Anders Thorup

On a Jacobi-Trudi identity for supersymmetric polynomials

In the very late eighties a new identity for symmetric polynomials was discovered. In the form presented here the identity is a generalization of the Jacobi-Trudi identity. The latter identity expresses the Schur polynomials in a finite set of variables as a certain symmetrizing operator applied to monomials in the variables. The new identity involves two sets of variables. It expresses the super Schur polynomials as a certain symmetrizing operator applied to very simple polynomials in the two sets of variables. It is a classical result that the Schur polynomials are the characters of the polynomial representations of SL,. Hence the Jacobi-Trudi identity may be viewed as a character formula for SL,. This approach was generalized to other algebraic groups by H. Weyl in his character formula. The new identity was in fact discovered as a Weyl-type formula for the characters of polynomial representations of the Lie superalgebra sZ(m/n). From one side the formula was conjectured by J. van der Jeugt, J. W. B. Hughes, R. C. King, and J. Thierry-Mieg [J-H-K-T, p. 22911. On the other side the identity was communicated without proof by A. Serge’ev to the first author who gave a proof of its validity in [PI. The proof, though elementary, rested on the characterization of J. Stembridge of super Schur polynomials via a certain cancellation property; therefore, the proof in [P] was not self-contained. The aim of the present note is to give a self-contained and elementary proof of the new identity. The method used gives a simple insight in the space of supersymmetric polynomials. Byproducts of the proof are the

Weierstrass points and gap sequences for families of curves

The theory of Weierstrass points and gap sequences for linear series on smooth curves is generalized to smooth families of curves with geometrically irreducible fibers, and over an arbitrary base scheme.

These are the differentials of order

We answer P.-A. Meyer’s question “Qu’est ce qu’une différentielle d’ordre n?”. In fact, we present a general theory of higher order differentials based upon a construction of universal objects for higher order differentials. Applied to successive tangent spaces on a differentiable manifold, our theory gives the higher order differentials of Meyer as well as several new results on differentials on differentiable manifolds. In addition our approach gives a natural explanation of the quite mysterious multiplicative structure on higher order differentials observed by Meyer. Applied to iterations of the first order Kähler differentials our theory gives an algebra of higher order differentials for any smooth scheme. We also observe that much of the recent work on higher order osculation spaces of varieties fits well into the framework of our theory.

Minimal Huffman trees

Abstract. For a given set of real weights, Huffman trees minimize the weighted external path length. Over the years, several algorithms have been proposed for constructing Huffman trees that minimize additional natural cost functions such as the external path length, the variance and, more generally, the central moments. We show that all these cost functions are minimized on exactly the same class of Huffman trees, and we characterize the class: it consists of all Huffman trees of minimal level set. It follows that a Huffman tree minimizing one of the cost functions in fact minimizes all of them, and has the minimal level set; in particular, it has minimum height. We show that the unique Huffman tree produced by the simplest construction method, the bottom-merge algorithm of Schwartz, belongs to the class. Finally, we prove that several natural variants of Huffmans algorithm, that appear to be nondeterministic, in fact all lead to the single Huffman tree obtained by Schwartzs algorithm.

Parameter spaces for quadrics

The parameter spaces for quadrics are reviewed. In addition, an explicit formula for the number of quadrics tangent to given linear subspaces is presented. 1. Schubert’s problem. 1.1. One century ago, in 1894, Schubert considered the following problem: Let P be a projective space. Assume there is given in P a finite number of linear subspaces in general position, say m1 hyperplanes, m2 codimension-2 planes, and in general, mi codimension-i planes. Then, how many quadrics in P are tangent to the given linear subspaces? In Schubert’s problem, the quadrics are assumed to be non-singular. Assume P = P(E) where E is a vector space of rank r. Then a non-singular quadric in P corresponds to a regular symmetric r×r matrix up to multiplication by a scalar. The symmetric matrices form a vector space of rank ( r+1 2 ) . Therefore, the set of non-zero symmetric matrices up to multiplication by a scalar is parametrized by a projective space of dimension N := ( r+1 2 ) −1. In this P , the matrices with non-zero determinant form an open subset U . By construction, the points of U correspond to the non-singular quadrics in P , that is, U is a parameter space for the set of non-singular quadrics in P . The set of quadrics that are tangent to a given linear subspace of P form, in the parameter space U , a hypersurface. Therefore, in Schubert’s problem it is natural to require that the number ∑ imi of given linear subspaces is equal to the dimension N of the parameter space. Then the quadrics tangent to the given linear subspaces correspond in the parameter space to the points in the intersection of N hypersurfaces. It could be hoped that the intersection is finite; Schubert’s problem is then to count the number of points in the intersection. To solve the counting problem by enumerative techniques, a closed (or complete) parameter space is needed. By construction, the space U is an open (dense) subset of P . A naive completion of U is then to take P as its closure. Clearly, the boundary points of U in P correspond to the singular quadrics in P . However, we cannot expect 1991 Mathematics Subject Classification: 14N10; 14M15. Supported in part by the Danish Natural Science Research Council, grant 11–7428. The paper is in final form and no version of it will be published elsewhere.

Generalized Plücker Formulas

The classical Plucker formula for a plane curve was generalized by Teissier to the case of a hypersurface with isolated singularities and further by Kleiman to the case of an arbitrary n-dimensional projective variety V with isolated singularities. The formula relates the zeroth rank of V (the degree of the dual variety) to the Segre numbers of the conormal module and certain Buchsbaum—Rim multiplicities associated to the singular points of V. A second generalization was obtained by Pohl. It relates the (n-1)th rank of V to the first Chern class of a desingularization of V and the degree of the cuspidal divisor. We describe, for a projective variety V with arbitrary singularities, a natural class in the Chow group of the singular locus whose top dimensional part is given by Buchsbaum—Rim multiplicities, and we obtain generalizations of both formulas. The formulas are equations in the Chow group of V. They imply numerical formulas for all the ranks of V.

Slender domains and compact domains

Abstract We prove that a one-dimensional Noetherian domain is slender if and only if it is not a local complete ring. The latter condition for a general Noetherian domain characterizes the domains that are not algebraically compact. For a general Noetherian domain R we prove that R is algebraically compact if and only if R satisfies a condition slightly stronger than not being slender. In addition we enlarge considerably the number of classes of rings for which the question of slenderness can be answered. For instance we prove that any domain, not a field, essentially of finite type over a field is slender.

What Costs Are Minimized by Huffman trees

We characterize those functions on weighted trees that are minimized at Huffman trees and those that are minimized at trees with the same level sequence as a Huffman tree. An important tool is a set of inequalities between weights of subtrees shown to be characteristic for Huffman trees. A byproduct is an algorithm transforming an arbitrary weighted tree into a Huffman tree; for a given tree, the maximal number of steps that may be taken by this algorithm is a numerical measure of how far the tree is from being a Huffman tree.

How Did the Camera Move

Abstract Given 5 points in 3-space and two snapshots of these points, taken with a camera at two different positions. Then, in general, there are 10 possibilities for the second position of the camera relative to its first position. The result is well known. We prove it using the Thom–Porteous Formula. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Symmetric matrices with alternating blocks

A statement in algebraic geometry over fields of arbitrary characteristic follows from the existence of matrices with integer entries of the type mentioned in the title. It is shown how these matrices can be built from a finite number of small matrices. It is reported how these small matrices, of which the largest is a 25 by 25 matrix, were found using computer algebra systems.