Capelli Operators for Spherical Superharmonics and the Dougall–ramanujan Identity

Sahi, Siddhartha; Salmasian, Hadi; Serganova, Vera

doi:10.1007/s00031-021-09655-y

Cited by 2 publications

(3 citation statements)

References 23 publications

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“…A similar extensive, but noncharacterizing, vanishing condition also holds for a different family of polynomials studied by Sahi-Salmasian-Serganova in[81].…”

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confidence: 55%

“…The authors thank Johnny Fonseca and Jason Saied for pointing out an inconsistency in the original definition of our fused weights. The authors are also grateful to Daniel Bump for bringing [17] to our attention; Anton Mellit for explaining his work to us; Siddhartha Sahi for bringing [81] to our attention; and the anonymous referees for their helpful suggestions.…”

Section: Acknowledgmentsmentioning

confidence: 99%

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Colored fermionic vertex models and symmetric functions

Borodin

Wheeler

2023

Comm. Amer. Math. Soc.

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In this text we introduce and analyze families of symmetric functions arising as partition functions for colored fermionic vertex models associated with the quantized affine Lie superalgebra 𝑈 𝑞 ( ŝ𝔩(1|𝑛)). We establish various combinatorial results for these vertex models and symmetric functions, which include the following.(1) We apply the fusion procedure to the fundamental 𝑅-matrix for 𝑈 𝑞 ( ŝ𝔩(1|𝑛)) to obtain an explicit family of vertex weights satisfying the Yang-Baxter equation.(2) We define families of symmetric functions as partition functions for colored, fermionic vertex models under these fused weights. We further establish several combinatorial properties for these symmetric functions, such as branching rules and Cauchy identities.(3) We show that the Lascoux-Leclerc-Thibon (LLT) polynomials arise as special cases of these symmetric functions. This enables us to show both old and new properties about the LLT polynomials, including Cauchy identities, contour integral formulas, stability properties, and branching rules under a certain family of plethystic transformations. (4) A different special case of our symmetric functions gives rise to a new family of polynomials called factorial LLT polynomials. We show they generalize the LLT polynomials, while also satisfying a vanishing condition reminiscent of that satisfied by the factorial Schur functions. ( 5) By considering our vertex model on a cylinder, we obtain fermionic partition function formulas for both the symmetric and nonsymmetric Macdonald polynomials. (6) We prove combinatorial formulas for the coefficients of the LLT polynomials when expanded in the modified Hall-Littlewood basis, as partition functions for a 𝑈 𝑞 ( ŝ𝔩(2|𝑛)) vertex model.

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“…A similar extensive, but noncharacterizing, vanishing condition also holds for a different family of polynomials studied by Sahi-Salmasian-Serganova in[81].…”

mentioning

confidence: 55%

Section: Acknowledgmentsmentioning

confidence: 99%

Colored fermionic vertex models and symmetric functions

Borodin

Wheeler

2023

Comm. Amer. Math. Soc.

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“…Supersymmetric spaces are homogeneous superspaces of the form G/K, where K is a symmetric subgroup of the supergroup G. Such spaces are natural in the study of super harmonic analysis, see for instance [2] and [5]. They also have important connections to interpolation polynomials (see [8], [9], and [10]), integrable systems (see [12]), and physics (see [22] and [11]). 1.7.…”

Section: Introductionmentioning

confidence: 99%

Ghost Distributions on Supersymmetric Spaces II: Basic Classical Superalgebras

Sherman

2023

International Mathematics Research Notices

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We study ghost distributions on supersymmetric spaces for the case of basic classical Lie superalgebras. We introduce the notion of interlaced pairs, which are those for which both $({\mathfrak{g}},{\mathfrak{k}})$ and $({\mathfrak{g}},{\mathfrak{k}}^{\prime})$ admit Iwasawa decompositions. For such pairs, we define a ghost algebra, generalizing the subalgebra of ${\mathcal{U}}{\mathfrak{g}}$ defined by Gorelik. We realize this algebra as an algebra of $G$-equivariant operators on the supersymmetric space itself, and for certain pairs, the “special” ones, we realize our operators as twisted-equivariant differential operators on $G/K$. We additionally show that the Harish-Chandra morphism is injective, compute its image for all rank one pairs, and provide a conjecture for the image when $({\mathfrak{g}},{\mathfrak{k}})$ is interlaced.

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Capelli Operators for Spherical Superharmonics and the Dougall–ramanujan Identity

Cited by 2 publications

References 23 publications

Colored fermionic vertex models and symmetric functions

Colored fermionic vertex models and symmetric functions

Ghost Distributions on Supersymmetric Spaces II: Basic Classical Superalgebras

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