Editing Symmetric polynomial (section)

== Special kinds of symmetric polynomials ==

There are a few types of symmetric polynomials in the variables ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub> that are fundamental.

=== Elementary symmetric polynomials ===
{{Main|Elementary symmetric polynomial}}
For each nonnegative [[integer]] ''k'', the elementary symmetric polynomial ''e''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) is the sum of all distinct products of ''k'' distinct variables.  (Some authors denote it by σ<sub>''k''</sub> instead.) For ''k''&nbsp;=&nbsp;0 there is only the [[empty product]] so ''e''<sub>0</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)&nbsp;=&nbsp;1, while for ''k''&nbsp;&gt;&nbsp;''n'', no products at all can be formed, so ''e''<sub>''k''</sub>(''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub>)&nbsp;=&nbsp;0 in these cases. The remaining ''n'' elementary symmetric polynomials are building blocks for all symmetric polynomials in these variables: as mentioned above, any symmetric polynomial in the variables considered can be obtained from these elementary symmetric polynomials using multiplications and additions only. In fact one has the following more detailed facts:
*any symmetric polynomial ''P'' in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> can be written as a [[polynomial expression]] in the polynomials ''e''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) with 1&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''n'';
*this expression is unique up to equivalence of polynomial expressions;
*if ''P'' has [[integer|integral]] coefficients, then the polynomial expression also has integral coefficients.
For example, for ''n'' = 2, the relevant elementary symmetric polynomials are ''e''<sub>1</sub>(''X''<sub>1</sub>, ''X''<sub>2</sub>) = ''X''<sub>1</sub> + ''X''<sub>2</sub>, and ''e''<sub>2</sub>(''X''<sub>1</sub>, ''X''<sub>2</sub>) = ''X''<sub>1</sub>''X''<sub>2</sub>. The first polynomial in the list of examples above can then be written as
:<math>X_1^3+X_2^3-7=e_1(X_1,X_2)^3-3e_2(X_1,X_2)e_1(X_1,X_2)-7</math>
(for a [[mathematical proof|proof]] that this is always possible see the [[fundamental theorem of symmetric polynomials]]).

=== Monomial symmetric polynomials === <!--[[Monomial symmetric polynomial]] redirects here -->
Powers and products of elementary symmetric polynomials work out to rather complicated expressions. If one seeks basic ''additive'' building blocks for symmetric polynomials, a more natural choice is to take those symmetric polynomials that contain only one type of [[monomial]], with only those copies required to obtain symmetry. Any monomial in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> can be written as ''X''<sub>1</sub><sup>α<sub>1</sub></sup>...''X''<sub>''n''</sub><sup>α<sub>''n''</sub></sup> where the exponents α<sub>''i''</sub> are [[natural number]]s (possibly zero); writing α&nbsp;=&nbsp;(α<sub>1</sub>,...,α<sub>''n''</sub>) this can be abbreviated to ''X''<sup>&thinsp;α</sup>. The '''monomial symmetric polynomial''' ''m''<sub>α</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) is defined as the sum of all monomials ''x''<sup>β</sup> where β ranges over all ''distinct'' permutations of (α<sub>1</sub>,...,α<sub>''n''</sub>). For instance one has
:<math>m_{(3,1,1)}(X_1,X_2,X_3)=X_1^3X_2X_3+X_1X_2^3X_3+X_1X_2X_3^3</math>,
:<math>m_{(3,2,1)}(X_1,X_2,X_3)=X_1^3X_2^2X_3+X_1^3X_2X_3^2+X_1^2X_2^3X_3+X_1^2X_2X_3^3+X_1X_2^3X_3^2+X_1X_2^2X_3^3.</math>
Clearly ''m''<sub>α</sub>&nbsp;=&nbsp;''m''<sub>β</sub> when β is a permutation of α, so one usually considers only those ''m''<sub>α</sub> for which α<sub>1</sub>&nbsp;≥&nbsp;α<sub>2</sub>&nbsp;≥&nbsp;...&nbsp;≥&nbsp;α<sub>''n''</sub>, in other words for which α is a [[partition (number theory)|partition of an integer]].
These monomial symmetric polynomials form a vector space [[basis (linear algebra)|basis]]: every symmetric polynomial ''P'' can be written as a [[linear combination]] of the monomial symmetric polynomials. To do this it suffices to separate the different types of monomial occurring in ''P''. In particular if ''P'' has integer coefficients, then so will the linear combination.

The elementary symmetric polynomials are particular cases of monomial symmetric polynomials: for 0&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''n'' one has
:<math>e_k(X_1,\ldots,X_n)=m_\alpha(X_1,\ldots,X_n)</math> where &alpha; is the partition of ''k'' into ''k'' parts&nbsp;1 (followed by ''n''&nbsp;−&nbsp;''k'' zeros).

