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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

The Hall–Littlewood polynomial P is defined by

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

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

Hall polynomial

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.