A Proof of the Jung-Kim Conjecture on the Centers of Walled Brauer Algebras

We give a unified account of the five families of Khovanov arc algebras $K_n^m$, $K_n^{-\mathrm{\infty }}$, $K_n^{+\mathrm{\infty }}$, $K_n^{\mathrm{\infty }}$ and $K_{\mathrm{\infty }}^{\mathrm{\infty }}$ 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 $K_{\mathrm{\infty }}^{\mathrm{\infty }}$. 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 $K_n^{-\mathrm{\infty }}$ and classical super duality, and identify $K_n^{-\mathrm{\infty }}$ and $K_n^{+\mathrm{\infty }}$ as Ringel duals. We also revisit mixed Schur–Weyl duality for general linear Lie superalgebras within the same framework.