mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Kostka polynomials, named after the mathematician Carl Kostka, are families of

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s that generalize the Kostka numbers. They are studied primarily in

algebraic combinatorics Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algeb ...

and

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

. 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 In number theory and combinatorics, a partition of a non-negative integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same ...

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 polynomial In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In ...

s ''s''_λ: :

s_\lambda(x_1,\ldots,x_n) =\sum_\mu K_(t)P_\mu(x_1,\ldots,x_n;t).\

These polynomials were

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

d to have non-negative

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...

s by Foulkes, and this was later proved in 1978 by

Alain Lascoux Alain Lascoux (17 October 1944 – 20 October 2013) was a French mathematician at Université de Paris VII, University of Marne la Vallée and Nankai University. His research was primarily in algebraic combinatorics, particularly Affine Hecke alge ...

and

Marcel-Paul Schützenberger Marcel-Paul "Marco" Schützenberger (24 October 1920 – 29 July 1996) was a French mathematician and Doctor of Medicine. He worked in the fields of formal language, combinatorics, and information theory.Herbert Wilf, Dominique Foata, ''et al.'', ...

. In fact, they show that :

K_(t) = \sum_ t^

where the sum is taken over all semi-standard

Young tableau In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups an ...

x 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 In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald origi ...

(also denoted by ''P''_μ) to

s ''s''_λ: :

s_\lambda(x_1,\ldots,x_n) =\sum_\mu K_(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_(1-q^t^).\

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

K_= K_(1)=K_(0,1).\

Examples

References

* * *{{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
Symmetric functions