arXiv subjects

Richard Rimanyi

Publications and source records attributed to Richard Rimanyi.

24 records · Page 2Linked to original sources

arXiv2018

Residues, Grothendieck polynomials and K-theoretic Thom polynomials

Grothendieck polynomials were introduced by Lascoux and Schützenberger, and they play an important role in K-theoretic Schubert calculus. In this paper, we give a new definition of double stable Grothendieck polynomials based on an iterated residue operation. We illustrate the power of our definition by calculating the Grothendieck expansion of K-theoretic Thom polynomials of $A_2$ singularities. We present the expansion in two versions: one displays its expected stabilization, while the other displays its expected finiteness property.

math.AT↗

arXiv2018

Quadric rank loci on moduli of curves and K3 surfaces

Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many applications to moduli theory: (i) we find a simple proof of Borcherds' result that the Hodge class on the moduli space of polarized K3 surfaces of fixed genus is of Noether-Lefschetz type, (ii) we construct an explicit canonical divisor on the Hurwitz space parametrizing degree k covers of the projective line from curves of genus 2k-1, (iii) we provide a closed formula for the Petri divisor on the moduli space of curves consisting of canonical curves which lie on a rank 3 quadric and (iv) construct myriads of effective divisors of small slope on M_g.

math.AG↗

arXiv2018

Motivic characteristic classes in cohomological Hall algebras

The equivariant Chern-Schwartz-MacPherson (CSM) class and the equivariant Motivic Chern (MC) class are important characteristic classes of singular varieties in cohomology and K theory---and their theory overlaps with the theory of Okounkov's stable envelopes. We study CSM and MC classes for the orbits of Dynkin quiver representations. We show that the problem of computing the CSM and MC classes of all these orbits can be reduced to some basic classes $c^o_β$, $C^o_β$ parameterized by positive roots $β$. We prove an identity in a deformed version of Kontsevich-Soibelman's Cohomological (and K-theoretic) Hall Algebra (CoHA, KHA), namely, that a product of exponentials of $c^o_β$ (or $C^o_β$) classes formally depending on a stability function Z, does not depend on Z. This identity---which encodes infinitely many identities among rational functions in growing number of variables---has the structure of Donaldson-Thomas type quantum dilogarithm identities. Using a wall-crossing argument we present the $c^o_β$, $C^o_β$ classes as certain commutators in the CoHA, KHA.

math.AG↗

arXiv2016

Partition Identities and Quiver Representations

We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy's Durfee square identity to multipartitions. We then use this result to give a new proof of M. Reineke's identity in the case of quivers $Q$ of Dynkin type $A$ of arbitrary orientation. Our identity is stated in terms of the lacing diagrams of S. Abeasis - A. Del Fra, which parameterize orbits of the representation space of $Q$ for a fixed dimension vector.

math.CO↗

arXiv2013

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.

math.AG↗

arXiv2003

Specializations of Grothendieck polynomials

We prove a formula for double Schubert and Grothendieck polynomials specialized to two rearrangements of the same set of variables. Our formula generalizes the usual formulas for Schubert and Grothendieck polynomials in terms of RC-graphs, and it gives immediate proofs of many other important properties of these polynomials.

math.AG↗