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?


There are two very famous instances of Jack polynomials in relationship to the Virasoro algebra (there are some others, but they very often seem to be related to one of these):

Katsuhisa Mimachi and Yasuhiko Yamada. Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials. Comm. Math. Phys. Volume 174, Number 2 (1995), 447-455.

B. Feigin, M. Jimbo, T. Miwa, E. Mukhin. A differential ideal of symmetric polynomials spanned by Jack polynomials at rβ = -(r=1)/(k+1). International Mathematics Research Notices, Volume 2002, Issue 23, 2002, Pages 1223–1237.

Perhaps you can explain a little bit more about what you want/expect to get further references!

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  • $\begingroup$ 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. $\endgroup$ – Tom Copeland Apr 6 at 16:23

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