Hall–Littlewood polynomials

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

The Hall–Littlewood polynomial P is defined by

P_\lambda(x_1,\ldots,x_n;t) = \left( \prod_{i\geq 0} \prod_{j=1}^{m(i)} \frac{1-t}{1-t^{j}} \right) {\sum_{w\in S_n}w\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod_{i<j}\frac{x_i-tx_j}{x_i-x_j}\right)},

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

Specializations

We have that $P_\lambda(x;1) = m_\lambda(x)$ , $P_\lambda(x;0) = s_\lambda(x)$ and $P_\lambda(x;-1) = P_\lambda(x)$ where the latter is the Schur P polynomials.

Properties

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

s_\lambda(x) = \sum_\mu K_{\lambda\mu}(t) P_\mu(x,t)

where $K_{\lambda\mu}(t)$ 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 were given by Lascoux and Schützenberger,

K_{\lambda \mu }(t)=\sum _{T\in SSYT(\lambda ,\mu )}t^{\mathrm {charge} (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

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

Weisstein, Eric Wolfgang. "Hall–Littlewood Polynomial". MathWorld.

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