Schur polynomial

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

${\displaystyle s_{\lambda }=\det(h_{\lambda _{i}+j-i})_{i,j=1}^{l(\lambda )}=\det \left[{\begin{matrix}h_{\lambda _{1}}&h_{\lambda _{1}+1}&\dots &h_{\lambda _{1}+n-1}\\h_{\lambda _{2}-1}&h_{\lambda _{2}}&\dots &h_{\lambda _{2}+n-2}\\\vdots &\vdots &\ddots &\vdots \\h_{\lambda _{n}-n+1}&h_{\lambda _{n}-n+2}&\dots &h_{\lambda _{n}}\end{matrix}}\right],}$

where h_i := s_(i).[1]

The second Jacobi-Trudi formula expresses the Schur polynomial as a determinant in terms of the elementary symmetric polynomials,

${\displaystyle s_{\lambda }=\det(e_{\lambda '_{i}+j-i})_{i,j=1}^{l(\lambda ')}=\det \left[{\begin{matrix}e_{\lambda '_{1}}&e_{\lambda '_{1}+1}&\dots &e_{\lambda '_{1}+l-1}\\e_{\lambda '_{2}-1}&e_{\lambda '_{2}}&\dots &e_{\lambda '_{2}+l-2}\\\vdots &\vdots &\ddots &\vdots \\e_{\lambda '_{l}-l+1}&e_{\lambda '_{l}-l+2}&\dots &e_{\lambda '_{l}}\end{matrix}}\right],}$

where e_i := s_(1ⁱ) and λ' is the conjugate partition to λ.[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

${\displaystyle (a_{1},\ldots ,a_{r}\mid b_{1},\ldots ,b_{r})}$

where, for each diagonal element in position ii, a_i denotes the number of boxes to the right in the same row and b_i denotes the number of boxes beneath it in the same column (the arm and leg lengths, respectively).

The Giambelli identity expresses the Schur function corresponding to this partition as the determinant

${\displaystyle s_{(a_{1},\ldots ,a_{r}\mid b_{1},\ldots ,b_{r})}=\det(s_{(a_{i}\mid b_{j})})}$

of those for hook partitions.

The Cauchy identity for Schur functions (now in infinitely many variables), and its dual state that

${\displaystyle \sum _{\lambda }s_{\lambda }(x)s_{\lambda }(y)=\sum _{\lambda }m_{\lambda }(x)h_{\lambda }(y)=\prod _{i,j}(1-x_{i}y_{j})^{-1},}$

and

${\displaystyle \sum _{\lambda }s_{\lambda }(x)s_{\lambda '}(y)=\sum _{\lambda }m_{\lambda }(x)e_{\lambda }(y)=\prod _{i,j}(1+x_{i}y_{j}),}$

where the sum is taken over all partitions λ, and ${\displaystyle h_{\lambda }(x)}$, ${\displaystyle e_{\lambda }(x)}$ denote the complete symmetric functions and elementary symmetric functions, respectively. If the sum is taken over products of Schur polynomials in ${\displaystyle n}$ variables ${\displaystyle (x_{1},\dots ,x_{n})}$, the sum includes only partitions of length ${\displaystyle \ell (\lambda )\leq n}$ since otherwise the Schur polynomials vanish.

There are many generalizations of these identities to other families of symmetric functions. For example, Macdonald polynomials, Schubert polynomials and Grothendieck polynomials admit Cauchy-like identities.

The Schur polynomial can also be computed via a specialization of a formula for Hall–Littlewood polynomials,

${\displaystyle s_{\lambda }(x_{1},\dotsc ,x_{n})=\sum _{w\in S_{n}/S_{n}^{\lambda }}w\left(x^{\lambda }\prod _{\lambda _{i}>\lambda _{j}}{\frac {x_{i}}{x_{i}-x_{j}}}\right)}$

where ${\displaystyle S_{n}^{\lambda }}$ is the subgroup of permutations such that ${\displaystyle \lambda _{w(i)}=\lambda _{i}}$ for all i, and w acts on variables by permuting indices.

The Murnaghan–Nakayama rule expresses a product of a power-sum symmetric function with a Schur polynomial, in terms of Schur polynomials:

${\displaystyle p_{r}\cdot s_{\lambda }=\sum _{\mu }(-1)^{ht(\mu /\lambda )+1}s_{\mu }}$

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.

The Littlewood–Richardson coefficients depend on three partitions, say ${\displaystyle \lambda ,\mu ,\nu }$, of which ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ describe the Schur functions being multiplied, and ${\displaystyle \nu }$ gives the Schur function of which this is the coefficient in the linear combination; in other words they are the coefficients ${\displaystyle c_{\lambda ,\mu }^{\nu }}$ such that

${\displaystyle s_{\lambda }s_{\mu }=\sum _{\nu }c_{\lambda ,\mu }^{\nu }s_{\nu }.}$

The Littlewood–Richardson rule states that ${\displaystyle c_{\lambda ,\mu }^{\nu }}$ is equal to the number of Littlewood–Richardson tableaux of skew shape ${\displaystyle \nu /\lambda }$ and of weight ${\displaystyle \mu }$.

