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 $K_n^m$, $K_n^{-\infty}$, $K_n^{+\infty}$, $K_n^\infty$ and $K_\infty^\infty$ 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 $K_\infty^\infty$. 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 $K_n^{-\infty}$ and classical super duality, and identify $K_n^{-\infty}$ and $K_n^{+\infty}$ as Ringel duals. We also revisit mixed Schur--Weyl duality for general linear Lie superalgebras within the same framework.