Schur polynomial
= Schur polynomial
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{title2=$s_\lambda$}
{wiki}
= Schur function
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{synonym}
For an <integer partition> $\lambda$ with at most $N$ parts, padded with zeros, the <Schur polynomial> in $N$ <variables> is
$$
s_\lambda(x_1,\ldots,x_N)=\frac{\det[x_j^{\lambda_i+N-i}]_{i,j=1}^N}{\det[x_j^{N-i}]_{i,j=1}^N}.
$$
Both <determinants> change sign when two <variables> are exchanged. Every factor of the denominator's <Vandermonde determinant> divides the numerator, so the quotient is a <symmetric polynomial>, including at repeated <variable> values. The <bialternant formula> and <Jacobi–Trudi identity> give equivalent descriptions; the latter writes it as $\det[h_{\lambda_i-i+j}]$ in <complete homogeneous symmetric functions>.<On Immanants of Jacobi-Trudi Matrices and Permutations with Restricted Position by Richard P. Stanley and John R. Stembridge>[§1, equation(1.1), printed page262]{ref}
Equivalently, $s_\lambda$ sums the weight <monomials> of all <Semistandard Young tableaux> of shape $\lambda$, with entries in $\{1,\ldots,N\}$, weakly increasing along rows and strictly increasing down columns. For example,
$$
s_{(2,1)}(x_1,x_2)=x_1^2x_2+x_1x_2^2.
$$
The two <monomials> correspond to the two possible fillings of this three-cell shape. A single-row shape gives $s_{(r)}=h_r$, while a single-column shape gives an <elementary symmetric polynomial>.
Adding a zero <variable> leaves these expressions unchanged, defining a <Schur function> in the <ring> of <symmetric functions>. In each fixed degree, the <Schur functions> form a <basis> indexed by the <integer partitions> of that degree. The <Frobenius characteristic map> identifies this <basis> with the <irreducible characters> of <symmetric groups>, connecting tableau enumeration to <representation theory>.