Jerzy Weyman

Schur Functors and Schur Complexes

The study of Schur functors has a relatively long history. Its main impetus derived from representation theory, originally in characteristic zero. Over the years, however, with the development of modular representations, and algebraic geometry over fields of positive characteristic, the need for a theory of universal polynomial functors increased and, since the mid-1960s, approaches to a characteristic-free treatment of Schur functors have been developing (see, for instance, the recent book of Green [ 1 l] in which the treatments by Carter and Lusztig [6], Higman [12], and Towber [20], among others, are discussed). Our own interest in such a treatment was awakened by the work of Lascoux [ 141 on resolutions of determinantal ideals. Although his thesis treated only the characteristic zero case, it suggested that a general and elementary theory of Schur functors could be developed using only the rudiments of multilinear algebra (involving the Hopf algebra structures of the symmetric, exterior, and divided power algebras). Moreover, this elementary development admitted of a natural generalization to the idea of Schur complexes, whose usefulness was demonstrated, for instance, in our construction of a universal minimal

Resultants and Chow forms via exterior syzygies

Let W be a vector space of dimension n+ 1 over a field K. The Chow divisor of a k-dimensional variety X in P = P(W ) is the hypersurface, in the Grassmannian Gk+1 of planes of codimension k+1 in P, whose points (over the algebraic closure of K) are the planes that meet X . The Chow form of X is the defining equation of the Chow divisor. For example, the resultant of k+ 1 forms of degree e in k + 1 variables is the Chow form of P embedded by the e-th Veronese mapping in P with n = ( k+e k ) − 1. More generally, the Chow divisor of a k-cycle ∑ i ni[Vi] on projective space is defined to be ∑ i niDi, where Di is the Chow divisor of Vi. The Chow divisor of a sheaf F with k-dimensional support is the Chow divisor of the associated k-cycle of F . In this paper we will give a new expression for the Chow divisor and apply it to give explicit formulas in many new cases. Starting with a sheaf F on P, we use exterior algebra methods to define a canonical and effectively computable Chow complex of F on each Grassmannian of planes in P. If F has k-dimensional support, we show that the Chow form of F is the determinant of the Chow complex of F on the Grassmannian of planes of codimension k + 1. The Beilinson monad of F [Beilinson 1978] is the Chow complex of F on the Grassmannian of 0-planes (that is, on P itself.) In particular, we are able to give explicit determinantal and Pfaffian formulas for resultants in some cases where no polynomial formulas were known. For example, the Horrocks-Mumford bundle gives rise to polynomial formulas for the resultant of five homogeneous forms of degrees 4, 6 or 8 in five variables. The easiest of our new formulas to write down is for the resultant of 3 quadratic forms in three variables, the Chow form of the Veronese surface in P. Using the tangent bundle of P, conclude that it can be written in “Bezout form” (described below) as the Pfaffian of the matrix  0 [245] [345] [135] [045] [035] [145] [235]

Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients

Σ(Q,α) is defined in the space of all weights by one homogeneous linear equation and by a finite set of homogeneous linear inequalities. In particular the set Σ(Q,α) is saturated, i.e., if nσ ∈ Σ(Q,α), then also σ ∈ Σ(Q,α). These results, when applied to a special quiver Q = Tn,n,n and to a special dimension vector, show that the GLn-module Vλ appears in Vμ ⊗ Vν if and only if the partitions λ, μ and ν satisfy an explicit set of inequalities. This gives new proofs of the results of Klyachko ([7, 3]) and Knutson and Tao ([8]). The proof is based on another general result about semi-invariants of quivers (Theorem 1). In the paper [10], Schofield defined a semi-invariant cW for each indecomposable representation W of Q. We show that the semi-invariants of this type span each weight space in SI(Q,α). This seems to be a fundamental fact, connecting semi-invariants and modules in a direct way. Given this fact, the results on sets of weights follow at once from the results in another paper of Schofield [11].

Formalizing classes of information fusion systems

This paper provides an outline of a formalization of classes of information fusion systems in terms of category theory and formal languages. The formalization captures both the inputs/outputs of a fusion system and the fusion processing algorithms. The paper also introduces a notion of subclass, which is used to compare classes of fusion systems, whether they are different or one is a special case of another. Two examples of classes of fusion systems formalized in the paper are data fusion and decision fusion; decision fusion is shown to be a subclass of data fusion. A number of other classes of fusion systems are defined. The formalization is extended by adding the notion of measure of effectiveness, which is then used to prove that one of the classes (so called overlapping system) is at least as efficient as a single-source system. And finally it is shown how data association can be formalized in this framework. While at first the formalization could be used by information fusion scientists to formally define various types of fusion systems and then to prove theorems about properties of such systems, it is expected that it should lead to the development of tools that could be used by software engineers to formally derive designs of fusion systems.

On the Littlewood-Richardson polynomials

We prove the equivalence of several descriptions of generators of rings of semiinvariants of quivers, due to Domokos and Zubkov, Schofield and van den Bergh, and our earlier work. We also show that the dimensions of semi-invariants of weights nσ depend polynomially on n.  2002 Elsevier Science (USA). All rights reserved.

