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 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 functions 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.
Schur polynomials correspond to integer partitions. Given a partition
Since they are alternating, they are all divisible by the Vandermonde determinant:
The Schur polynomials are defined as the ratio:
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.
The degree d Schur polynomials in n variables are a linear basis for the space of homogeneous degree d symmetric polynomials in n variables.
For a partition λ = (λ1, λ2, ..., λn), the Schur polynomial is a sum of monomials:
where the summation is over all semistandard Young tableaux T of shape λ. The exponents t1, ..., tn give the weight of T, in other words each ti counts the occurrences of the number i in T. 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).
The second Jacobi-Trudi formula expresses the Schur polynomial as a determinant in terms of the elementary symmetric polynomials:
and is the dual partition to .
These two formulae are known as "determinantal identities". Another such identity is the Giambelli formula, which expresses the Schur function for an arbitrary partition in terms of those for the "hook partitions" contained within the Young diagram. In Frobenius' notation, the partition is denoted
where, for each diagonal element in position ii, ai denotes the number of boxes to the right in the same row and bi denotes the number of boxes beneath it in the same column (the "arm" and "leg" lengths, respectively).
The Giambelli identity expresses the partition as the determinant
Evaluating the Schur polynomial Sλ in (1,1,...,1) gives the number of semi-standard Young tableaux of shape λ with entries in 1,2,...n. One can show, by using the Weyl character formula for example, that
In this formula, λ, the tuple indicating the width of each row of the Young diagram, is implicitly extended with zeros until it has length n. The sum of the elements λi is d.
The following extended example should help clarify these ideas. Consider the case n = 3, d = 4. Using Ferrers diagrams or some other method, we find that there are just four partitions of 4 into at most three parts. We have
and so on. Summarizing:
Every homogeneous degree-four symmetric polynomial in three variables can be expressed as a unique linear combination of these four Schur polynomials, and this combination can again be found using a Gröbner basis for an appropriate elimination order. For example,
is obviously a symmetric polynomial which is homogeneous of degree four, and we have
Relation to representation theory
The Schur polynomials occur in the representation theory of the symmetric groups, general linear groups, and unitary groups. The Weyl character formula implies that the Schur polynomials are the characters of finite-dimensional irreducible representations of the general linear groups, and helps to generalize Schur's work to other compact and semisimple Lie groups.
Several expressions arise for this relation, one of the most important being the expansion of the Schur functions sλ in terms of the symmetric power functions . If we write χλ
ρ for the character of the representation of the symmetric group indexed by the partition λ evaluated at elements of cycle type indexed by the partition ρ, then
where ρ = (1r1, 2r2, 3r3, ...) means that the partition ρ has rk parts of length k.
Skew Schur functions
Skew Schur functions sλ/μ depend on two partitions λ and μ, and can be defined by the property
Similar to the ordinary Schur polynomials, there are numerous ways to compute these. The corresponding Jacobi-Trudi identities are
There is also a combinatorial interpretation of the skew Schur polynomials, namely it is a sum over all semi-standard Young tableaux (or column-strict tableaux) of the skew shape .
- Littlewood–Richardson rule, where one finds some identities involving Schur polynomials.
- Schubert polynomials, a generalization of Schur polynomials.
- Macdonald, I. G. (1995). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs (2nd ed.). The Clarendon Press Oxford University Press. ISBN 978-0-19-853489-1. MR 1354144.
- Sagan, Bruce E. (2001), "Schur functions in algebraic combinatorics", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Sturmfels, Bernd (1993). Algorithms in Invariant Theory. New York: Springer. ISBN 0-387-82445-6.
- Formula A.5 in Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6
- Formula A.6 in Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6