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Symmetric and nonsymmetric Hall-Littlewood polynomials of type BC
Representation Theory · Submitted 13 Sep 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim As a consequence of this work, we obtain direct arguments for the constant term evaluations and norms in both the symmetric and nonsymmetric cases.
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The abstract, sentence by sentence
Koornwinder polynomials are a 6-parameter BC_n-symmetric family of Laurent polynomials indexed by partitions, from which Macdonald polynomials can be recovered in suitable limits of the parameters. As in the Macdonald polynomial case, standard constructions via difference operators do not allow one to directly control these polynomials at q=0. In the first part of this paper, we provide an explicit construction for these polynomials in this limit, using the defining properties of Koornwinder polynomials. Our formula is a first step in developing the analogy between Hall-Littlewood polynomials and Koornwinder polynomials at q=0. Next, we provide an analogous construction for the nonsymmetric Koornwinder polynomials in the same limiting case. The method employed in this paper is a BC-type adaptation of techniques used in an earlier work of the author, which gave a combinatorial method for proving vanishing results of Rains and Vazirani at the Hall-Littlewood level. As a consequence of this work, we obtain direct arguments for the constant term evaluations and norms in both the symmetric and nonsymmetric cases.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Koornwinder polynomials are a 6-parameter BC_n-symmetric family of Laurent polynomials indexed by partitions, from which Macdonald polynomials can be recovered in suitable limits of the parameters. | background 1.00 | 0.06 | 0.04 | asserted 1.00 |
| s1 | As in the Macdonald polynomial case, standard constructions via difference operators do not allow one to directly control these polynomials at q=0. | gap 0.54 | 0.09 | 0.02 | asserted 1.00 |
| s2 | In the first part of this paper, we provide an explicit construction for these polynomials in this limit, using the defining properties of Koornwinder polynomials. | contribution 0.98 | 0.96 | 0.24 | asserted 1.00 |
| s3 | Our formula is a first step in developing the analogy between Hall-Littlewood polynomials and Koornwinder polynomials at q=0. | interpretation 0.84 | 0.66 | 0.10 | asserted 0.94 |
| s4 | Next, we provide an analogous construction for the nonsymmetric Koornwinder polynomials in the same limiting case. | contribution 1.00 | 0.95 | 0.10 | asserted 1.00 |
| s5 | The method employed in this paper is a BC-type adaptation of techniques used in an earlier work of the author, which gave a combinatorial method for proving vanishing results of Rains and Vazirani at the Hall-Littlewood level. | method 0.92 | 0.40 | 0.03 | asserted 1.00 |
| s6 | As a consequence of this work, we obtain direct arguments for the constant term evaluations and norms in both the symmetric and nonsymmetric cases. | result 0.97 | 0.93 | 0.45 | asserted 1.00 |
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The paper
Original abstract
Koornwinder polynomials are a 6-parameter BC_n-symmetric family of Laurent polynomials indexed by partitions, from which Macdonald polynomials can be recovered in suitable limits of the parameters. As in the Macdonald polynomial case, standard constructions via difference operators do not allow one to directly control these polynomials at q=0. In the first part of this paper, we provide an explicit construction for these polynomials in this limit, using the defining properties of Koornwinder polynomials. Our formula is a first step in developing the analogy between Hall-Littlewood polynomials and Koornwinder polynomials at q=0. Next, we provide an analogous construction for the nonsymmetric Koornwinder polynomials in the same limiting case. The method employed in this paper is a BC-type adaptation of techniques used in an earlier work of the author, which gave a combinatorial method for proving vanishing results of Rains and Vazirani at the Hall-Littlewood level. As a consequence of this work, we obtain direct arguments for the constant term evaluations and norms in both the symmetric and nonsymmetric cases.
Human review
How the literature uses it
8 arXiv papers cite this in 24 sentences
Papers citing it · 16 sentences without a confident label
Uses its method or tool 1 paper
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Following Venkateswaran [ref] , let $λ$ be a partition with $n$ or fewer parts, and define the $BC_n$-symmetric Hall–Littlewood polynomial indexed by $λ$ as
Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices · $BC_n$-symmetric Hall–Littlewood polynomials · 0.88
Uses its data or values 1 paper
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Note that for Cases 1, 2, and 4, the $V_λ$ are determined explicitly in [ref] for general parameters $t_0, …, t_3$ of the Koornwinder $q=0$ polynomials and in [ref] for these choices of parameters.
A p-adic interpretation of some integral identities for Hall-Littlewood polynomials · $P$-adic represention theory · 0.80
Compares with it 2 papers
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In so doing, we will find that [ref] coincides with a known formula for $BC_n$ Hall–Littlewood polynomials [ref] , proving that [ref] is indeed a valid expression for a Koornwinder polynomial at $q=0$.
Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons · $BC_n$ Hall–Littlewood polynomials: sum over hyperoctahedral group · 0.94
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Instead, the equivalence of the two formulas (for this specialization of the parameters) is not obvious and rather follows from the fact that both expressions represent the same polynomials of the form in equations [ref] and $Δ$ given by equation [ref] with $t_3=t_4=0$ [ref] .
Boundary Interactions for the Semi-Infinite q-Boson System and Hyperoctahedral Hall-Littlewood Polynomials · Two-parameter reduction · 0.91
Cites it as background 2 papers
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On the symmetric polynomials side, we will focus attention on Schur polynomials and their Hall–Littlewood generalization, as well as symplectic characters ($BC_n$-symmetric Laurent polynomials that are the irreducible characters of the symplectic group $Sp(2n)$) and their generalization – the $BC_n$-symmetric Hall–Littlewood polynomials (see [ref] ).
Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices · Introduction · 0.94
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where $P_λ$ and $K_λ$ denote Hall–Littlewood polynomials of type $A_n$ [ref] and $BC_n$ [ref] , respectively.
Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures · Introduction · 0.99
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