arXiv subjects

R. Rimanyi

Publications and source records attributed to R. Rimanyi.

23 records · Page 2Linked to original sources

arXiv2006

A formula for non-equioriented quiver orbits of type A

We prove a positive combinatorial formula for the equivariant class of an orbit closure in the space of representations of an arbitrary quiver of type $A$. Our formula expresses this class as a sum of products of Schubert polynomials indexed by a generalization of the minimal lace diagrams of Knutson, Miller, and Shimozono. The proof is based on the interpolation method of Fehér and Rimányi. We also conjecture a more general formula for the equivariant Grothendieck class of an orbit closure.

math.AG↗

arXiv2005

Combinatorics of rational functions and Poincare-Birkhoff-Witt expansions of the canonical U(n-)-valued differential form

We study the canonical U(n-)-valued differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of KZ-type differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie albegras A_r, B_r, C_r, D_r in a conviniently chosen Poincare-Birkhoff-Witt basis. As a byproduct we obtain results on the combinatorics of rational functions, namely non-trivial identities are proved between certain rational functions with partial symmetries.

math.CO↗

arXiv2005

Poincare-Birkhoff-Witt expansions of the canonical elliptic differential form

We study the canonical U(\n)-valued elliptic differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of elliptic KZ differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie algebras A_r, B_r, C_r, D_r in a conveniently chosen Poincare-Birkhoff-Witt basis. As an application we give a new formula for eigenfunctions of Hamiltonians of the Calogero-Moser model.

math.RT↗

arXiv2003

Coincident root loci of binary forms

Coincident root loci are subvarieties of $S^d(C^2)$--the space of binary forms of degree $d$--labelled by partitions of $d$. Given a partition $λ$, let $X_λ$ be the set of forms with root multiplicity corresponding to $λ$. There is a natural action of $GL_2(C)$ on $S^d(C^2)$ and the coincident root loci are invariant under this action. We calculate their equivariant Poincaré 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 (stable points$/G$, semistable points$/G$, link of the singularity) using geometrically defined relations.

math.AG↗

arXiv2003

Positivity of quiver coefficients through Thom polynomials

Let r be an orbit of the quiver representation of type A_n (equioriented case). In this paper we study the Poincare dual of the closure of r (a.c.a. Thom polynomial/degeneracy loci formula) in equivariant cohomology. Using general Thom polynomial theory we prove the component formula and its stable version of [Knutson-Miller-Shimozono] as well as the positivity of the quiver coefficients of [Buch-Fulton]. We also show how the component formula follows from Grobner degeneration easily. In relation with the K-theory counterpart of the formula we give a new description of KMS factorizations based on transformations of lace diagrams.

math.AG↗