AHA! RSK

Abstract

We give a spectral realization of the Robinson-Schensted-Knuth (RSK) correspondence in terms of the representation theory of the symmetric group $S_n$ and the degenerate affine Hecke algebra (AHA) $H_n$. We view RSK, which builds a pair of standard Young tableaux from a permutation, as a special case of rectification, also known as Jeu de Taquin, which turns skew tableaux into straight ones. In this framing, the initial permutation corresponds to a skew tableau of staircase shape. To interpret this in terms of representation theory, take permutations to label weight vectors in a generic $H_n$-module $V(a_1, \ldots , a_n)$, which is isomorphic to $\mathbb{C}[S_n]$ as an $S_n$-module. Writing permutations as staircases amounts to placing these weight vectors inside the regular representation of a larger symmetric group containing $S_n$; more geometrically, we push $S_n$ to the right toward infinity so its Jucys-Murphy (JM) elements have enough room to represent the external translations of $H_n$. Then, rectification corresponds to squeezing out this extra room from the left, leaving only $S_n$ and its regular JM elements as the limit of the external translations. By expressing slides via sequences of exchanges of consecutive values inside the tableau, we can model rectification by an operator acting on the regular representation. This lets us explicitly write down the change of basis between $H_n$-weight vectors and $S_n$-weight vectors, where the latter are eigenvectors of the JM elements in $S_n$ acting both on the left and on the right, and hence labeled by pairs of standard tableaux. The resulting correspondence between the labels of the weight vectors is exactly RSK.