Resolutions of determinantal ideals: The submaximal minors

For many years there has been considerable interest in finding a resolution of the ideal generated by the minors of order p of a generic m x n matrix. To put the problem more precisely, suppose R, is a commutative ring and X, are variables with 1 < i < m and 1 < j Q n. If we let R = R, [X,] be the polynomial ring over R,, then we have the “generic” matrix (X,) and we may form the ideal ZP in R generated by the p X p minors of this matrix. The problem, then, is to find an explicit free resolution of the ideal ZP over the ring R. It was proved by Eagon and Hochster [lo] that R/I, has a resolution of length (m-p + l)(n -p + l), but their proof consisted in showing that the ideal ZP is perfect; it did not provide a construction of the resolution. In fact, it is not known whether the Betti numbers of the ideal ZP depend on the characteristic of the ground ring R,. In [ 121, Lascoux succeeded in giving an explicit resolution provided that the ground ring R, contained the field, Q, of rational numbers. His construction rests heavily on the theory of Schur functors and the fact that in characteristic zero the Schur functors are the irreducible representations of the general linear group. Over the integers, however, the construction breaks down despite the fact that one can define the Schur functors over an arbitrary commutative ring (see [ 1,2, 14, 151 for various constructions of Schur functors). In analyzing the work of Lascoux and its subsequent reworking by Nielsen [ 131, some basic facts seemed to clamor for attention. One was that within a resolution of R/Z,, there appeared to be two types of boundary maps: one of degree 1 and one of degree p. The maps of degree 1 were maps between sums of Schur functors of fixed Durfee square k (see Section 2 for definitions), while the maps of degree p were from sums of Schur functors of

On the Canonical Decomposition of Quiver Representations

Kac introduced the notion of the canonical decomposition for a dimension vector of a quiver. Here we will give an efficient algorithm to compute the canonical decomposition. Our study of the canonical decomposition for quivers with three vertices gives us fractal-like pictures.

Multiplicities of points on a Schubert variety in a minuscule GP

In this paper we prove the results announced in [13]. Let G be a semi- simple, simply connected algebraic group defined over an algebraically closed field k. Let T be a maximal torus, B a Bore1 subgroup, B 3 T. Let W be the Weyl group of G. Let R (resp. R+ ) be the set of roots (resp. positive roots) relative to T (resp. B). Let S be the set of simple roots in R+. Let P be a maximal parabolic subgroup in G with associated fundamental weight w. Let W, be the Weyl group of P, and Wp be the set of minimal representatives of W/W,. For w E Wp, let e(w) be the point and X(w) the Schubert variety in G/P associated to w. In this paper we deter- mine the multiplicity m,(w) of X(w) at e(z), where e(z) E X(w), for all minuscule P’s and also for P = Pgn, G being of type C, (here Pun denotes the maximal parabolic subgroup obtained by omitting a,). The determina- tion of m,(w) is done as follows. Let L be the ample generator of Pic(G/P). A basis has been constructed for @(X(w), L”) in terms of standard monomials on X(w) (cf. [ 16, 11 I). Let U; be the unipotent subgroup of G generated by U-,, /?ET(R+ - Rp+) (here R, denotes the set of roots of P and U, denotes the unipotent subgroup of G, associated to tl E R). Then U; e(r) gives an affme neighborhood of e(z) in G/P. Let A, be the affine algebra of U, e(z) and A,.. = A,/&, where & is the ideal of elements of A, that vanish on X(w) n U; e(z). Let M,,, be the maximal ideal in A, H, corresponding to e(r). Then using the results of [ 16, 111, we obtain a basis of M;, &f:,+,,’ . This enables us to obtain an inductive formula for F,,,, the Hilbert polynomial of X(w) at e(T) (cf. Corollaries 3.8 and 4.11), and also express m, (w) in terms of m, (w’)‘s, X(w’)‘s being the Schubert divisors in X(w) such that e(T)E X(w’) (cf. Theorems 3.7 and 4.10). Using this we

Graded Characters of Modules Supported in the Closure of a Nilpotent Conjugacy Class

This is a combinatorial study of the Poincare polynomials of isotypic components of a natural family of graded GL(n)-modules supported in the closure of a nilpotent conjugacy class. These polynomials generalize the Kostka?Foulkes polynomials and are q -analogues of Littlewood?Richardson coefficients. The coefficients of two-column Macdonald?Kostka polynomials also occur as a special case. It is conjectured that these q -analogues are the generating function of so-called catabolizable tableaux with the charge statistic of Lascoux and Schutzenberger. A general approach for a proof is given, and is completed in certain special cases including the Kostka?Foulkes case. Catabolizable tableaux are used to prove a characterization of Lascoux and Schutzenberger for the image of the tableaux of a given content under the standardization map that preserves the cyclage poset.

The algebras of semi-invariants of quivers

We show that the algebras of semi-invariants of a finite connected quiverQ are complete intersections if and only ifQ is of Dynkin or Euclidean type. Moreover, we give a uniform description of the algebras of semi-invariants of Euclidean quivers.