Elliptic Stable Envelopes and Finite-dimensional Representations of Elliptic Quantum Group

Abstract

We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $s{l}_N$ on the Gelfand-Tsetlin basis of the tensor product of the $n$-vector representations. The result is described in a combinatorial way by using the partitions of $[1,n]$. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper[Konno17]. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we show a correspondence of the Gelfand-Tsetlin bases (resp. the standard bases) to the fixed point classes (resp. the stable classes) in the equivariant elliptic cohomology $E_T(X)$ of the cotangent bundle $X$ of the partial flag variety. As a result we obtain a geometric representation of the elliptic quantum group on $E_T(X)$.

48 pages, corrected formulas for quasi-periodicity in 3.4.4., to appear in Journal of Integrable Systems

References in corpus (3)

• Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology
• Dynamical R Matrices of Elliptic Quantum Groups and Connection Matrices for the q-KZ Equations
• Elliptic and K-theoretic stable envelopes and Newton polytopes

Cited by in corpus (1)

• 3d Mirror Symmetry and Elliptic Stable Envelopes