Jack Content Operators and the Deformed 𝒲1+∞ Algebra
Abstract
Frenkel and Wang obtained a representation of the Virasoro algebra by commuting Goulden's cut-and-join operator with the Heisenberg generators. A vertex-operator construction by Lascoux and the author extends this representation to 𝒲1+∞ by means of differential operators whose eigenvalues are the power sums of the contents of a Young diagram. We develop a Jack deformation in the spherical degenerate double affine Hecke algebra and its stable limit. Starting from the Heckman--Polychronakos integrals, we isolate operators whose eigenvalues are the power sums of the α-contents. Because the Goulden--Jackson product is defined in the convention dual to the usual Calogero--Sutherland Hamiltonians, the multiplication operators Δμ(α) are obtained by taking Hall adjoints. This gives conceptual derivations of the Jack cut-and-join operator and of the stable $3$-cycle operator. A normal-ordering construction due to Sergeev and Veselov makes the latter calculation explicit and suggests an integral form over ℤ[α]. The commutators of the cut-and-join operator contain one half of the usual Feigin--Fuchs realization, but this Virasoro completion is not the deformation of the Frenkel--Wang construction. The latter takes place in the deformed 𝒲1+∞ algebra SHc, equivalently in the affine Yangian of gl1: in our normalization its first nontrivial Cartan mode is ψ3=3Δ2(α)+2(α−1)E, and its commutators with the first raising and lowering modes recursively generate the remaining currents. At α=1 these relations specialize to the central-charge-one 𝒲1+∞ representation used by Lascoux and the author.
Related
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BibTeX
@misc{thibon2026jack,
title = {{Jack Content Operators and the Deformed $\mathcal W_1+\infty$ Algebra}},
author = {Jean-Yves Thibon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10284},
eprint = {2609.10284},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10284},
}
RIS
TY - UNPB
TI - Jack Content Operators and the Deformed 𝒲_1+∞ Algebra
AU - Jean-Yves Thibon
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10284
UR - https://arxiv.org/abs/2609.10284
AB - Frenkel and Wang obtained a representation of the Virasoro algebra by commuting Goulden's cut-and-join operator with the Heisenberg generators. A vertex-operator construction by Lascoux and the author extends this representation to 𝒲_1+∞ by means of differential operators whose eigenvalues are the power sums of the contents of a Young diagram. We develop a Jack deformation in the spherical degenerate double affine Hecke algebra and its stable limit. Starting from the Heckman--Polychronakos integrals, we isolate operators whose eigenvalues are the power sums of the α-contents. Because the Goulden--Jackson product is defined in the convention dual to the usual Calogero--Sutherland Hamiltonians, the multiplication operators Δ_μ(α) are obtained by taking Hall adjoints. This gives conceptual derivations of the Jack cut-and-join operator and of the stable $3$-cycle operator. A normal-ordering construction due to Sergeev and Veselov makes the latter calculation explicit and suggests an integral form over ℤ[α]. The commutators of the cut-and-join operator contain one half of the usual Feigin--Fuchs realization, but this Virasoro completion is not the deformation of the Frenkel--Wang construction. The latter takes place in the deformed 𝒲_1+∞ algebra SH^c, equivalently in the affine Yangian of gl_1: in our normalization its first nontrivial Cartan mode is ψ_3=3Δ_2(α)+2(α−1)E, and its commutators with the first raising and lowering modes recursively generate the remaining currents. At α=1 these relations specialize to the central-charge-one 𝒲_1+∞ representation used by Lascoux and the author.
ER -