Quantum affine Gelfand–Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces

Tsymbaliuk, Alexander

doi:10.1007/s00029-009-0013-3

Cited by 16 publications

(22 citation statements)

References 17 publications

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“…Alternatively, Theorem 3.9 can be proved by direct calculations using the explicit matrix coefficients of the Chevalley generators given in Theorem 3.11 below. These calculations, again, are the rational versions of the trigonometric calculations of [23].…”

Section: The Action Of Generatorsmentioning

confidence: 99%

Yangians and cohomology rings of Laumon spaces

Feigin

Finkelberg

Neguţ

et al. 2011

Sel. Math. New Ser.

113

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Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of G L n . We construct the action of the Yangian of sl n in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of sl n [s ±1 , t]) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analog of the Gelfand-Tsetlin basis. The affine analog of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space M n,d of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is 574 B. Feigin et al.naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center Z of the Yangian of gl n naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on M n,d is the image of a noncommutative power sum in Z .

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Section: The Action Of Generatorsmentioning

confidence: 99%

Yangians and cohomology rings of Laumon spaces

Feigin

Finkelberg

Neguţ

et al. 2011

Sel. Math. New Ser.

113

View full text Add to dashboard Cite

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“…It should be noted that analogues of GT-patterns for the (quantum) affine scenario have been considered in papers [12][13][14]. Our construction, however, differs substantially.…”

Section: The Main Resultsmentioning

confidence: 67%

“…Of course, (14) needs to be formalized in order to make sense. This is done routinely so we will not go into detail.…”

Section: Theorem 71mentioning

confidence: 99%

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A combinatorial formula for affine Hall–Littlewood functions via a weighted Brion theorem

Feigin

Makhlin

2016

Sel. Math. New Ser.

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We present a new combinatorial formula for Hall-Littlewood functions associated with the affine root system of typeÃ n−1 , i.e., corresponding to the affine Lie algebra sl n . Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation. Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensional convex polyhedron. We derive a weighted version of Brion's theorem and then apply it to our polyhedron to prove the formula.

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“…Yangian analogue of this result was obtained in [26]. In [8,9,51], certain geometric actions of the universal enveloping algebra U (gl N ) on the Laumon spaces, of the affine Yangian of type A Let A denote E q,p ( gl N ) or U q,p ( gl N ). The A is bi-graded over H * by…”

Section: Geometric Representationmentioning

confidence: 81%

Elliptic stable envelopes and finite-dimensional representations of elliptic quantum group

Konno

2018

Journal of Integrable Systems

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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 [33]. 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).

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Quantum affine Gelfand–Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces

Cited by 16 publications

References 17 publications

Yangians and cohomology rings of Laumon spaces

Yangians and cohomology rings of Laumon spaces

A combinatorial formula for affine Hall–Littlewood functions via a weighted Brion theorem

Elliptic stable envelopes and finite-dimensional representations of elliptic quantum group

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