5,241 research outputs found
Words and polynomial invariants of finite groups in non-commutative variables
Let V be a complex vector space with basis {x_1,x_2,...,x_n} and G be a
finite subgroup of GL(V). The tensor algebra T(V) over the complex is
isomorphic to the polynomials in the non-commutative variables x_1, x_2,...,
x_n with complex coefficients. We want to give a combinatorial interpretation
for the decomposition of T(V) into simple G-modules. In particular, we want to
study the graded space of invariants in T(V) with respect to the action of G.
We give a general method for decomposing the space T(V) into simple modules in
terms of words in a Cayley graph of the group G. To apply the method to a
particular group, we require a homomorphism from a subalgebra of the group
algebra into the character algebra. In the case of G as the symmetric group, we
give an example of this homomorphism from the descent algebra. When G is the
dihedral group, we have a realization of the character algebra as a subalgebra
of the group algebra. In those two cases, we have an interpretation for the
graded dimensions of the invariant space in term of those words
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
Combinatorial Representation Theory
We attempt to survey the field of combinatorial representation theory,
describe the main results and main questions and give an update of its current
status. We give a personal viewpoint on the field, while remaining aware that
there is much important and beautiful work that we have not been able to
mention
Set families and Foulkes modules
We construct a new family of homomorphisms from Specht modules into Foulkes
modules for the symmetric group. These homomorphisms are used to give a
combinatorial description of the minimal partitions (in the dominance order)
which label irreducible characters appearing as summands of the characters of
Foulkes modules. The homomorphisms are defined using certain families of
subsets of the natural numbers. These families are of independent interest; we
prove a number of combinatorial results concerning them.Comment: 22 pages, 3 figures, final published versio
- …