Kostka polynomial

{{short description|Certain family of polynomials}}

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:

: $s\_\backslash lambda(x\_1,\backslash ldots,x\_n)\; =\backslash sum\_\backslash mu\; K\_\{\backslash lambda\backslash mu\}(t)P\_\backslash mu(x\_1,\backslash ldots,x\_n;t).\backslash$

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.

{{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}}

In fact, they show that

: $K\_\{\backslash lambda\backslash mu\}(t)\; =\; \backslash sum\_\{T\; \backslash in\; SSYT(\backslash lambda,\backslash mu)\}\; t^\{charge(T)\}$

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:

: $s\_\backslash lambda(x\_1,\backslash ldots,x\_n)\; =\backslash sum\_\backslash mu\; K\_\{\backslash lambda\backslash mu\}(q,t)J\_\backslash mu(x\_1,\backslash ldots,x\_n;q,t)\backslash$

where

: $J\_\backslash mu(x\_1,\backslash ldots,x\_n;q,t)\; =\; P\_\backslash mu(x\_1,\backslash ldots,x\_n;q,t)\backslash prod\_\{s\backslash in\backslash mu\}(1-q^\{arm(s)\}t^\{leg(s)+1\}).\backslash$

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

: $K\_\{\backslash lambda\backslash mu\}=\; K\_\{\backslash lambda\backslash mu\}(1)=K\_\{\backslash lambda\backslash mu\}(0,1).\backslash$