Kostka polynomial

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:

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

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:

s_\lambda(x_1,\ldots,x_n) =\sum_\mu K_{\lambda\mu}(q,t)J_\mu(x_1,\ldots,x_n;q,t)\

where

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 1 or 2 variable Kostka polynomials:

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

References

↑ Lascoux, A.; Scützenberger, M.P. "Sur une conjecture de H.O. Foulkes". Comptes Rendus de l'Académie des Sciences, Série A-B. 286 (7): A323–A324.

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., 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

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[1] Lascoux, A.; Scützenberger, M.P. "Sur une conjecture de H.O. Foulkes". Comptes Rendus de l'Académie des Sciences, Série A-B. 286 (7): A323–A324.