In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and representation theory.
The two-variable Kostka polynomials Kλμ(q, t) are known by several names including Kostka–Foulkes polynomials, Macdonald–Kostka polynomials or q,t-Kostka polynomials. Here the indices λ and μ are integer partitions and Kλμ(q, t) is polynomial in the variables q and t. Sometimes one considers single-variable versions of these polynomials that arise by setting q = 0, i.e., by considering the polynomial Kλμ(t) = Kλμ(0, t).
There are two slightly different versions of them, one called transformed Kostka polynomials.
The one-variable specializations of the Kostka polynomials can be used to relate Hall-Littlewood polynomials Pμ to Schur polynomials sλ:
sλ(x1,\ldots,xn)=\sum\muKλ\mu(t)P\mu(x1,\ldots,xn;t).
These polynomials were conjectured to have non-negative integer coefficients by Foulkes, and this was later proved in 1978 by Alain Lascoux and Marcel-Paul Schützenberger.[1] In fact, they show that
Kλ\mu(t)=\sumTtcharge(T)
The Macdonald–Kostka polynomials can be used to relate Macdonald polynomials (also denoted by Pμ) to Schur polynomials sλ:
sλ(x1,\ldots,xn)=\sum\muKλ\mu(q,t)J\mu(x1,\ldots,xn;q,t)
where
J\mu(x1,\ldots,xn;q,t)=P\mu(x1,\ldots,xn;q,t)\prods\in\mu(1-qarm(s)tleg(s)+1).
Kostka numbers are special values of the one- or two-variable Kostka polynomials:
Kλ\mu=Kλ\mu(1)=Kλ\mu(0,1).