37 research outputs found
Equivariant Chern-Schwartz-MacPherson classes in partial flag varieties: interpolation and formulae
Thom series of contact singularities
Thom polynomials measure how global topology forces singularities. The power
of Thom polynomials predestine them to be a useful tool not only in
differential topology, but also in algebraic geometry (enumerative geometry,
moduli spaces) and algebraic combinatorics. The main obstacle of their
widespread application is that only a few, sporadic Thom polynomials have been
known explicitly. In this paper we develop a general method for calculating
Thom polynomials of contact singularities. Along the way, relations with the
equivariant geometry of (punctual, local) Hilbert schemes, and with iterated
residue identities are revealed
Elliptic and K-theoretic stable envelopes and Newton polytopes
In this paper we consider the cotangent bundles of partial flag varieties. We
construct the -theoretic stable envelopes for them and also define a version
of the elliptic stable envelopes. We expect that our elliptic stable envelopes
coincide with the elliptic stable envelopes defined by M. Aganagic and A.
Okounkov. We give formulas for the -theoretic stable envelopes and our
elliptic stable envelopes. We show that the -theoretic stable envelopes are
suitable limits of our elliptic stable envelopes. That phenomenon was predicted
by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms
of the elliptic and trigonometric weight functions which originally appeared in
the theory of integral representations of solutions of qKZ equations twenty
years ago. (More precisely, the elliptic weight functions had appeared earlier
only for the case.) We prove new properties of the trigonometric
weight functions. Namely, we consider certain evaluations of the trigonometric
weight functions, which are multivariable Laurent polynomials, and show that
the Newton polytopes of the evaluations are embedded in the Newton polytopes of
the corresponding diagonal evaluations. That property implies the fact that the
trigonometric weight functions project to the -theoretic stable envelopes.Comment: Latex, 37 pages; v.2: Appendix and Figure 1 added; v.3: missing shift
in Theorem 2.9 added and a proof of Theorem 2.9 adde
Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra
We interpret the equivariant cohomology algebra
H^*_{GL_n\times\C^*}(T^*F_\lambda;\C) of the cotangent bundle of a partial flag
variety F_\lambda parametrizing chains of subspaces 0=F_0\subset
F_1\subset\dots\subset F_N =\C^n, \dim F_i/F_{i-1}=\lambda_i, as the Yangian
Bethe algebra of the gl_N-weight subspace of a gl_N Yangian module. Under this
identification the dynamical connection of [TV1] turns into the quantum
connection of [BMO] and [MO]. As a result of this identification we describe
the algebra of quantum multiplication on H^*_{GL_n\times\C^*}(T^*F_\lambda;\C)
as the algebra of functions on fibers of a discrete Wronski map. In particular
this gives generators and relations of that algebra. This identification also
gives us hypergeometric solutions of the associated quantum differential
equation. That fact manifests the Landau-Ginzburg mirror symmetry for the
cotangent bundle of the flag variety.Comment: Latex, 45 pages, references added, Conjecture 7.10 is now Theorem
7.10, Theorem 7.13 adde
Quiver polynomials in iterated residue form
Degeneracy loci polynomials for quiver representations generalize several important polynomials in algebraic combinatorics. In this paper we give a nonconventional generating sequence description of these polynomials when the quiver is of Dynkin type
Elliptic dynamical quantum groups and equivariant elliptic cohomology
We define an elliptic version of the stable envelope of Maulik and Okounkov for the equivariant elliptic cohomology of cotangent bundles of Grassmannians. It is a version of the construction proposed by Aganagic and Okounkov and is based on weight functions and shuffle products. We construct an action of the dynamical elliptic quantum group associated with on the equivariant elliptic cohomology of the union of cotangent bundles of Grassmannians. The generators of the elliptic quantum groups act as difference operators on sections of admissible bundles, a notion introduced in this paper
Cohomology of a flag variety as a Bethe algebra
We interpret the equivariant cohomology H∗GLn (Fλ,C) of a partial flag variety Fλ parametrizing chains of subspaces 0 = F0 ⊂ F1 ⊂ · · · ⊂ FN = Cn, dim Fi/Fi−1 = λi, as the Bethe algebra B∞(V±λ) of the glN-weight subspace V±λ of a glN[t]-module V±
Coincident root loci of binary forms
Coincident root loci are subvarieties of SdC2 — the space of binary forms of degree d — labelled by partitions of d. Given a partition λ, let Xλ be the set of forms with root multiplicity corresponding to λ. There is a natural action of GL2(C) on SdC2 and the coincident root loci are invariant under this action. We calculate their equivariant Poincare duals generalizing formulas of Hilbert and Kirwan. In the second part we apply these results to present the cohomology ring of the corresponding moduli spaces (in the GIT sense) by geometrically defined relations
Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra
We interpret the equivariant cohomology algebra H∗GLn×C*(T*Fλ;C) of the cotangent bundle of a partial flag variety Fλ parametrizing chains of subspaces 0 = F0 ⊂ F1 ⊂ · · · ⊂ FN = Cn, dim Fi/Fi−1 = λi, as the Yangian Bethe algebra B∞( 1DV−λ) of the glN-weight subspace 1/DV−λ of a Y (glN)-module 1/DV−. Under this identification the dynamical connection of [TV1] turns into the quantum connection of [BMO] and [MO]. As a result of this identification we describe the algebra of quantum multiplication on H∗GLn×C*(T ∗Fλ;C) as the algebra of functions on fibers of a discrete Wronski map. In particular this gives generators and relations of that algebra. This identification also gives us hypergeometric solutions of the associated quantum differential equation. That fact manifests the Landau-Ginzburg mirror symmetry for the cotangent bundle of the flag variety