give examples
showing that even if parameters are chosen such that a B-M-W algebra
is semisimple, the Jucys-Murphy elements may fail to distinguish between
distinct cell modules of the given algebra.
Finally we show how our construction may be applied to the
representation theory of Brauer's centraliser algebras.