Schur polynomial: Difference between revisions

In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables with integral coefficients. The elementary symmetric polynomials and the complete homogeneous symmetric polynomials are special cases of Schur polynomials. Schur polynomials depend on a partition of natural number d into at most n parts, and are then homogeneous of degree d. The importance of Schur polynomials stems for a large part from their role in representation theory, notably of symmetric groups or of general linear groups, where they correspond to irreducible representations. The set of all Schur polynomials for fixed values of d form a linear basis for the space of all symmetric polynomials that are homogeneous of degree d. As a consequence every symmetric polynomial P can be written in a unique way as a linear combination of Schur polynomials; moreover if P has integral coefficients, then the linear combination has so as well. 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.

Schur polynomials correspond to integer partitions. Given a partition

${\displaystyle d=d_{1}+d_{2}+\cdots +d_{n},\;\;d_{1}\geq d_{2}\geq \cdots \geq d_{n}}$

(where each ${\displaystyle d_{j}}$ is a non-negative integer), we can compute the corresponding Schur polynomial by expanding determinants

${\displaystyle s_{(d_{1},d_{2},\dots d_{n})}(x_{1},x_{2},\dots x_{n})={\frac {\det \left[{\begin{matrix}x_{1}^{d_{1}+n-1}&x_{2}^{d_{1}+n-1}&\dots &x_{n}^{d_{1}+n-1}\\x_{1}^{d_{2}+n-2}&x_{2}^{d_{2}+n-2}&\dots &x_{n}^{d_{2}+n-2}\\\vdots &\vdots &\ddots &\vdots \\x_{1}^{d_{n}}&x_{2}^{d_{n}}&\dots &x_{n}^{d_{n}}\end{matrix}}\right]}{\det \left[{\begin{matrix}x_{1}^{n-1}&x_{2}^{n-1}&\dots &x_{n}^{n-1}\\x_{1}^{n-2}&x_{2}^{n-2}&\dots &x_{n}^{n-2}\\\vdots &\vdots &\ddots &\vdots \\1&1&\dots &1\end{matrix}}\right].}}}$

This gives a symmetric function because the numerator and denominator are each determinants which change sign under any transposition of the variables. Furthermore, the denominator is a Vandermonde determinant:

${\displaystyle \Delta =\det \left[{\begin{matrix}x_{1}^{n-1}&x_{2}^{n-1}&\dots &x_{n}^{n-1}\\x_{1}^{n-2}&x_{2}^{n-2}&\dots &x_{n}^{n-2}\\\vdots &\vdots &\ddots &\vdots \\1&1&\dots &1\end{matrix}}\right]=\prod _{1\leq j<k\leq n}(x_{j}-x_{k}).}$

Each factor divides the determinant in the numerator, so the quotient is a polynomial, which can moreover be seen to be homogeneous of degree ${\displaystyle d}$.

Because we can readily enumerate the distinct partitions of d into n parts using Ferrers diagrams, using this formula we can write down all the degree d Schur polynomials in n variables, giving a linear basis for the space of homogeneous degree d symmetric polynomials in n variables.

Each Schur polynomial in n variables is a polynomial function of the elementary symmetric polynomials

${\displaystyle e_{0}(x_{1},x_{2},\dots ,x_{n})=1,}$

${\displaystyle e_{1}(x_{1},x_{2},\dots ,x_{n})=\sum _{1\leq j\leq n}x_{j},}$

${\displaystyle e_{2}(x_{1},x_{2},\dots ,x_{n})=\sum _{1\leq j<k\leq n}x_{j}\,x_{k},}$

and so forth, down to

${\displaystyle e_{n}(x_{1},x_{2},\dots ,x_{n})=x_{1}\,x_{2}\cdots x_{n}.}$

Explicit expressions can be found using computational techniques from elimination theory, perhaps the most elementary of which are Gröbner bases using an elimination order.

For a partition ${\displaystyle \lambda }$, the Schur function can be expanded as a sum of monomials as

${\displaystyle s_{\lambda }(x_{1},x_{2},\ldots ,x_{n})=\sum _{T}x_{1}^{t_{1}}\cdots x_{n}^{t_{n}}}$

where the summation is over all semistandard Young tableaux ${\displaystyle T}$ of shape ${\displaystyle \lambda }$; the exponents ${\displaystyle t_{1},\ldots ,t_{n}}$ give the weight of ${\displaystyle T}$, in other words each ${\displaystyle t_{i}}$ counts the occurrences of the number ${\displaystyle i}$ in ${\displaystyle T}$.

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

${\displaystyle s_{(2,1,1)}(x_{1},x_{2},x_{3})={\frac {1}{\Delta }}\;\det \left[{\begin{matrix}x_{1}^{4}&x_{2}^{4}&x_{3}^{4}\\x_{1}^{2}&x_{2}^{2}&x_{3}^{2}\\x_{1}&x_{2}&x_{3}\end{matrix}}\right]=x_{1}\,x_{2}\,x_{3}\,(x_{1}+x_{2}+x_{3})}$

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

and so forth. Summarizing:

1. ${\displaystyle s_{(2,1,1)}=e_{1}\,e_{3}}$
2. ${\displaystyle s_{(2,2,0)}=e_{2}^{2}-e_{1}\,e_{3}}$
3. ${\displaystyle s_{(3,1,0)}=e_{1}^{2}\,e_{2}-e_{2}^{2}-e_{1}\,e_{3}}$
4. ${\displaystyle s_{(4,0,0)}=e_{1}^{4}-3\,e_{1}^{2}\,e_{2}+2\,e_{1}\,e_{3}+e_{2}^{2}}$

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,

${\displaystyle \phi (x_{1},x_{2},x_{3})=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}}$

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

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

The Schur polynomials occur in the representation theory of the symmetric groups, general linear groups, and unitary groups, and in fact this is how they arose. The Weyl character formula helps to generalize Schur's work to other compact and semisimple Lie groups.

• symmetric polynomial
• elementary symmetric polynomial
• Galois theory
• Issai Schur

• Sagan, Bruce E. (2001) [1994], "Schur functions in algebraic combinatorics", Encyclopedia of Mathematics, EMS Press
• Sturmfels, Bernd (1993). Algorithms in Invariant Theory. New York: Springer. ISBN 0-387-82445-6. is a beautiful introduction to computational methods in invariant theory.
• Tignol, Jean-Pierre (2001). Galois's Theory of Algebraic Equations. Singapore: World Scientific. ISBN 981-02-4541-6. Offers some nice historical background.