Schur-Weyl duality for tensor powers of the Burau representation

Artin's braid group B-n is generated by sigma(1), ... , sigma(n-1) subject to the relations sigma(i)sigma(i+1)sigma(i) = sigma(i+1)sigma(i)sigma(i+1), sigma(i)sigma(j) = sigma(j)sigma(i) if |i - j| > 1. For complex parameters q(1), q(2) such that q(1)q(2) not equal 0, the group B-n acts on the vector space E = Sigma(i) Ce-i with basis e(1), ... , e(n) by sigma(i) center dot e(i) = (q(1) + q(2))e(i) + q(1)ei(+1), sigma(i) center dot e(i+ 1) = -q(2)e(i), sigma(i) center dot e(j) = q(1)e(j) if j not equal i, i + 1. This representation is (a slight generalization of) the Burau representation. If q = -q(2)/q(1) is not a root of unity, we show that the algebra of all endomorphisms of E-circle times r commuting with the B-n-action is generated by the place-permutation action of the symmetric group S-r and the operator p(1), given by p(1)(e(j1) circle times e(j2) circle times ... e(jr)) = q(j1-1) Sigma(n)(i=1) e(i) circle times e(j2) circle times ... circle times e(jr). Equivalently, as a ( CBn, P'(r) ([n](q)))-bimodule, E-circle times r satisfies Schur-Weyl duality, where P'(r) ([n](q)) is a certain subalgebra of the partition algebra P-r ([n](q)) on 2r nodes with parameter [n](q) = 1 + q+ ... + q(n-1), isomorphic to the semigroup algebra of the "rook monoid" studied by W. D. Munn, L. Solomon, and others.