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representations of a Schur cover of the corresponding group of algebraic automorphisms.Comment: This paper has been withdrawn by one of the authors, since it requires more wor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 $X \subset \mathbb{P}^d$ of an arbitrary codimension $\ell$ with respect to a real $\ell - 1$-dimensional linear subspace $V \subset \mathbb{P}^d$ 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 ($\ell=1$), 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 $\mathbb{P}^d$ hyperbolic with respect to some real $d-2$-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