# Jack polynomials and the Witt algebra

The symmetric Jack polynomials $J_n^{\alpha}(x_1,x_2,..,x_{n+1})$, a special subset of the symmetric Jack functions presented in Stanley's paper in equation a) on page 80, can be represented by the action of the operator $x^{1+\alpha}d/dx$, which specializes to reps of the Witt algebra, or centerless Virasoro algebra, for integer exponents.

Can someone provide some Web-accessible references on the relation of any subsets of the general Jack polynomials to the Witt algebra, particularly if couched in terms of Witt diff ops?

• Thanks. I'm continually exploring the interplay of umbral calculus/Sheffer sequences, encompassing symmetric polynomials; operator calculus, particularly vectors of the form $g(x)d/dx$; combinatorics; geometry/topology, including trees, graphs, simplicial complexes, polytopes, Riemann surfaces, and differential manifolds; and associated physics, particularly quantum field theory--rather broad interests. – Tom Copeland Apr 6 at 16:23