An infinite family of higher-order difference operators that commute with Ruijsenaars operators of type A

Noumi, Masatoshi; Sano, Ayako

doi:10.1007/s11005-021-01435-9

Cited by 4 publications

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References 12 publications

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“…For X and X ! in this paper, we are led to introduce a family of q-difference operators commuting with the Macdonald difference operators, closely related to those studied in [14] and [15]. Once the fundamental property of the difference operators is known, the remainder of the proof can be completed by direct computation.…”

Section: 1mentioning

confidence: 99%

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Symplectic Duality of $T^*Gr(k,n)$

Dinkins

2020

Preprint

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In this paper, we explore a consequence of symplectic duality (also known as 3d mirror symmetry) in the setting of enumerative geometry. The theory of quasimaps allows one to associate hypergeometric functions called vertex functions to quiver varieties. In this paper, we prove a formula which relates the vertex functions of T * Gr(k, n) and its symplectic dual. In the course of the proof, we study a family of q-difference operators which act diagonally on Macdonald polynomials. Our results may be interpreted from a combinatorial perspective as providing an evaluation formula for a q-Selberg type integral.

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Section: 1mentioning

confidence: 99%

“…In particular, Ole Warnaar brought the paper [15] to our attention. We are grateful to Masatoshi Noumi for sharing the unpublished work [14] with us.…”

Section: Acknowledgementsmentioning

confidence: 99%

Symplectic Duality of $T^*Gr(k,n)$

Dinkins

2020

Preprint

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From Kajihara’s transformation formula to deformed Macdonald–Ruijsenaars and Noumi–Sano operators

Hallnäs

Langmann

Noumi

et al. 2021

Sel. Math. New Ser.

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Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with A-type root systems of different ranks. By specialisations of his formula, we deduce kernel identities for deformed Macdonald–Ruijsenaars (MR) and Noumi–Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by the super-Macdonald polynomials. We also provide an explicit description of the algebra generated by the deformed MR and/or NS operators by a Harish-Chandra type isomorphism and show that the deformed MR (NS) operators can be viewed as restrictions of inverse limits of ordinary MR (NS) operators.

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Higher Order Deformed Elliptic Ruijsenaars Operators

Hallnäs

Langmann

Noumi

et al. 2022

Commun. Math. Phys.

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We present four infinite families of mutually commuting difference operators which include the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators was previously introduced by Feigin and Silantyev. They provide a quantum mechanical description of two kinds of relativistic quantum mechanical particles which can be identified with particles and anti-particles in an underlying quantum field theory. We give direct proofs of the commutativity of our operators and of some other fundamental properties such as kernel function identities. In particular, we give a rigorous proof of the quantum integrability of the deformed Ruijsenaars model.

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An infinite family of higher-order difference operators that commute with Ruijsenaars operators of type A

Abstract: We introduce a new infinite family of higher-order difference operators that commute with the elliptic Ruijsenaars difference operators of type A. These operators are related to Ruijsenaars’ operators through a formula of Wronski type.

Cited by 4 publications

References 12 publications

Symplectic Duality of $T^*Gr(k,n)$

Symplectic Duality of $T^*Gr(k,n)$

From Kajihara’s transformation formula to deformed Macdonald–Ruijsenaars and Noumi–Sano operators

Higher Order Deformed Elliptic Ruijsenaars Operators

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