Jacobians of Symmetric Polynomials

Annals of Combinatorics - We give the Jacobian of any family of complete symmetric functions, or of power sums, in a finite number of variables.     

Formulas for Lagranigian and orthogonal degeneracy loci; \widetilde Q-polynomial approach

The main goal of the paper is to give explicit formulas for the fundamental classes of Schubert subschemes in Lagrangian and orthogonal Grassmannians of maximal isotropic subbundles as well as some globalizations of them. The used geometric tools overlap appropriate desingularizations of such Schubert subschemes and Gysin maps for such Grassmannian bundles. The main algebraic tools are provided by the families of and -polynomials introduced and investigated in the present paper. The key technical result of the paper is the computation of the class of the (relative) diagonal in isotropic Grassmannian bundles based on the orthogonality property of and polynomials. Some relationships with quaternionic Schubert varieties and Schubert polynomials for classical groups are also discussed.     

A note on the Chow groups of projective determinantal varieties; an appendix to the paper by Grzegorz Kapustka and Michal Kapustka on “A cascade of determinantal Calabi‐Yau threefolds”

We investigate the Chow groups of projective determinantal varieties and those of their strata of matrices of fixed rank, using Chern class computations (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bezoutians, Euclidean Algorithm, and Orthogonal Polynomials

We prove a quadratic expression for the Bezoutian of two univariate polynomials in terms of the remainders for the Euclidean algorithm. In case of two polynomials of the same degree, or of consecutive degrees, this allows us to interpret their Bezoutian as the Christoffel- Darboux kernel for a finite family of orthogonal polynomials, arising from the Euclidean algorithm. We give orthogonality properties of remainders, and reproducing properties of Bezoutians. Received December 13, 2004     

Schubert functors and Schubert polynomials

We construct a family of functors assigning an R-module to a flag of R-modules, where R is a commutative ring. As particular instances, we get flagged Schur functors and Schubert functors, the latter family being indexed by permutations. We identify Schubert functors for vexillary permutations with some flagged Schur functors, thus establishing a functorial analogue of a theorem of Lascoux and Schützenberger from C. R. Acad. Sci. Paris Sér. I Math. 294 (1982) 447 and of Wachs from J. Combin. Theory Ser. A 40 (1985) 276. Over an infinite field, we study the trace of a Schubert module, which is a cyclic module over a Borel subgroup B, restricted to the maximal torus. The main result of the paper says that this trace is equal to the corresponding Schubert polynomial of Lascoux and Schützenberger (C. R. Acad. Sci. Paris Sér. I Math. 294 (1982) 447). We also investigate filtrations of B-modules associated with the Monk formula (Proc. London Math. Soc. 9 (1959) 253) and transition formula from Lett. Math. Phys. 10 (1985) 111.