Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2004-07-30
Nucl.Phys. B704 (2005) 490-509
Physics
High Energy Physics
High Energy Physics - Theory
23 pages, references added
Scientific paper
10.1016/j.nuclphysb.2004.10.005
We use the Jack symmetric functions as a basis of the Fock space, and study the action of the Virasoro generators $L_n$. We calculate explicitly the matrix elements of $L_n$ with respect to the Jack-basis. A combinatorial procedure which produces these matrix elements is conjectured. As a limiting case of the formula, we obtain a Pieri-type formula which represents a product of a power sum and a Jack symmetric function as a sum of Jack symmetric functions. Also, a similar expansion was found for the case when we differentiate the Jack symmetric functions with respect to power sums. As an application of our Jack-basis representation, a new diagrammatic interpretation is presented, why the singular vectors of the Virasoro algebra are proportional to the Jack symmetric functions with rectangular diagrams. We also propose a natural normalization of the singular vectors in the Verma module, and determine the coefficients which appear after bosonization in front of the Jack symmetric functions.
Arnaudon Daniel
Frappat Luc
Ragoucy Eric
Sakamoto Reiho
Shiraishi Jun'ichi
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