arXiv subjects

R. Rimányi

Publications and source records attributed to R. Rimányi.

4 recordsLinked to original sources

arXiv2018

Elliptic and K-theoretic stable envelopes and Newton polytopes

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $K$-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 $K$-theoretic stable envelopes and our elliptic stable envelopes. We show that the $K$-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 $\frak{gl}_2$ 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 $K$-theoretic stable envelopes.

math.AG↗

arXiv2012

Extended Joseph polynomials, quantized conformal blocks, and a q-Selberg type integral

We consider the tensor power $V=(C^N)^{\otimes n}$ of the vector representation of $gl_N$ and its weight decomposition $V=\oplus_{λ=(λ_1,...,λ_N)}V[λ]$. For $λ= (λ_1 \geq ... \geq λ_N)$, the trivial bundle $V[λ]\times \C^n\to\C^n$ has a subbundle of q-conformal blocks at level l, where $l = λ_1-λ_N$ if $λ_1-λ_N> 0$ and l=1 if $λ_1-λ_N=0$. We construct a polynomial section $I_λ(z_1,...,z_n,h)$ of the subbundle. The section is the main object of the paper. We identify the section with the generating function $J_λ(z_1,...,z_n,h)$ of the extended Joseph polynomials of orbital varieties, defined in [DFZJ05,KZJ09]. For l=1, we show that the subbundle of q-conformal blocks has rank 1 and $I_λ(z_1,...,z_n,h)$ is flat with respect to the quantum Knizhnik-Zamolodchikov discrete connection. For N=2 and l=1, we represent our polynomial as a multidimensional q-hypergeometric integral and obtain a q-Selberg type identity, which says that the integral is an explicit polynomial.

math-ph↗

arXiv2011

Conformal blocks and equivariant cohomology

We show that the conformal blocks constructed in the previous article by the first and the third author may be described as certain integrals in equivariant cohomology. When the bundles of conformal blocks have rank one, this construction may be compared with the old integral formulas of the second and the third author. The proportionality coefficients are some Selberg type integrals which are computed. Finally, a geometric construction of the tensor products of vector representations of the Lie algebra $\frak{gl}(m)$ is proposed.

math-ph↗

arXiv2010

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.

math.AG↗