Hall–Littlewood polynomials

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In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t is 0 and monomial symmetric functions when t is 1 and are special cases of Macdonald polynomials. They were first defined indirectly by Philip Hall using the Hall algebra, and later defined directly by Littlewood (1961).

Definition[edit]

The Hall–Littlewood polynomial P is defined by

where λ is a partition of at most n with elements λi, and m(i) elements equal to i, and Sn is the symmetric group of order n!.


As an example,

Specializations[edit]

We have that , and where the latter is the Schur P polynomials.

Properties[edit]

Expanding the Schur polynomials in terms of the Hall–Littlewood polynomials, one has

where are the Kostka–Foulkes polynomials. Note that as , these reduce to the ordinary Kostka coefficients.

A combinatorial description for the Kostka–Foulkes polynomials was given by Lascoux and Schützenberger,

where "charge" is a certain combinatorial statistic on semistandard Young tableaux, and the sum is taken over all semi-standard Young tableaux with shape λ and type μ.

See also[edit]

References[edit]

  • I.G. Macdonald (1979). Symmetric Functions and Hall Polynomials. Oxford University Press. pp. 101–104. ISBN 0-19-853530-9.
  • D.E. Littlewood (1961). "On certain symmetric functions". Proceedings of the London Mathematical Society. 43: 485–498. doi:10.1112/plms/s3-11.1.485.

External links[edit]