Abstract
We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $sl_N$ 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 $E_T(X)$.
Highlights
It has long been conjectured that there is a parallelism between the infinite dimensional algebras and cohomology, K-theory, and elliptic cohomology [17, 19]
In [16,39,40], finite-dimensional representations of symmetrizable Kac-Moody algebras g were constructed in terms of homology groups of quiver varieties
Rimanyi, Tarasov and Varchenko found an identification of rational weight functions with stable envelopes for torus-equivariant cohomology of the partial flag variety T âFλ [21] and extended this to the trigonometric ones for the equivariant K-theory [44]
Summary
It has long been conjectured that there is a parallelism between the infinite dimensional (quantum) algebras and (equivariant) cohomology, K-theory, and elliptic cohomology [17, 19]. Rimanyi, Tarasov and Varchenko found an identification of rational weight functions with stable envelopes for torus-equivariant cohomology of the partial flag variety T âFλ [21] and extended this to the trigonometric ones for the equivariant K-theory [44] They succeeded to construct a geometric representation of the Yangian Y (glN ) [21] and the quantum affine algebra Uq(glN ) [44] on the equivariant cohomology and the equivariant K-theory, respectively. It turns out that in the trigonometric and non-dynamical limit their combinatorial structures coincide with those of Uq(slN ) on the equivariant K-theory obtained by Ginzburg and Vasserot [20, 55] and by Nakajima [41] We lift these representations to the geometric ones by identifying the elliptic weight functions with the elliptic stable envelopes. In Appendix C we present a direct check of Corollary 4.8 for the relation (2.33)
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