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

Abstract

We consider the cotangent bundle $T^{\ast }{F}_{\lambda }$ of a $G{L}_n$ partial flag variety, $\lambda =({\lambda }_1,...,{\lambda }_N)$, $|\lambda |={\sum }_i{\lambda }_i=n$, and the torus $T=(\C^\times)^{n+1}$ equivariant K-theory algebra $K_T(T^{\ast }{F}_{\lambda })$. We introduce K-theoretic stable envelope maps $\Stab_σ: \oplus_{|λ|=n} K_T((T^*F_λ)^T)\to\oplus_{|λ|=n}K_T(T^*F_λ)$, where $\sigma \in S_n$. Using these maps we define a quantum loop algebra action on ${\oplus }_{|\lambda |=n}{K}_T(T^{\ast }{F}_{\lambda })$. We describe the associated Bethe algebra $B^q(K_T(T^{\ast }{F}_{\lambda }))$ by generators and relations in terms of a discrete Wronski map. We prove that the limiting Bethe algebra $B^q(K_T(T^{\ast }{F}_{\lambda }))$, called the Gelfand-Zetlin algebra, coincides with the algebra of multiplication operators of the algebra $K_T(T^{\ast }{F}_{\lambda })$. We conjecture that the Bethe algebra $B^q(K_T(T^{\ast }{F}_{\lambda }))$ coincides with the algebra of quantum multiplication on $K_T(T^{\ast }{F}_{\lambda })$ introduced by Givental and Lee. The stable envelope maps are defined with the help of Newton polygons of Laurent polynomials representing elements of $K_T(T^{\ast }{F}_{\lambda })$ and with the help of the trigonometric weight functions introduced in [TV1, TV3] to construct q-hypergeometric solutions of trigonometric qKZ equations. The paper has five appendices. In particular, in Appendix 5 we describe the Bethe algebra of the XXZ model by generators and relations.

Latex, 56 pages, in the new Appendix 5 the Bethe algebra of the XXZ model is described by generators and relations

References in corpus (9)

• Quantum Groups and Quantum Cohomology
• Geometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras
• Combinatorial Formulae for Nested Bethe Vectors
• Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra
• Quantum cohomology of partial flag manifolds
• Partial flag varieties, stable envelopes and weight functions
• Grothendieck classes of quiver cycles as iterated residues
• Spaces of quasi-exponentials and representations of the Yangian Y(gl_N)
• An iterated residue perspective on stable Grothendieck polynomials

Cited by in corpus (27)

• Defects and Quantum Seiberg-Witten Geometry
• Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells
• Motivic Chern classes and K-theoretic stable envelopes
• Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology
• Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K theory
• $(SL(N),q)$-opers, the $q$-Langlands correspondence, and quantum/classical duality
• On the K-theory stable bases of the Springer resolution
• Elliptic and K-theoretic stable envelopes and Newton polytopes
• New quiver-like varieties and Lie superalgebras
• Quantum K-theory of grassmannians and non-abelian localization
• Bow varieties---geometry, combinatorics, characteristic classes
• Landau-Ginzburg mirror, quantum differential equations and qKZ difference equations for a partial flag variety
• Wall-crossings and a categorification of $K$-theory stable bases of the Springer resolution
• From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes
• Perfect Integrability and Gaudin Models
• Topologically Twisted SUSY Gauge Theory, Gauge-Bethe Correspondence and Quantum Cohomology
• Higher order Hamiltonians for the trigonometric Gaudin model
• Grassmann-Grassmann conormal varieties, integrability, and plane partitions
• Elliptic canonical bases for toric hyper-Kahler manifolds
• Higher rank elliptic partition functions and multisymmetric elliptic functions
• A class of partition functions associated with $E_{\tau ,\eta }(g{l}_3)$ by Izergin-Korepin analysis
• Loop Models and $K$-Theory
• Stable bases of the Springer resolution and representation theory
• Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical $R$-Matrices for Superspin Chains from the Bethe/Gauge Correspondence
• Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes
• Explicit Formulas for h-Deformed Structure Constants of Grassmannians
• Integrable models and $K$-theoretic pushforward of Grothendieck classes