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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).

Contents

Definition

The Hall–Littlewood polynomial P is defined by

P λ ( x 1 , … , x n ; t ) = ( ∏ i ≥ 0 ∏ j = 1 m ( i ) 1 − t 1 − t j ) ∑ w ∈ S n w ( x 1 λ 1 ⋯ x n λ n ∏ i < j x i − t x j x i − x j ) ,

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,

P 42 ( x 1 , x 2 ; t ) = x 1 4 x 2 2 + x 1 2 x 2 4 + ( 1 − t ) x 1 3 x 2 3

Specializations

We have that P λ ( x ; 1 ) = m λ ( x ) , P λ ( x ; 0 ) = s λ ( x ) and P λ ( x ; − 1 ) = P λ ( x ) where the latter is the Schur P polynomials.

Properties

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

s λ ( x ) = ∑ μ K λ μ ( t ) P μ ( x , t )

where K λ μ ( t ) are the Kostka–Foulkes polynomials. Note that as t = 1 , these reduce to the ordinary Kostka coefficients.

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

K λ μ ( t ) = ∑ T ∈ S S Y T ( λ , μ ) t c h a r g e ( T )

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.

References

Hall–Littlewood polynomials Wikipedia
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