Pieri's formula is a special case of the Littlewood-Richardson rule, which expresses the product ${\displaystyle h_{r}s_{\lambda }}$ in terms of Schur polynomials. The dual version expresses ${\displaystyle e_{r}s_{\lambda }}$ in terms of Schur polynomials.

Evaluating the Schur polynomial s_λ 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

$${\displaystyle s_{\lambda }(1,1,\dots ,1)=\prod _{1\leq i<j\leq n}{\frac {\lambda _{i}-\lambda _{j}+j-i}{j-i}}.}$$

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 λ_i is d. See also the Hook length formula which computes the same quantity for fixed λ.

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. The Edelman–Greene correspondence and the Robinson–Schensted–Knuth correspondence are examples of such bijections.

A bijection with more structure is a proof using so called crystals. This method can be described as defining a certain graph structure described with local rules on the underlying combinatorial objects.

A similar idea is the notion of dual equivalence. This approach also uses a graph structure, but on the objects representing the expansion in the fundamental quasisymmetric basis. It is closely related to the RSK-correspondence.

Skew Schur functions s_λ/μ depend on two partitions λ and μ, and can be defined by the property

${\displaystyle \langle s_{\lambda /\mu },s_{\nu }\rangle =\langle s_{\lambda },s_{\mu }s_{\nu }\rangle .}$

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

${\displaystyle s_{\lambda /\mu }=\det(h_{\lambda _{i}-\mu _{j}-i+j})_{i,j=1}^{l(\lambda )}}$

${\displaystyle s_{\lambda '/\mu '}=\det(e_{\lambda _{i}-\mu _{j}-i+j})_{i,j=1}^{l(\lambda )}}$

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 ${\displaystyle \lambda /\mu }$.

The skew Schur polynomials expands positively in Schur polynomials. A rule for the coefficients is given by the Littlewood-Richardson rule.

The double Schur polynomials[3] can be seen as a generalization of the shifted Schur polynomials. These polynomials are also closely related to the factorial Schur polynomials. Given a partition λ, and a sequence a₁, a₂,… one can define the double Schur polynomial s_λ(x || a) as

$${\displaystyle s_{\lambda }(x||a)=\sum _{T}\prod _{\alpha \in \lambda }(x_{T(\alpha )}-a_{T(\alpha )-c(\alpha )})}$$

where the sum is taken over all reverse semi-standard Young tableaux T of shape λ, and integer entries in 1, …, n. Here T(α) denotes the value in the box α in T and c(α) is the content of the box.

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.

The shifted Schur polynomials s^*_λ(y) can be obtained from the double Schur polynomials by specializing a_i = −i and y_i = x_i + i.

The double Schur polynomials are special cases of the double Schubert polynomials.

The factorial Schur polynomials may be defined as follows. Given a partition λ, and a doubly infinite sequence …,a₋₁, a₀, a₁, … one can define the factorial Schur polynomial s_λ(x|a) as

$${\displaystyle s_{\lambda }(x|a)=\sum _{T}\prod _{\alpha \in \lambda }(x_{T(\alpha )}-a_{T(\alpha )+c(\alpha )})}$$

where the sum is taken over all semi-standard Young tableaux T of shape λ, and integer entries in 1, …, n. Here T(α) denotes the value in the box α in T and c(α) is the content of the box.

There is also a determinant formula,

$${\displaystyle s_{\lambda }(x|a)={\frac {\det[(x_{j}|a)^{\lambda _{i}+n-i}]_{i,j=1}^{l(\lambda )}}{\prod _{i<j}(x_{i}-x_{j})}}}$$

where (y|a)^k = (y − a₁) ... (y − a_k). It is clear that if we let a_i = 0 for all i, we recover the usual Schur polynomial s_λ.

The double Schur polynomials and the factorial Schur polynomials in n variables are related via the identity s_λ(x||a) = s_λ(x|u) where a_n−i+1 = u_i.

There are numerous generalizations of Schur polynomials:

• Hall–Littlewood polynomials
• Shifted Schur polynomials
• Flagged Schur polynomials
• Schubert polynomials
• Stanley symmetric functions (also known as stable Schubert polynomials)
• Key polynomials (also known as Demazure characters)
• Quasi-symmetric Schur polynomials
• Row-strict Schur polynomials
• Jack polynomials
• Modular Schur polynomials
• Loop Schur functions
• Macdonald polynomials
• Schur polynomials for the symplectic and orthogonal group.
• k-Schur functions
• Grothendieck polynomials (K-theoretical analogue of Schur polynomials)
• LLT polynomials