Giandomenico Boffi

Characteristic-free decomposition of skew Schur functors

This note is a generalization of our thesis, Brandeis University 1984, whose material can be found in [2]. Our aim is to show that by the methods presented there, one can produce a characteristic-free filtration of the skew Schur functor L,, F, whose associated graded module is the direct sum of Schur modules L,F one would expect from the characteristic zero theory. As in [2], notations and facts are freely borrowed from [l].

On the resolutions of the powers of the pfaffian ideal

(X is a generic skew-symmetric matrix), and we let S stand for the polynomial ring R[X] (=R[Z,]). The pfaffians of the 2k-order principal submatrices of X generate an ideal of S, the “pfalfian ideal,” denoted by I. It is well known that I is a generically perfect Gorenstein prime ideal of grade 3, in fact a prototype of all the ideals of this kind (cf. [B-E, 31). Hence, I has a finite free S-resolution of length 3, a resolution which looks the same regardless of the ring R.

Computer algebra for fingerprint matching

We show in this paper how some algebraic methods can be used for fingerprint matching. The described technique is able to compute the score of a match also when the template and test fingerprints have been not correctly acquired. In particular, the match is independent of translations, rotations and scaling transformations of the template. The technique is also able to compute a match score when part of the fingerprint image is incorrect or missed. The algorithm is being implemented in CoCoA, a computer algebra system for doing computations in Commutative Algebra.

Lexicographic Gröbner bases for transportation problems of format r×3×3

By means of suitable sequences of graphs, we describe the reduced lexicographic Grobner basis of the toric ideal associated with the 3-dimensional transportation problem of format rx3x3 (r any integer > 1). In particular, we prove that the bases for r=2,3,4,5 determine all others.

Homotopy equivalence of two families of complexes

An explicit homotopy equivalence is established between two families of complexes, both of which generalize the classical Koszul complex.

Border bases for lattice ideals

Abstract The main ingredient to construct an O -border basis of an ideal I ⊆ K [ x 1 , … , x n ] is the order ideal O , which is a basis of the K-vector space K [ x 1 , … , x n ] / I . In this paper we give a procedure to find all the possible order ideals associated with a lattice ideal I M (where M is a lattice of Z n ). The construction can be applied to ideals of any dimension (not only zero-dimensional) and shows that the possible order ideals are always in a finite number. For lattice ideals of positive dimension we also show that, although a border basis is infinite, it can be defined in finite terms. Furthermore we give an example which proves that not all border bases of a lattice ideal come from Grobner bases. Finally, we give a complete and explicit description of all the border bases for ideals I M in case M is a 2-dimensional lattice contained in Z 2 .

On the Littlewood-Richardson rule for almost skew-shapes

We describe combinatorially the coefficients occurring in the irreducible decomposition of the Weyl module associated with an almost skewshape belonging to the family J. The proof uses the fundamental exact sequence for almost skew-shapes to initiate an inductive procedure which ultimately reduces to the classical Littlewood-Richardson rule for skew partitions.

ON THE COORDINATE RING OF PAIRS OF ALTERNATING MATRICES WITH PRODUCT ZERO

Given an integer n ≥ 2, let X = (xi j ) and Y = (yi j ) be two generic alternating n × n matrices over a commutative ring k. Denote by k[X, Y ] the polynomial ring with indeterminates the entries of X and Y . Moreover denote by I1(XY ) the ideal generated by the entries of the product of X and Y . The ring k[X, Y ]/I1(XY ) is the coordinate ring of the variety of pairs (U, V ) of alternating n × n matrices with entries in k, such that U V = 0. In this note we give a k-basis of that coordinate ring, under the assumption that n! is a unit in k. We use some ideas of De Concini, Procesi and Strickland (1–4), ideas which have been also applied in (5–7), papers beneficial to us.