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 K<sub>ûü</sub>(q, t) are known by several names including KostkaâÂÂFoulkes polynomials, MacdonaldâÂÂKostka polynomials or q,t-Kostka polynomials. Here the indices û and ü are integer partitions and K<sub>ûü</sub>(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 K<sub>ûü</sub>(t) = K<sub>ûü</sub>(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 P<sub>ü</sub> to Schur polynomials s<sub>û</sub>:
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.
In fact, they show that
where the sum is taken over all semi-standard Young tableaux with shape û and weight ü. Here, charge is a certain combinatorial statistic on semi-standard Young tableaux.
The MacdonaldâÂÂKostka polynomials can be used to relate Macdonald polynomials (also denoted by P<sub>ü</sub>) to Schur polynomials s<sub>û</sub>:
where
Kostka numbers are special values of the one- or two-variable Kostka polynomials: