“Kostka polynomial”的意思、由来-开放百科全书

词条

Kostka polynomial

释义

Examples
References
External links

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.{{citation needed|date=April 2012}}

The one variable specializations of the Kostka polynomials can be used to relate Hall-Littlewood polynomials P_μ to Schur polynomials s_λ:

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

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:

where

Kostka numbers are special values of the 1 or 2 variable Kostka polynomials:

Examples

References

1. ^{{cite journal|last1=Lascoux|first1=A.|last2=Scützenberger|first2=M.P.|title=Sur une conjecture de H.O. Foulkes|journal=Comptes Rendus de l'Académie des Sciences, Série A-B|volume=286|issue=7|pages=A323-A324}}

{{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 }}{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}
{{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|bibcode=2004math......1298N}}

{{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

Short tables of Kostka polynomials
Long tables of Kostka polynomials

1 : Symmetric functions

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。