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Abstract: A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haar-distributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices that, when used for conjugation, lead to freeness. We do so by first proving a general result asserting “asymptotic liberation” under quite mild conditions, and then we explain how to specialize these general results in a striking way by exploiting Hadamard matrices. In particular, we recover and generalize results of the second-named author and of Tulino, Caire, Shamai and Verdú. [Copyright &y& Elsevier]

Abstract: Let X be a nonsingular projective curve of genus one defined over an algebraically closed field of characteristic 0. Let D be a divisor of X of degree and let O be a (closed) point of X. As is well known, there exists a unique morphism such that if and only if the divisor is principal. Our main result is a simple explicit description of the map in terms of Wronskians and certain Wronskian-like determinants lacunary in the sense that derivatives of some orders are skipped. Further, for we interpret our main result as a syzygy from classical invariant theory, thus reconciling our work with a circle of ideas treated in two papers by Weil and a recent paper by An, Kim, Marshall, Marshall, McCallum and Perlis. [Copyright &y& Elsevier]

The expansion in Schur functions of the product ∏i is well known. It is more or less equivalent to Weyl''s identity for root-systems of type Dn. In this paper we obtain the expansion in Schur functions of the product ∏i0aℓxixj(xiℓ-xjℓ)/(xi-xj)), thus generalizing Weyl''s identity. We obtain this result by systematic calculation in fermionic Fock space. But familiarity with the latter is not a prerequisite for reading this paper. We develop from scratch the modest amount of theory that we need in elementary and purely algebraic fashion, taking pains to integrate the theory with classical symmetric function theory. The tools developed in this paper ought to have many further applications, e.g., to random matrix theory and to computational abelian function theory. [Copyright &y& Elsevier]

The abeliant is a polynomial rule which to each n×n by n+2 array with entries in a commutative ring with unit associates an n×n matrix with entries in the same ring. The theory of abeliants, first introduced in an earlier paper of the author, is simplified and extended here. Now let J be the Jacobian of a nonsingular projective algebraic curve defined over an algebraically closed field. With the aid of the theory of abeliants we obtain explicit defining equations for J and its group law. [Copyright &y& Elsevier]