Piotr Pragacz

Symmetric polynomials and divided differences in formulas of intersection theory

The goal of the paper is two-fold. At first, we attempt to give a survey of some recent applications of symmetric polynomials and divided differences to intersection theory. We discuss: polynomials universally supported on degeneracy loci; some explicit formulas for the Chern and Segre classes of Schur bundles with applications to enumerative geometry; flag degeneracy loci; fundamental classes, diagonals and Gysin maps; intersection rings of G/P and formulas for isotropic degeneracy loci; numerically positive polynomials for ample vector bundles. Apart of surveyed results, the paper contains also some new results as well as some new proofs of earlier ones: how to compute the fundamental class of a subvariety from the class of the diagonal of the ambient space; how to compute the class of the relative diagonal using Gysin maps; a new formula for pushing forward Schurs Q- polynomials in Grassmannian bundles; a new formula for the total Chern class of a Schur bundle; another proof of Schuberts and Giambellis enumeration of complete quadrics; an operator proof of the Jacobi-Trudi formula; a Schur complex proof of the Giambelli-Thom-Porteous formula.

Ribbon Schur functions

Abstract We present a new determinantal expression for Schur functions. Previous expressions were due to Jacobi, Trudi, Giambelli and others (see [7]) and involved elementary symmetric functions or hook functions. We give, in Theorem 1.1, a decomposition of a Schur function into ribbon functions (also called skew hook functions, new functions by MacMahon, and MacMahon functions by others). We provide two different proofs of this result in Sections 2 and 3. In Section 2, we use Bazins formula for the minors of a general matrix, as we already did in [6], to decompose a skew Schur function into hooks. In Section 3, we show how to pass from hooks to ribbons and conversely. In Section 4, we generalize to skew Schur functions. In Section 5, we give some applications, and show how such constructions, in the case of staircase partitions, generalize the classical continued fraction for the tangent function due to Euler.

Complexes associated with trace and evaluation. Another approach to Lascoux's resolution

+ I/?l)th syzygy. Hence the length of a minimal free resolution of D, is (m - p + 1 )(n - 1). Although Lascoux described explicitly maps between the consecutive syzygies he was unable to prove that this leads to a complex. Proofs of his Lemmas 3.7.1 and 3.7.2 in [S] are incomplete. We illustrate essential dif- ficulties on the example of 1 x 1 minors. In this case the Lascoux’s con- struction gives the rth syzygy as C,,, = r LIF@ LxG. Lascoux describes the differential as being composed from the maps LA F@ LxG -+ L,FQ L,G for p c 1, J,u = I1(- 1. Each of those maps is specified up to scalar and he does not say how to adjust them in order to get a complex. Even in the case p = 1, when the rth syzygy equals A’(F@ G) isomorphism A’(F@ G) = Cll, = I L,F@ LxG is quite complicated and it is difficult to compute these scalars. Roberts in his preprint [ 123 gave more down-to-earth construction of a complex but he was unable to prove that it is a resolution of D, except in some special cases. The purpose of this paper is to overcome these difficulties by presenting still another approach. That allows us to get a complex and to prove easily that it is a minimal free resolution of D,. Here are some essential features of our method. The basic idea can be explained in case

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

Ideals generated by Pfaffians

Influenced by the structure theorem of Buchsbaum and Eisenbud for Gorenstein ideals of depth 3, [4], we study in this paper ideals generated by Pfaffians of given order of an alternating matrix. Let ,x7 be an n by n alternating matrix (i.e., xi9 = --xji for i < j and xii = 0) with entries in a commutative ring R with identity. One can associate with X an element Pf(-k-) of R called Pfaffian of X (see [I] or [2] for the definition). For n odd det S ~-= 0 and Pf(X) = 0; for n even det X is a square in R and Pf(X)z = det X. For a basis-free account and basic properties ofpfaffians we send interested readers to Chapter 2 of [4] or to [IO]. For a sequence i1 ,..., &, 1 < i r < n, the matrix obtained from X by omitting rows and columns with indices i1 ,..., ik is again alternating; we write Pf il*....ik(X) for its Pfaffian and call it the (n k)-order Pfaffian of X. We are interested in ideals Pf,,(X) generated by all the 2p-order Pfaffians of X, 0 ,( 2p < n. After some preliminaries in Section 1 we prove in Section 2 that the height and the depth of E-“&X) are bounded by the number

THE GEOMETRY OF T-VARIETIES

This survey paper is based on my IMPANGA lectures given in the Banach Center, Warsaw in January 2011. We study the moduli of holomorphic map germs from the complex line into complex compact manifolds with applications in global singularity theory and the theory of hyperbolic algebraic varieties.We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring.We introduce a variety

Schubert functors and Schubert polynomials

\hat{G}_2

Double Sylvester sums for subresultants and multi-Schur functions

parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety

S-function series

G_2

Jacobians of Symmetric Polynomials

, the adjoint variety of the simple Lie group