Schur polynomial

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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.

Definition[edit]

Schur polynomials correspond to integer partitions. Given a partition

 d = d_1 + d_2 + \cdots + d_n, \; \; d_1 \geq d_2 \geq \cdots \ge d_n

(where each dj is a non-negative integer), the following functions are alternating polynomials (in other words they change sign under any transposition of the variables):

 a_{(d_1+n-1, d_2+n-2, \dots , d_n)} (x_1, x_2, \dots , x_n) =
\det \left[ \begin{matrix} x_1^{d_1+n-1} & x_2^{d_1+n-1} & \dots & x_n^{d_1+n-1} \\
x_1^{d_2+n-2} & x_2^{d_2+n-2} & \dots & x_n^{d_2+n-2} \\
\vdots & \vdots & \ddots & \vdots \\
x_1^{d_n} & x_2^{d_n} & \dots & x_n^{d_n} \end{matrix} \right]

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_{(d_1, d_2, \dots , d_n)} (x_1, x_2, \dots , x_n) =
\frac{ a_{(d_1+n-1, d_2+n-2, \dots , d_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[edit]

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:


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 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 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 + n - 1} \\
e_{\lambda'_2-1} & e_{\lambda'_2} & \dots & e_{\lambda'_2+n-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)^j}

and \lambda' = \{\lambda_1', \lambda_2', \ldots, \lambda_l'\} is the dual partition to \lambda.

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

 (a_{1}, ... a_{r}| b_{1}, ... b_{r})

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

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

Schur polynomials can be expressed as linear combinations of monomial symmetric functions mμ with non-negative integer coefficients Kλμ called Kostka numbers:

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

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

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 λi is d.

Example[edit]

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:

  1.  S_{(2,1,1)} = e_1 \, e_3
  2.  S_{(2,2,0)} = e_2^2 - e_1 \, e_3
  3.  S_{(3,1,0)} = e_1^2 \, e_2 - e_2^2 - e_1 \, e_3
  4.  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[edit]

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 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_{\rho=(1^{r_1},2^{r_2},3^{r_3},\dots)}\chi^\lambda_\rho \prod_k \frac{p^{r_k}_k}{r_k!},

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

Skew Schur functions[edit]

Skew Schur functions sλ/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.

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.

See also[edit]

References[edit]

  1. ^ 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 
  2. ^ 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