Schur polynomial explained

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 representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials.

Definition (Jacobi's bialternant formula)

Schur polynomials are indexed by integer partitions. Given a partition,where, and each is a non-negative integer, the functions

are alternating polynomials by properties of the determinant. A polynomial is alternating if it changes sign under any transposition of the variables.

Since they are alternating, they are all divisible by the Vandermonde determinantThe Schur polynomials are defined as the ratio

$$s\_\; (x\_1,\; x\_2,\; \backslash dots,\; x\_n)\; =\backslash frac.$$

This is known as the bialternant formula of Jacobi. It is a special case of the Weyl character formula.

This is a symmetric function because the numerator and denominator are both alternating, and a polynomial since all alternating polynomials are divisible by the Vandermonde determinant.

Properties

The degree Schur polynomials in variables are a linear basis for the space of homogeneous degree symmetric polynomials in variables. For a partition, the Schur polynomial is a sum of monomials,

s _{λ(x} _{1,x} _{2,\ldots,x} _{n)=\sum} _{T} x ^{T} = \sum _{T} x ^{t _{1}} _{1} … x ^{t _{n}} _{n}

where the summation is over all semistandard Young tableaux of shape . The exponents give the weight of, in other words each counts the occurrences of the number in . This can be shown to be equivalent to the definition from the first Giambelli formula using the Lindström–Gessel–Viennot lemma (as outlined on that page).

Schur polynomials can be expressed as linear combinations of monomial symmetric functions with non-negative integer coefficients called Kostka numbers,

s _{λ=} \sum _{\mu} K _{λ\mu} m _{\mu.}

The Kostka numbers are given by the number of semi-standard Young tableaux of shape λ and weight μ.

Jacobi−Trudi identities

The first Jacobi−Trudi formula expresses the Schur polynomial as a determinantin terms of the complete homogeneous symmetric polynomials,

s _{λ} = \det(h _{λ _{i} + j - i} ) ^{l( λ )} _{i,j =1} = \det\left[ \begin{matrix} h _{λ _{1}} & h _{λ _{1} +1} & ... & h _{λ _{1} + n -1} \\ h _{λ _{2-1}} & h _{λ _{2}} & ... & h _{λ _{2+n-2}} \\ \vdots & \vdots & \ddots & \vdots \\ h _{λ _{n-n+1}} & h _{λ _{n-n+2}} & ... & h _{λ _{n}} \end{matrix} \right], where . The second Jacobi-Trudi formula expresses the Schur polynomial as a determinant in terms of the elementary symmetric polynomials,s _{λ} = \det(e _{λ ' _{i} + j - i} ) ^{l( λ ')} _{i,j =1} = \det\left[ \begin{matrix} e _{λ ' _{1}} & e _{λ ' _{1} +1} & ... & e _{λ ' _{1} + l -1} \\ e _{λ ' _{2-1}} & e _{λ ' _{2}} & ... & e _{λ ' _{2+} l -2} \\ \vdots & \vdots & \ddots & \vdots \\ e _{λ ' _{l-l+1}} & e _{λ ' _{l-l+2}} & ... & e _{λ ' _{l}} \end{matrix} \right], where and