2 research outputs found
Methods for the construction of generators of algebraic curvature tensors
We demonstrate the use of several tools from Algebraic Combinatorics such as
Young tableaux, symmetry operators, the Littlewood-Richardson rule and discrete
Fourier transforms of symmetric groups in investigations of algebraic curvature
tensors.Comment: 11 page
Canonical Algebraic Curvature Tensors of Symmetric and Anti-Symmetric Builds
We relate canonical algebraic curvature tensors that are built from a
self-adjoint () or skew adjoint () linear operator A.
Several authors have proven that any algebraic curvature tensor may be
expressed as a sum of , or as a sum of . This motivates
our interest in relating them as well as in the linear independence of sets of
canonical algebraic curvature tensors. We develop an identity that relates
to , which will allow us to employ previous methods used
for to the case of as well as use them interchangeably
in some instances. We compute the structure group of , and
develop methods for determining the linear independence of sets which contain
both and . We consider cases where the operators are
arranged in chain complexes and find that this greatly restricts the linear
independence of the curvature tensors with those operators. Moreover, if one of
the operators has a nontrivial kernel, we develop a method for reducing the
bound on the least number of canonical algebraic curvature tensors that it
takes to write a canonical algebraic curvature tensor