On q-Deformed $${{\mathfrak{gl}}_{\ell+1}}$$ -Whittaker Function III

Gerasimov, A.; Лебедев, Д. В.; Oblezin, Sergey

doi:10.1007/s11005-011-0468-y

Cited by 32 publications

(65 citation statements)

References 14 publications

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“…In this Section we collect basic facts on the Macdonald symmetric polynomials and their degenerate versions given by Jack polynomials and class one q-Whittaker functions. For details on Macdonald and Jack's polynomials see [M]; for class one q-deformed Whittaker functions see [GLO2], [GLO3], [GLO4].…”

Section: Preliminaries On Symmetric Polynomialsmentioning

confidence: 99%

Baxter Operator Formalism for Macdonald Polynomials

2013

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We develop basic constructions of the Baxter operator formalism for the Macdonald polynomials associated with root systems of type A. Precisely we construct a dual pair of mutually commuting Baxter operators such that the Macdonald polynomials are their common eigenfunctions. The dual pair of Baxter operators is closely related to the dual pair of recursive operators for Macdonald polynomials leading to various families of their integral representations. We also construct the Baxter operator formalism for the q-deformed gl ℓ+1 -Whittaker functions and the Jack polynomials obtained by degenerations of the Macdonald polynomials associated with the type A ℓ root system. This note provides a generalization of our previous results on the Baxter operator formalism for the Whittaker functions. It was demonstrated previously that Baxter operator formalism for the Whittaker functions has deep connections with representation theory. In particular the Baxter operators should be considered as elements of appropriate spherical Hecke algebras and their eigenvalues are identified with local Archimedean L-factors associated with admissible representations of reductive groups over R. We expect that the Baxter operator formalism for the Macdonald polynomials has an interpretation in representation theory of higher-dimensional arithmetic fields.

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Section: Preliminaries On Symmetric Polynomialsmentioning

confidence: 99%

Baxter Operator Formalism for Macdonald Polynomials

2013

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“…Thus, taking into account constructions proposed in this paper one can expect that (q-deformed) gl ℓ+1 -Whittaker functions (encoding Gromov-Witten invariants and their K-theory generalizations) can be expressed in terms of representation theory of affine Lie algebras (see [GiL] for a related conjecture and [FFJMM] for a recent progress in this direction). The paper [GLO2] deals with a relation of our results with the representation theory of (quantum) affine Lie groups.…”

Section: Introductionmentioning

confidence: 96%

On q-Deformed $${\mathfrak{gl}_{\ell+1}}$$ -Whittaker Function I

Gerasimov

Лебедев

Oblezin

2009

Commun. Math. Phys.

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Abstract. A representation of a specialization of a q-deformed class one lattice gl ℓ+1 -Whittaker function in terms of cohomology groups of line bundles on the space QM d (P ℓ ) of quasi-maps P 1 → P ℓ of degree d is proposed. For ℓ = 1, this provides an interpretation of non-specialized q-deformed gl 2 -Whittaker function in terms of QM d (P 1 ). In particular the (q-version of) MellinBarnes representation of gl 2 -Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of Γ-function as a substitute of topological genus in semi-infinite geometry. A relation with Givental-Lee universal solution (J-function) of q-deformed gl 2 -Toda chain is also discussed.

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“…We show that the distribution of weights of the pair of tableaux obtained when one applies the insertion algorithm to a random word or permutation takes a particularly simple form and is closely related to q-Whittaker functions. These are functions defined on integer partitions which are eigenfunctions the relativistic Toda chain [Rui90,Rui99,Eti99,GLO10] and simply related to Macdonald polynomials (as a function of the index) with the parameter t = 0 [GLO11]. When q = 0, they are given by Schur polynomials.…”

Section: Introductionmentioning

confidence: 99%

A $q$-weighted version of the Robinson-Schensted algorithm

O’Connell

Pei

2013

Electron. J. Probab.

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We introduce a q-weighted version of the Robinson-Schensted (column insertion) algorithm which is closely connected to q-Whittaker functions (or Macdonald polynomials with t = 0) and reduces to the usual Robinson-Schensted algorithm when q = 0. The q-insertion algorithm is 'randomised', or 'quantum', in the sense that when inserting a positive integer into a tableau, the output is a distribution of weights on a particular set of tableaux which includes the output which would have been obtained via the usual column insertion algorithm. There is also a notion of recording tableau in this setting. We show that the distribution of weights of the pair of tableaux obtained when one applies the q-insertion algorithm to a random word or permutation takes a particularly simple form and is closely related to q-Whittaker functions. In the case 0 ≤ q < 1, the q-insertion algorithm applied to a random word also provides a new framework for solving the q-TASEP interacting particle system introduced (in the language of q-bosons) by Sasamoto and Wadati [SW98] and yields formulas which are equivalent to some of those recently obtained by Borodin and Corwin [BC11] via a stochastic evolution on discrete Gelfand-Tsetlin patterns (or semistandard tableaux) which is coupled to the q-TASEP process. We show that the sequence of P -tableaux obtained when one applies the q-insertion algorithm to a random word defines another, quite different, evolution on semistandard tableaux which is also coupled to the q-TASEP process.

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On q-Deformed $${{\mathfrak{gl}}_{\ell+1}}$$ -Whittaker Function III

Cited by 32 publications

References 14 publications

Baxter Operator Formalism for Macdonald Polynomials

Baxter Operator Formalism for Macdonald Polynomials

On q-Deformed $${\mathfrak{gl}_{\ell+1}}$$ -Whittaker Function I

A $q$-weighted version of the Robinson-Schensted algorithm

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