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In [[mathematics]], '''Kostka polynomials''', named after the mathematician [[Carl Kostka]], are families of [[polynomial]]s 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 [[Partition (number theory)|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'''.{{cn|date=April 2012}}

The one variable specializations of the Kostka polynomials can be used to express [[Hall-Littlewood polynomial]]s ''P''μ as a linear combination of [[Schur polynomial]]s ''s''λ:

: <math>P_\mu(x_1,\ldots,x_n;t) =\sum_\lambda K_{\lambda\mu}(t)s_\lambda(x_1,\ldots,x_n).\ </math>

The Macdonald–Kostka polynomials can be used to express [[Macdonald polynomial]]s (also denoted by ''P''μ) as a linear combination of [[Schur polynomial]]s ''s''λ:

: <math>J_\mu(x_1,\ldots,x_n;q,t) =\sum_\lambda K_{\lambda\mu}(q,t)s_\lambda(x_1,\ldots,x_n)\ </math>

where
: <math>J_\mu(x_1,\ldots,x_n;q,t) = P_\mu(x_1,\ldots,x_n;q,t)\prod_{s\in\mu}(1-q^{a(s)}t^{l(s)}).\ </math>

[[Kostka number]]s are special values of the 1 or 2 variable Kostka polynomials:
: <math>K_{\lambda\mu}= K_{\lambda\mu}(1)=K_{\lambda\mu}(0,1).\ </math>

==Examples==

{{Empty section|date=July 2010}}

==References==

*{{Citation | last1=Macdonald | first1=I. G. | author1-link=Ian G. Macdonald | title=Symmetric functions and Hall polynomials | url=http://www.oup.com/uk/catalogue/?ci=9780198504504 | publisher=The Clarendon Press Oxford University Press | edition=2nd | series=Oxford Mathematical Monographs | isbn=978-0-19-853489-1 | mr=1354144 | year=1995}}
*{{citation|mr=2011741
|last=Nelsen|first= Kendra|last2= Ram|first2=Arun
|chapter=Kostka-Foulkes polynomials and Macdonald spherical functions|title= Surveys in combinatorics, 2003 (Bangor)|pages= 325–370,
|series=London Math. Soc. Lecture Note Ser.|volume= 307|publisher= Cambridge Univ. Press|place=Cambridge|year= 2003
|arxiv= math/0401298}}
*{{citation|first=J. R.|last= Stembridge|title=Kostka-Foulkes Polynomials of General Type|series=lecture notes from AIM workshop on Generalized Kostka polynomials|year= 2005|url=http://www.aimath.org/WWN/kostka}}

==External links==
*[http://garsia.math.yorku.ca/MPWP/qtanalogs/sttab/index.html Short tables of Kostka polynomials]
*[http://garsia.math.yorku.ca/MPWP/qtTEXtables.html Long tables of Kostka polynomials]

[[Category:Symmetric functions]]

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129.206.28.121: /* Static Limit */ bigger parenthesis