Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety

By Richárd Rimányi, V. Tarasov, A. Varchenko. 2014

66 citations recorded by the source corpus.

We consider the cotangent bundle T∗Fλ of a GLn partial flag variety, λ=(λ1,…,λN), |λ|=∑iλi=n, and the torus T=(C×)n+1 equivariant K-theory algebra KT(T∗Fλ). We introduce K-theoretic stable envelope maps Stabσ:⨁|λ|=nKT((T∗Fλ)T)→⨁|λ|=nKT(T∗Fλ), where σ∈Sn. Using these maps we define a quantum loop algebra action on ⨁|λ|=nKT(T∗Fλ). We describe the associated Bethe algebra Bq(KT(T∗Fλ)) by generators and relations in terms of a discrete Wronski map. We prove that the limiting Bethe algebra B∞(KT(T∗Fλ)), called the Gelfand–Zetlin algebra, coincides with the algebra of multiplication operators of the algebra KT(T∗Fλ). We conjecture that the Bethe algebra Bq(KT(T∗Fλ)) coincides with the algebra of quantum multiplication on KT(T∗Fλ) introduced by Givental (2000), Givental and Lee (2003). The stable envelope maps are defined with the help of Newton polygons of Laurent polynomials representing elements of KT(T∗Fλ) and with the help of the trigonometric weight functions introduced in Varchenko and Tarasov (1994), Tarasov and Varchenko (2013) to construct q-hypergeometric solutions of trigonometric qKZ equations. The paper has five appendices. In particular, in Appendix E we describe the Bethe algebra of the XXZ model by generators and relations.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e16466baa41299bc3e9a0fcca4debc7b50e858d5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells (2017, methodology): “This equality was proved earlier by Rimányi and Varchenko [48, Thm. 8.1 and Rmk. 8.2] (see also [59]), using interpolation properties of CSM classes stemming from Weber’s work [61], and the defining localization properties of the stable envelopes recalled (cf. §6.2 below).”
Defects and quantum Seiberg-Witten geometry (2014, background): “Recently we became aware of reference [31] where quantum K-rings of flag varieties were discussed.”