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_λμ(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 K_λμ(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_λμ(t) = K_λμ(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_μ to Schur polynomials s_λ:

${\displaystyle s_{\lambda }(x_{1},\ldots ,x_{n})=\sum _{\mu }K_{\lambda \mu }(t)P_{\mu }(x_{1},\ldots ,x_{n};t).\ }$

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

${\displaystyle K_{\lambda \mu }(t)=\sum _{T\in SSYT(\lambda ,\mu )}t^{charge(T)}}$

where the sum is taken over all semi-standard Young tableaux with shape λ and weight μ. Here, charge is a certain combinatorial statistic on semi-standard Young tableaux.

The Macdonald–Kostka polynomials can be used to relate Macdonald polynomials (also denoted by P_μ) to Schur polynomials s_λ:

${\displaystyle s_{\lambda }(x_{1},\ldots ,x_{n})=\sum _{\mu }K_{\lambda \mu }(q,t)J_{\mu }(x_{1},\ldots ,x_{n};q,t)\ }$

where

${\displaystyle J_{\mu }(x_{1},\ldots ,x_{n};q,t)=P_{\mu }(x_{1},\ldots ,x_{n};q,t)\prod _{s\in \mu }(1-q^{arm(s)}t^{leg(s)+1}).\ }$

Kostka numbers are special values of the one- or two-variable Kostka polynomials:

${\displaystyle K_{\lambda \mu }=K_{\lambda \mu }(1)=K_{\lambda \mu }(0,1).\ }$

Examples

References

• Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR 1354144
• Nelsen, Kendra; Ram, Arun (2003), "Kostka-Foulkes polynomials and Macdonald spherical functions", Surveys in combinatorics, 2003 (Bangor), London Math. Soc. Lecture Note Ser., vol. 307, Cambridge: Cambridge Univ. Press, pp. 325–370, arXiv:math/0401298, Bibcode:2004math......1298N, MR 2011741
• Stembridge, J. R. (2005), Kostka-Foulkes Polynomials of General Type, lecture notes from AIM workshop on Generalized Kostka polynomials

External links

• Short tables of Kostka polynomials
• Long tables of Kostka polynomials