Enrich
We study the combinatorial properties of vexillary signed permutations, which are signed analogues of the vexillary permutations first considered by Lascoux and Schutzenberger. We give several equivalent characterizations of vexillary signed permutat
ions, including descriptions in terms of essential sets and pattern avoidance, and we relate them to the vexillary elements introduced by Billey and Lam.
In previous work, we employed a geometric method of Kazarian to prove Pfaffian formulas for a certain class of degeneracy loci in types B, C, and D. Here we refine that approach to obtain formulas for more general loci, including those coming from al
l isotropic Grassmannians. In these cases, the formulas recover the remarkable theta- and eta-polynomials of Buch, Kresch, Tamvakis, and Wilson. The streamlined geometric approch yields simple and direct proofs, which proceed in parallel for all four classical types. In an appendix, we develop some foundational algebra and prove several Pfaffian identities. Another appendix establishes a basic formula for classes in quadric bundles.
We compute the Picard group of the moduli stack of elliptic curves and its canonical compactification over general base schemes.
This is an expository lecture, for the Abel bicentennial (Oslo, 2002), describing some recent work on the (small) quantum cohomology ring of Grassmannians and other homogeneous varieties.
Answering a question raised by S. Friedland, we show that the possible eigenvalues of Hermitian matrices (or compact operators) A, B, and C with C <= A + B are given by the same inequalities as in Klyachkos theorem for the case where C = A + B, excep
t that the equality corresponding to tr(C) = tr(A) + tr(B) is replaced by the inequality corresponding to tr(C) <= tr(A) + tr(B). The possible types of finitely generated torsion modules A, B, and C over a discrete valuation ring such that there is an exact sequence B -> C -> A are characterized by the same inequalities.
