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Wreath Macdonald polynomials as eigenstates
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abstract
We show that the wreath Macdonald polynomials for $\mathbb{Z}/\ell\mathbb{Z}\wr\Sigma_n$, when naturally viewed as elements in the vertex representation of the quantum toroidal algebra $U_{\mathfrak{q},\mathfrak{d}}(\ddot{\mathfrak{sl}}_\ell)$, diagonalize its horizontal Heisenberg subalgebra. Our proof makes heavy use of shuffle algebra methods, and we also obtain a new proof of existence of wreath Macdonald polynomials.
Forward citations
Cited by 2 Pith papers
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Quantum Trigonometric Spin Ruijsenaars-Schneider Models from $K$-theoretic Coulomb Branches
The trigonometric spin Ruijsenaars-Schneider model is quantized from K-theoretic Coulomb branch data, with commuting Hamiltonians and quantum spin commutation relations derived.
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Tesler identities for wreath Macdonald polynomials
For r>2, an explicit operator identity (Tesler identity) relates each wreath Macdonald polynomial to a delta function and yields Macdonald-Koornwinder duality, evaluation, interpolation, Kostka, and bispectral results.
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