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.
A polynomial is alternating if it changes sign under any transposition of the variables.
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).
Schur polynomials can be expressed as linear combinations of monomial symmetric functions mμ with non-negative integer coefficients Kλμ called Kostka numbers, The Kostka numbers Kλμ are given by the number of semi-standard Young tableaux of shape λ and weight μ.
[2] In both identities, functions with negative subscripts are defined to be zero.
Another determinantal identity is Giambelli's 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 Cauchy identity for Schur functions (now in infinitely many variables), and its dual state that and where the sum is taken over all partitions λ, and
The Murnaghan–Nakayama rule expresses a product of a power-sum symmetric function with a Schur polynomial, in terms of Schur polynomials: where the sum is over all partitions μ such that μ/λ is a rim-hook of size r and ht(μ/λ) is the number of rows in the diagram μ/λ.
Pieri's formula is a special case of the Littlewood-Richardson rule, which expresses the product
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. See also the Hook length formula which computes the same quantity for fixed λ.
Using Ferrers diagrams or some other method, we find that there are just four partitions of 4 into at most three parts.
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.
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.
If we write χλρ 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. A proof of this can be found in R. Stanley's Enumerative Combinatorics Volume 2, Corollary 7.17.5.
This decomposition reflects how a permutation module is decomposed into irreducible representations.
There are several approaches to prove Schur positivity of a given symmetric function F. If F is described in a combinatorial manner, a direct approach is to produce a bijection with semi-standard Young tableaux.
This approach also uses a graph structure, but on the objects representing the expansion in the fundamental quasisymmetric basis.
Skew Schur functions sλ/μ depend on two partitions λ and μ, and can be defined by the property Here, the inner product is the Hall inner product, for which the Schur polynomials form an orthonormal basis.
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
Given a partition λ, and a sequence a1, a2,... one can define the double Schur polynomial sλ(x || a) as
A combinatorial rule for the Littlewood-Richardson coefficients (depending on the sequence a) was given by A.I Molev.
[3] In particular, this implies that the shifted Schur polynomials have non-negative Littlewood-Richardson coefficients.
Given a partition λ, and a doubly infinite sequence ...,a−1, a0, a1, ... one can define the factorial Schur polynomial sλ(x|a) as
It is clear that if we let ai = 0 for all i, we recover the usual Schur polynomial sλ.
The double Schur polynomials and the factorial Schur polynomials in n variables are related via the identity sλ(x||a) = sλ(x|u) where an−i+1 = ui.