Hall–Littlewood polynomials

The Hall–Littlewood polynomial P is defined by

:$P\_\backslash lambda(x\_1,\backslash ldots,x\_n;t)\; =\; \backslash left(\; \backslash prod\_\{i\backslash geq\; 0\}\; \backslash prod\_\{j=1\}^\{m(i)\}\; \backslash frac\{1-t\}\{1-t^\{j\}\}\; \backslash right)$

{\sum_{w\in S_n}w\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod_{i

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

As an example,

:$P\_\{42\}(x\_1,x\_2;t)\; =\; x\_1^4\; x\_2^2\; +\; x\_1^2\; x\_2^4\; +\; (1-t)\; x\_1^3\; x\_2^3$

We have that $P\_\backslash lambda(x;1)\; =\; m\_\backslash lambda(x)$, $P\_\backslash lambda(x;0)\; =\; s\_\backslash lambda(x)$ and

$P\_\backslash lambda(x;-1)\; =\; P\_\backslash lambda(x)$ where the latter is the Schur P polynomials.

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

:$s\_\backslash lambda(x)\; =\; \backslash sum\_\backslash mu\; K\_\{\backslash lambda\backslash mu\}(t)\; P\_\backslash mu(x,t)$

where $K\_\{\backslash lambda\backslash mu\}(t)$ 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 was given by Lascoux and Schützenberger,

:$K\_\{\backslash lambda\backslash mu\}(t)\; =\; \backslash sum\_\{T\; \backslash in\; SSYT(\backslash lambda,\backslash mu)\}\; t^\{\backslash mathrm\{charge\}(T)\}$

where "charge" is a certain combinatorial statistic on semistandard Young tableaux,

and the sum is taken over the set $SSYT(\backslash lambda,\backslash mu)$ of all semi-standard Young tableaux T with shape λ and type μ.

• Hall polynomial

• {{cite book | author=I.G. Macdonald | authorlink=Ian G. Macdonald | title=Symmetric Functions and Hall Polynomials | publisher=Oxford University Press | pages=101–104 | year=1979 | isbn=0-19-853530-9 }}
• {{cite journal | authorlink=Dudley E. Littlewood | author=D.E. Littlewood | title=On certain symmetric functions | journal= Proceedings of the London Mathematical Society | volume=43 | year=1961 | pages=485–498 | doi=10.1112/plms/s3-11.1.485 }}

• {{MathWorld |title=Hall–Littlewood Polynomial |urlname=Hall-LittlewoodPolynomial}}

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Category:Orthogonal polynomials

Category:Algebraic combinatorics

Category:Symmetric functions