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).


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,


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


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 were 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.

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