Schur polynomial

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 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.

Definition

Schur polynomials are indexed by integer partitions. Given a partition $λ = (λ 1, λ 2, \dots, λ n)$ , where $λ 1 \geq λ 2 \geq \dots \geq λ n$ , and each $λ j$ is a non-negative integer, the functions

a_{(\lambda_1+n-1, \lambda_2+n-2, \dots , \lambda_n)} (x_1, x_2, \dots , x_n) = \det \left[ \begin{matrix} x_1^{\lambda_1+n-1} & x_2^{\lambda_1+n-1} & \dots & x_n^{\lambda_1+n-1} \\ x_1^{\lambda_2+n-2} & x_2^{\lambda_2+n-2} & \dots & x_n^{\lambda_2+n-2} \\ \vdots & \vdots & \ddots & \vdots \\ x_1^{\lambda_n} & x_2^{\lambda_n} & \dots & x_n^{\lambda_n} \end{matrix} \right]

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 determinant,

a_{(n-1, n-2, \dots , 0)} (x_1, x_2, \dots , x_n) = \det \left[ \begin{matrix} x_1^{n-1} & x_2^{n-1} & \dots & x_n^{n-1} \\ x_1^{n-2} & x_2^{n-2} & \dots & x_n^{n-2} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \dots & 1 \end{matrix} \right] = \prod_{1 \leq j < k \leq n} (x_j-x_k).

The Schur polynomials are defined as the ratio

s_{\lambda} (x_1, x_2, \dots , x_n) = \frac{ a_{(\lambda_1+n-1, \lambda_2+n-2, \dots , \lambda_n+0)} (x_1, x_2, \dots , x_n)} {a_{(n-1, n-2, \dots , 0)} (x_1, x_2, \dots , x_n) }.

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 $d$ Schur polynomials in $n$ variables are a linear basis for the space of homogeneous degree $d$ symmetric polynomials in $n$ variables. For a partition $λ = (λ 1, λ 2, ..., λ n)$ , the Schur polynomial is a sum of monomials,

s_\lambda(x_1,x_2,\ldots,x_n)=\sum_T x^T = \sum_T x_1^{t_1}\cdots x_n^{t_n}

where the summation is over all semistandard Young tableaux $T$ of shape $λ$ . The exponents $t 1, ..., t n$ give the weight of $T$ , in other words each $t i$ 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 $m μ$ with non-negative integer coefficients $K λμ$ called Kostka numbers,

s_\lambda= \sum_\mu K_{\lambda\mu}m_\mu.\

The Kostka numbers $K λμ$ 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 determinant in terms of the complete homogeneous symmetric polynomials,

s_{\lambda} = \det_{ij} h_{\lambda_{i} + j - i}, 1 \le i,j \le n = \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|,

^[1]

where $h i := s (i)$ .

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

s_{\lambda} = \det_{ij} e_{\lambda'_{i} + j - i}, 1 \le i,j \le l = \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|,

^[2]

where $e i := s (1 i)$ . and $λ'$ is the conjugate partition to $λ$ .

These two formulae are known as determinantal identities.

The Giambelli identity

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

(a_{1}, ... a_{r}| b_{1}, ... 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 partition as the determinant

s_{ (a_{1}, ... a_{r}| b_{1}, ... b_{r})} = \det ( s_{(a_{i} | b_{j})})

The Cauchy identity

The Cauchy identities for the Schur functions (now in infinitely many variables), states that

\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

\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. There are many generalizations of these identities, for example, Hall-Littlewood polynomials, Schubert polynomials and Grothendieck polynomials admit Cauchy-like identities.

The Murnaghan−Nakayama rule

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

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 rule and Pieri's formula

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

s_\lambda s_\mu=\sum_\nu c_{\lambda,\mu}^\nu s_\nu.

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

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

Specializations

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

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.

Example

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

s_{(2,1,1)} (x_1, x_2, x_3) = \frac{1}{\Delta} \; \det \left[ \begin{matrix} x_1^4 & x_2^4 & x_3^4 \\ x_1^2 & x_2^2 & x_3^2 \\ x_1 & x_2 & x_3 \end{matrix} \right] = x_1 \, x_2 \, x_3 \, (x_1 + x_2 + x_3)

s_{(2,2,0)} (x_1, x_2, x_3) = \frac{1}{\Delta} \; \det \left[ \begin{matrix} x_1^4 & x_2^4 & x_3^4 \\ x_1^3 & x_2^3 & x_3^3 \\ 1 & 1 & 1 \end{matrix} \right]= x_1^2 \, x_2^2 + x_1^2 \, x_3^2 + x_2^2 \, x_3^2 + x_1^2 \, x_2 \, x_3 + x_1 \, x_2^2 \, x_3 + x_1 \, x_2 \, x_3^2

and so on. Summarizing:

$s_{(2,1,1)} = e_1 \, e_3$
$s_{(2,2,0)} = e_2^2 - e_1 \, e_3$
$s_{(3,1,0)} = e_1^2 \, e_2 - e_2^2 - e_1 \, e_3$
$s_{(4,0,0)} = e_1^4 - 3 \, e_1^2 \, e_2 + 2 \, e_1 \, e_3 + e_2^2.$

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,

\phi(x_1, x_2, x_3) = x_1^4 + x_2^4 + x_3^4

is obviously a symmetric polynomial which is homogeneous of degree four, and we have

\phi = s_{(2,1,1)} - s_{(3,1,0)} + s_{(4,0,0)}.\,\!

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 s_λ in terms of the symmetric power functions $p_k=\sum_i x_i^k$ . 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

s_\lambda = \sum_{\nu} \frac{\chi^\lambda_\nu}{z_\nu} p_\nu = \sum_{\rho=(1^{r_1},2^{r_2},3^{r_3},\dots)}\chi^\lambda_\rho \prod_k \frac{p^{r_k}_k}{r_k! k^{r_k} },

where ρ = (1^r₁, 2^r₂, 3^r₃, ...) means that the partition ρ has r_k parts of length k.

A proof of this can be found in R. Stanley's Enumerative combinatoric II, Corollary 7.17.5.

The integers χλ
ρ can be computed using the Murnaghan–Nakayama rule.

Skew Schur functions

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

\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

s_{\lambda/\mu} = (h_{\lambda_i - \mu_j -i + j}), 1\leq i,j \leq l(\lambda)

s_{\lambda'/\mu'} = (e_{\lambda_i - \mu_j -i + j}), 1\leq i,j \leq 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 $\lambda/\mu$ .

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

Generalizations

There are numerous generalizations of Schur polynomials:

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

Double Schur polynomials

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 parititon $λ$ , and a sequence $a 1, a 2, \dots$ one can define the double Schur polynomial $s λ (x || a)$ as

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 $1, \dots, n$ . 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), is given by A.I Molev in.^[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.

Factorial Schur polynomials

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

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,

s_\lambda(x|a) = \frac{\det[(x_j|a)^{\lambda_i+n-i}]_{1\leq i,j\leq n}}{\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.

References

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
1 2 Molev, A.I. (June 2009). "Littlewood–Richardson polynomials". Journal of Algebra 321 (11): 3450–3468. doi:10.1016/j.jalgebra.2008.02.034.

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