17 research outputs found
On algebraic fusions of association schemes
We give a complete description of the irreducible representations of
algebraic fusions of association schemes, in terms of the irreducible
representations of a Schur cover of the corresponding group of algebraic
automorphisms.Comment: This paper has been withdrawn by one of the authors, since it
requires more wor
Livsic-type Determinantal Representations and Hyperbolicity
Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic
real projective hypersurfaces, and their determinantal representations, play a
key role in the emerging field of convex algebraic geometry. In this paper we
consider a natural notion of hyperbolicity for a real subvariety of an arbitrary codimension with respect to a real -dimensional linear subspace and study its basic
properties. We also consider a special kind of determinantal representations
that we call Livsic-type and a nice subclass of these that we call \vr{}. Much
like in the case of hypersurfaces (), the existence of a definite
Hermitian \vr{} Livsic-type determinantal representation implies hyperbolicity.
We show that every curve admits a \vr{} Livsic-type determinantal
representation. Our basic tools are Cauchy kernels for line bundles and the
notion of the Bezoutian for two meromorphic functions on a compact Riemann
surface that we introduce. We then proceed to show that every real curve in
hyperbolic with respect to some real -dimensional linear
subspace admits a definite Hermitian, or even real symmetric, \vr{} Livsic-type
determinantal representation
Non-Commutative Functions on the Non-Commutative Ball
In this talk we will discuss nc-functions on the unit nc-ball \mathfrak{B}_d. The focus of the talk will be the algebra H^{\infty}(\mathfrak{B}_d) of multipliers of the nc-RKHS on the unit ball obtained from the non-commutative Szego kernel. We will give a new proof for the fact that the non-commutative Szego kernel is completely Pick. Then we will consider subvarieties of \mathfrak{B}_d and quotients of H^{\infty}(\mathfrak{B}_d) arising as multipliers on those varieties. We are interested in determining when the multiplier algebras of two varieties are completely isometrically isomorphic. It is natural to conjecture that two such algebras are completely isometrically isomorphic if and only if there is an automorphism of the nc ball that maps one variety onto the other. We present several partial results in this direction