A Proof of the Jung-Kim Conjecture on the Centers of Walled Brauer Algebras
Abstract
We give a unified account of the five families of Khovanov arc algebras Knm, Kn−∞, Kn+∞, Kn∞ and K∞∞ in the framework of level-two tensor product categorifications, making explicit a perspective familiar to specialists. Our main application is a proof of the Jung--Kim conjecture: over an algebraically closed field of characteristic zero, the centre of every walled Brauer algebra is generated by supersymmetric polynomials in its Jucys--Murphy elements. We also give a homogeneous basis of the centre in terms of arc diagrams, for the grading induced by K∞∞. The proof combines a calculation of the arc-algebra centre with an explicit comparison of central actions under the Brundan--Stroppel Morita equivalence and Brundan-Kleshchev isomorphism. Along the way, we explain the relation between Kn−∞ and classical super duality, and identify Kn−∞ and Kn+∞ as Ringel duals. We also revisit mixed Schur--Weyl duality for general linear Lie superalgebras within the same framework.
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BibTeX
@misc{hirota2026proof,
title = {{A Proof of the Jung-Kim Conjecture on the Centers of Walled Brauer Algebras}},
author = {Shunsuke Hirota},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21576},
eprint = {2610.21576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21576},
}
RIS
TY - UNPB
TI - A Proof of the Jung-Kim Conjecture on the Centers of Walled Brauer Algebras
AU - Shunsuke Hirota
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21576
UR - https://arxiv.org/abs/2610.21576
AB - We give a unified account of the five families of Khovanov arc algebras K_n^m, K_n^−∞, K_n^+∞, K_n^∞ and K_∞^∞ in the framework of level-two tensor product categorifications, making explicit a perspective familiar to specialists. Our main application is a proof of the Jung--Kim conjecture: over an algebraically closed field of characteristic zero, the centre of every walled Brauer algebra is generated by supersymmetric polynomials in its Jucys--Murphy elements. We also give a homogeneous basis of the centre in terms of arc diagrams, for the grading induced by K_∞^∞. The proof combines a calculation of the arc-algebra centre with an explicit comparison of central actions under the Brundan--Stroppel Morita equivalence and Brundan-Kleshchev isomorphism. Along the way, we explain the relation between K_n^−∞ and classical super duality, and identify K_n^−∞ and K_n^+∞ as Ringel duals. We also revisit mixed Schur--Weyl duality for general linear Lie superalgebras within the same framework.
ER -