Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular K
hovanov homology carries a commuting action of the symmetric group S_n. One therefore obtains a knotted Schur-Weyl representation that agrees with classical sl_2 Schur-Weyl duality when K is the Seifert-framed unknot.
The trace (or zeroth Hochschild homology) of Khovanovs Heisenberg category is identified with a quotient of the algebra W_{1+infty}. This induces an action of W_{1+infty} on symmetric functions.
We prove a Littlewood-Richardson type formula for $(s_{lambda/mu},s_{ u/kappa})_{t^k,t}$, the pairing of two skew Schur functions in the MacDonald inner product at $q = t^k$ for positive integers $k$. This pairing counts graded decomposition numbers
in the representation theory of wreath products of the algebra $C[x]/x^k$ and symmetric groups.
