Szpiro's conjecture

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

$P_{\lambda }(x_{1},\ldots ,x_{n};t)=\prod _{i}{\frac {1-t}{1-t^{m(i)}}}{\sum _{w\in S_{n}}w\left(x_{1}^{\lambda _{1}}\cdots x_{n}^{\lambda _{n}}\prod _{i<j}{\frac {x_{i}-tx_{j}}{x_{i}-x_{j}}}\right)},$

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

See also

• Hall polynomial

References

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