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:
s λ ( x 1 , … , x n ) = ∑ μ K λ μ ( t ) P μ ( x 1 , … , x n ; t ) . {\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.
In fact, they show that
K λ μ ( t ) = ∑ T ∈ S S Y T ( λ , μ ) t c h a r g e ( T ) {\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 Pμ) to Schur polynomials sλ:
s λ ( x 1 , … , x n ) = ∑ μ K λ μ ( q , t ) J μ ( x 1 , … , x n ; q , t ) {\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
J μ ( x 1 , … , x n ; q , t ) = P μ ( x 1 , … , x n ; q , t ) ∏ s ∈ μ ( 1 − q a r m ( s ) t l e g ( s ) + 1 ) . {\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:
K λ μ = K λ μ ( 1 ) = K λ μ ( 0 , 1 ) . {\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{{citation}}: CS1 maint: deprecated archival service (link) 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
