mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Kostka polynomials, named after the mathematician

Carl Kostka Carl Gottfried Franz Albert Kostka (3 December 1846 in Lyck – 28 December 1921 in Insterburg) was a mathematician who introduced Kostka numbers in 1882. He lived and worked in Insterburg. See also *Kostka polynomial In mathematics, Kostka pol ...

, are families of

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

s that generalize the

Kostka number In mathematics, the Kostka number ''K''λμ (depending on two Partition (number theory), integer partitions λ and μ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape λ and weight μ. They were intro ...

s. 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. 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 positive 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 parti ...

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 polynomials ''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 (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

d to have non-negative

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...

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 the University of Marne la Vallée and Nankai University. His research was primarily in algebraic combinatorics, particularly Hecke algebras and Young tableaux. L ...

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 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_\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^).\

s 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