  We introduce an algebra model to study higher order sum rules for orthogonal
polynomials on the unit circle. We build the relation between the algebra model
and sum rules, and prove an equivalent expression on the algebra side for the
sum rules, involving a Hall-Littlewood type polynomial. By this expression, we
recover an earlier result by Golinskii and Zlat\v{o}s, and prove a new case -
half of the Lukic conjecture in the case of a single critical point with
arbitrary order.
