Hall-Littlewood Polynomial

Let  be an integer such that , where  is a partition of  if , where  are a sequence of positive integers stabilizing 0 such that . Also let  be the number of parts of  of size . Then the permutation , where  is the symmetric group, acts on the variables , ...,  by sending  to . Letting  be a complex number, the Hall-Littlewood polynomials are defined by

These polynomials interpolate between the Schur functions (with ) and the monomial symmetric functions (with ; Fulman 1999).

REFERENCES:

Fulman, J. "The Rogers-Ramanujan Identities, the Finite General Linear Groups, and the Hall-Littlewood Polynomials." Proc. Amer. Math. Soc. 128, 17-25, 1999.

Macdonald, I. G. Symmetric Functions and Hall Polynomials, 2nd ed. Oxford, England: Oxford University Press, p. 208, 1995.