Duality for Bethe algebras acting on polynomials in anticommuting variables

Tarasov, Vitaly; Uvarov, Filipp

doi:10.1007/s11005-020-01329-2

Cited by 4 publications

(16 citation statements)

References 11 publications

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“…The functions in the second line are the same except the function α x i x j is omitted. Using an induction similar to what we used in the proof of Lemma 6.5 in [13], we obtain the following formulas:…”

Section: 4mentioning

confidence: 99%

“…We will use the following two facts about linear differential operators. For proofs, see for example, [13].…”

Section: Recall the Mapmentioning

confidence: 99%

“…The results of this work and our previous works [14], [13] are devoted to the (gl k , gl n )duality in quantum integrable models on the space P kn of polynomials in anticommuting variables. The parallel results for the space P kn of polynomials in commuting variables were obtained earlier, see works [11], [7], [5], and [6].…”

Section: Introductionmentioning

confidence: 99%

“…Both U( gl k ) and Y (gl n ) act on the space P kn , and it is expected that the images of the two Bethe algebras in End P kn are the same. The corresponding result for the rational Gaudin model was established in [13], where we developed and used the differential analogs of the results for the quotient difference operator studied here. Therefore, the results of this paper can be considered as the first steps in establishing the duality between the Bethe algebras of the trigonometric Gaudin model and the XXX-type spin chain model.…”

Section: Introductionmentioning

confidence: 99%

“…

We study the difference analog of the quotient differential operator from [13].Starting with a space of quasi-exponentials W = α x i p ij (x), i = 1, . .

…”

mentioning

confidence: 99%

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Difference operators and duality for trigonometric Gaudin and Dynamical Hamiltonians

Uvarov¹

2022

Preprint

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We study the difference analog of the quotient differential operator from [13].Starting with a space of quasi-exponentials W = α x i p ij (x), i = 1, . . . , n, j = 1, . . . , n i , where α i ∈ C * and p ij (x) are polynomials, we consider the formal conjugate Š † W of the quotient difference operator ŠW satisfying S = ŠW S W . Here, S W is a linear difference operator of order dim W annihilating W , and S is a linear difference operator with constant coefficients depending on α i and deg p ij (x) only. We show that ker Š † W is also a space of quasi-exponentials, describe its basis and discrete exponents.We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form x z q(x), where z ∈ C and q(x) is a polynomial.Combining our results with the results on the bispectral duality obtained in [6], we relate the construction of the quotient difference operator to the (gl k , gl n )-duality of the trigonometric Gaudin Hamiltonians and trigonometric Dynamical Hamiltonians acting on the space of polynomials in kn anticommuting variables.

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Section: 4mentioning

confidence: 99%

“…We will use the following two facts about linear differential operators. For proofs, see for example, [13].…”

Section: Recall the Mapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

We study the difference analog of the quotient differential operator from [13].Starting with a space of quasi-exponentials W = α x i p ij (x), i = 1, . .

…”

mentioning

confidence: 99%

See 3 more Smart Citations

Difference operators and duality for trigonometric Gaudin and Dynamical Hamiltonians

Uvarov¹

2022

Preprint

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Gaudin model and Deligne’s category

Feigin,

Rybnikov,

Uvarov

2023

Lett Math Phys

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An identity in the Bethe subalgebra of C[Sn]$\mathbb {C}[\mathfrak {S}_n]$

Purbhoo

2023

Proceedings of London Math Soc

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As part of the proof of the Bethe ansatz conjecture for the Gaudin model for , Mukhin, Tarasov, and Varchenko described a correspondence between inverse Wronskians of polynomials and eigenspaces of the Gaudin Hamiltonians. Notably, this correspondence afforded the first proof of the Shapiro–Shapiro conjecture. In this paper, we give an identity in the group algebra of the symmetric group, which allows one to establish the correspondence directly, without using the Bethe ansatz.

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Duality for Bethe algebras acting on polynomials in anticommuting variables

Cited by 4 publications

References 11 publications

Difference operators and duality for trigonometric Gaudin and Dynamical Hamiltonians

Difference operators and duality for trigonometric Gaudin and Dynamical Hamiltonians

Gaudin model and Deligne’s category

An identity in the Bethe subalgebra of C[Sn]$\mathbb {C}[\mathfrak {S}_n]$

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