15 research outputs found
Jucys-Murphy elements and a presentation for partition algebras
We give a new presentation for the partition algebras. This presentation was
discovered in the course of establishing an inductive formula for the partition
algebra Jucys-Murphy elements defined by Halverson and Ram [European J. Combin.
26 (2005), 869-921]. Using Schur-Weyl duality we show that our recursive
formula and the original definition of Jucys-Murphy elements given by Halverson
and Ram are equivalent. The new presentation and inductive formula for the
partition algebra Jucys-Murphy elements given in this paper are used to
construct the seminormal representations for the partition algebras in a
separate paper.Comment: 39 pages, 9 figures. Typos corrected and editorial changes made from
v1-3. The final publication is available at springerlink.co
Cellular bases for the Brauer and Birman–Murakami–Wenzl algebras
AbstractAn explicit combinatorial construction is given for cellular bases (in the sense of Graham and Lehrer) for the Birman–Murakami–Wenzl and Brauer algebra. We provide cell modules for the Birman–Murakami–Wenzl and Brauer algebras with bases index by certain bitableaux, generalising the Murphy basis for the Specht modules of the Iwahori–Hecke algebra of the symmetric group. The bases for the cell modules given here are constructed non-diagrammatically and hence are relatively amenable to computation
Cellular bases for algebras with a Jones basic construction
We define a method which produces explicit cellular bases for algebras
obtained via a Jones basic construction. For the class of algebras in question,
our method gives formulas for generic Murphy--type cellular bases indexed by
paths on branching diagrams and compatible with restriction and induction on
cell modules. The construction given here allows for a uniform combinatorial
treatment of cellular bases and representations of the Brauer,
Birman-Murakami-Wenzl, Temperley-Lieb, and partition algebras, among others.Comment: 43 pages. Includes figures created with tik
Specht modules and semisimplicity criteria for Brauer and Birman--Murakami--Wenzl Algebras
A construction of bases for cell modules of the Birman--Murakami--Wenzl (or
B--M--W) algebra by lifting bases for cell modules of
is given. By iterating this procedure, we produce cellular bases for B--M--W
algebras on which a large abelian subalgebra, generated by elements which
generalise the Jucys--Murphy elements from the representation theory of the
Iwahori--Hecke algebra of the symmetric group, acts triangularly. The
triangular action of this abelian subalgebra is used to provide explicit
criteria, in terms of the defining parameters and , for B--M--W algebras
to be semisimple. The aforementioned constructions provide generalisations, to
the algebras under consideration here, of certain results from the Specht
module theory of the Iwahori--Hecke algebra of the symmetric group
A seminormal form for partition algebras
Using a new presentation for partition algebras (J. Algebraic Combin.
37(3):401-454, 2013), we derive explicit combinatorial formulae for the
seminormal representations of the partition algebras. These results generalise
to the partition algebras the classical formulae given by Young for the
symmetric group.Comment: Published version. 51 pages, includes figures and table
The cellular second fundamental theorem of invariant theory for classical groups
We construct explicit integral bases for the kernels and the images of diagram algebras (including the symmetric groups, orthogonal and symplectic Brauer algebras) acting on tensor space. We do this by providing an axiomatic framework for studying quotients of diagram algebras
Diagram algebras, dominance triangularity and skew cell modules
We present an abstract framework for the axiomatic study of diagram algebras. Algebras that fit this framework possess analogues of both the Murphy and seminormal bases of the Hecke algebras of the symmetric groups. We show that the transition matrix between these bases is dominance unitriangular. We construct analogues of the skew Specht modules in this setting. This allows us to propose a natural tableaux theoretic framework in which to study the infamous Kronecker problem