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Register a new userK-theory of quiver varieties, q-Fock space and nonsymmetric Macdonald polynomials
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Authors
Kentaro Nagao
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We have two constructions of the level-$(0,1)$ irreducible representation of the quantum toroidal algebra of type $A$. One is due to Nakajima and Varagnolo-Vasserot. They constructed the representation on the direct sum of the equivariant K-groups of the quiver varieties of type $hat{A}$. The other is due to Saito-Takemura-Uglov and Varagnolo-Vasserot. They constructed the representation on the q-deformed Fock space introduced by Kashiwara-Miwa-Stern. In this paper we give an explicit isomorphism between these two constructions. For this purpose we construct simultaneous eigenvectors on the q-Fock space using nonsymmetric Macdonald polynomials. Then the isomorphism is given by corresponding these vectors to the torus fixed points on the quiver varieties.
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Let $lambda in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary coset representative of minimal length for the cosets in $W/W_{lambda}$, where $W_{lambda}$ is the stabilizer of $lambda$ in a finite Weyl group $W$. In this paper, we give a module $mathbb{K}_{w}(lambda)$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at $t = infty$ of the nonsymmetric Macdonald polynomial $E_{w lambda}(q,,t)$ multiplied by a certain explicit finite product of rational functions of $q$ of the form $(1 - q^{-r})^{-1}$ for a positive integer $r$. This module $mathbb{K}_{w}(lambda)$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $V_{w}^{-}(lambda)$ by the sum of the submodules $V_{z}^{-}(lambda)$ for all those coset representatives $z$ of minimal length for the cosets in $W/W_{lambda}$ such that $z > w$ in the Bruhat order $<$ on $W$.
Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac-Moody algebra. The underlying set consists of the irreducible components of Lusztigs quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac-Moody algebras by replacing Lusztigs preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.
Let $mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$ $mathcal{N}=4$ quiver gauge theory with quiver $Gamma$, and $mathscr{A}_q subseteq mathscr{A}_q$ be the subalgebra generated by the equivariant $K$-theory of a point together with the dressed minuscule monopole operators $M_{varpi_{i,1},f}$ and $M_{varpi^*_{i,1},f}$. In this paper, we construct an associated cluster algebra quiver $mathcal{Q}_Gamma$ and provide an embedding of the subalgebra $mathscr{A}_q$ into the quantized algebra of regular functions on the corresponding cluster variety.
In the 90s a collection of Plethystic operators were introduced in [3], [7] and [8] to solve some Representation Theoretical problems arising from the Theory of Macdonald polynomials. This collection was enriched in the research that led to the results which appeared in [5], [6] and [9]. However since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished. As a consequence of very recent publications [4], [11], [19], [20], [21], a truly remarkable expansion of this theory has taken place. However most of this work has appeared in a language that is virtually inaccessible to practitioners of Algebraic Combinatorics. Yet, these developments have led to a variety of new conjectures in [2] in the Combinatorics and Symmetric function Theory of Macdonald Polynomials. The present work results from an effort to obtain in an elementary and accessible manner all the background necessary to construct the symmetric function side of some of these new conjectures. It turns out that the above mentioned unpublished results provide precisely the tools needed to carry out this project to its completion.
We provide elementary identities relating the three known types of non-symmetric interpolation Macdonald polynomials. In addition we derive a duality for non-symmetric interpolation Macdonald polynomials. We consider some applications of these results, in particular for binomial formulas involving non-symmetric interpolation Macdonald polynomials.
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