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In [[mathematics]], the '''Hall–Littlewood polynomials''' are [[symmetric function]]s depending on a parameter ''t'' and a [[partition (number theory)|partition]] λ.  They are [[Schur function]]s when ''t'' is 0 and monomial symmetric functions when ''t'' is 1 and are special cases of [[Macdonald polynomial]]s.
They were first defined indirectly by  [[Philip Hall]] using the [[Hall algebra]], and later defined directly by {{harvtxt|Littlewood|1961}}.


==Definition==
The Hall–Littlewood polynomial ''P'' is defined by
:<math>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)},
</math>
where &lambda; is a partition of length at most ''n'' with elements &lambda;<sub>''i''</sub>, and ''m''(''i'') elements equal to ''i'',  and ''S''<sub>''n''</sub> is the [[symmetric group]] of order ''n''!.


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==See also==
* [[Hall polynomial]]
 
==References==
* {{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 }}
 
==External links==
*{{MathWorld |title=Hall-Littlewood Polynomial |urlname=Hall-LittlewoodPolynomial}}
 
{{DEFAULTSORT:Hall-Littlewood polynomials}}
[[Category:Orthogonal polynomials]]
[[Category:Algebraic combinatorics]]
[[Category:Symmetric functions]]

Revision as of 01:21, 9 March 2013

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 Template:Harvtxt.

Definition

The Hall–Littlewood polynomial P is defined by

Pλ(x1,,xn;t)=i1t1tm(i)wSnw(x1λ1xnλni<jxitxjxixj),

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

See also

References

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