The Krein condition, spherical designs, Norton algebras and permutation groups

I . Permutation Groups and Coherent ( Cellular ) Algebras

Finitely Presented Algebras and Groups Defined by Permutation Relations *

The class of finitely presented algebras over a field K with a set of generators a1, . . . , an and defined by homogeneous relations of the form a1a2 · · · an = aσ(a)aσ(2) · · · aσ(n), where σ runs through a subset H of the symmetric group Sym n of degree n, is introduced. The emphasis is on the case of a cyclic subgroup H of Sym n of order n. A normal form of elements of the algebra is obtaine...

Primitivity of Permutation Groups, Coherent Algebras and Matrices

A coherent algebra is F-primitive if each of its non-identity basis matrices is primitive in the sense of Frobenius. We investigate the relationship between the primitivity of a permutation group, the primitivity of its centralizer algebra, and F-primitivity. The results obtained are applied to give new proofs of primitivity criteria for the exponentiations of permutation groups and of coherent...

Spherical 2-Designs and Lattices from Abelian Groups

We consider lattices generated by finite Abelian groups. The main result says that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a unifo...

The Krein signature, Krein eigenvalues, and the Krein Oscillation Theorem

In this paper the problem of locating eigenvalues of negative Krein signature is considered for operators of the form JL, where J is skew-symmetric with bounded inverse and L is self-adjoint. A finite-dimensional matrix, hereafter referred to as the Krein matrix, associated with the eigenvalue problem JLu = λu is constructed with the property that if the Krein matrix has a nontrivial kernel for...