Schurfüggvények

Schurfüggvények, also known as Schur functions, are a class of symmetric polynomials that arise in the representation theory of the general linear group. They are named after Issai Schur, who introduced them in his doctoral thesis in 1901. Schur functions form a basis for the vector space of symmetric polynomials, and they play a crucial role in the study of characters of the symmetric group and the general linear group.

The definition of a Schur function is closely related to Young diagrams and Young tableaux. A Young diagram is a graphical representation of a partition of a non-negative integer. A standard Young tableau is a filling of a Young diagram with the numbers from 1 to n, where n is the number of cells in the diagram, such that the numbers increase along rows and down columns. For a given partition lambda, the Schur function s_lambda(x_1, x_2, ...) is defined as the sum of monomials corresponding to all standard Young tableaux of shape lambda.

Alternatively, Schur functions can be defined using determinants. They are the unique symmetric polynomials that satisfy certain properties related to the characters of the general linear group. Their relationship to characters means that Schur functions can be used to express the irreducible representations of the symmetric group and the general linear group. This connection makes them a fundamental tool in algebraic combinatorics and representation theory. They also find applications in areas such as random matrix theory and statistical mechanics.