=== Power-sum symmetric polynomials ===
{{Main|Power sum symmetric polynomial}}
For each integer ''k''&nbsp;≥&nbsp;1, the monomial symmetric polynomial ''m''<sub>(''k'',0,...,0)</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) is of special interest. It is the power sum symmetric polynomial, defined as
:<math>p_k(X_1,\ldots,X_n) = X_1^k + X_2^k + \cdots + X_n^k .</math> 
All symmetric polynomials can be obtained from the first ''n'' power sum symmetric polynomials by additions and multiplications, possibly involving [[rational number|rational]] coefficients.  More precisely,
:Any symmetric polynomial in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> can be expressed as a polynomial expression with rational coefficients in the power sum symmetric polynomials ''p''<sub>1</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>), ..., ''p''<sub>''n''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>).
In particular, the remaining power sum polynomials ''p''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) for ''k''&nbsp;&gt;&nbsp;''n'' can be so expressed in the first ''n'' power sum polynomials; for example
:<math>p_3(X_1,X_2)=\textstyle\frac32p_2(X_1,X_2)p_1(X_1,X_2)-\frac12p_1(X_1,X_2)^3.</math>

In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in ''n'' variables with ''integral'' coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials.
For an example, for ''n''&nbsp;=&nbsp;2, the symmetric polynomial
:<math>m_{(2,1)}(X_1,X_2) = X_1^2 X_2 + X_1 X_2^2</math>
has the expression
:<math> m_{(2,1)}(X_1,X_2)= \textstyle\frac12p_1(X_1,X_2)^3-\frac12p_2(X_1,X_2)p_1(X_1,X_2).</math>
Using three variables one gets a different expression
:<math>\begin{align}m_{(2,1)}(X_1,X_2,X_3) &= X_1^2 X_2 + X_1 X_2^2 + X_1^2 X_3 + X_1 X_3^2 + X_2^2 X_3 + X_2 X_3^2\\
  &= p_1(X_1,X_2,X_3)p_2(X_1,X_2,X_3)-p_3(X_1,X_2,X_3).
\end{align}</math>
The corresponding expression was valid for two variables as well (it suffices to set ''X''<sub>3</sub> to zero), but since it involves ''p''<sub>3</sub>, it could not be used to illustrate the statement for ''n''&nbsp;=&nbsp;2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first ''n'' power sum polynomials involves rational coefficients may depend on ''n''. But rational coefficients are ''always'' needed to express elementary symmetric polynomials (except the constant ones, and ''e''<sub>1</sub> which coincides with the first power sum) in terms of power sum polynomials. The [[Newton identities]] provide an explicit method to do this; it involves division by integers up to ''n'', which explains the rational coefficients. Because of these divisions, the mentioned statement fails in general when coefficients are taken in a [[field (mathematics)|field]] of finite [[characteristic (algebra)|characteristic]]; however, it is valid with coefficients in any [[ring (mathematics)|ring]] containing the rational numbers.

=== Complete homogeneous symmetric polynomials ===
{{Main|Complete homogeneous symmetric polynomial}}
For each nonnegative integer ''k'', the complete homogeneous symmetric polynomial ''h''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) is the sum of all distinct [[monomial]]s of [[degree of a polynomial|degree]] ''k'' in the variables ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>. For instance
:<math>h_3(X_1,X_2,X_3) = X_1^3+X_1^2X_2+X_1^2X_3+X_1X_2^2+X_1X_2X_3+X_1X_3^2+X_2^3+X_2^2X_3+X_2X_3^2+X_3^3.</math>
The polynomial ''h''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) is also the sum of all distinct monomial symmetric polynomials of degree ''k'' in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>, for instance for the given example
:<math>\begin{align}
 h_3(X_1,X_2,X_3)&=m_{(3)}(X_1,X_2,X_3)+m_{(2,1)}(X_1,X_2,X_3)+m_{(1,1,1)}(X_1,X_2,X_3)\\
 &=(X_1^3+X_2^3+X_3^3)+(X_1^2X_2+X_1^2X_3+X_1X_2^2+X_1X_3^2+X_2^2X_3+X_2X_3^2)+(X_1X_2X_3).\\
\end{align}</math>

All symmetric polynomials in these variables can be built up from complete homogeneous ones: any symmetric polynomial in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> can be obtained from the complete homogeneous symmetric polynomials ''h''<sub>1</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>), ..., ''h''<sub>''n''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) via multiplications and additions. More precisely:
:Any symmetric polynomial ''P'' in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> can be written as a polynomial expression in the polynomials ''h''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) with 1&nbsp;&le;&nbsp;''k''&nbsp;&le;&nbsp;''n''.
:If ''P'' has integral coefficients, then the polynomial expression also has integral coefficients.
For example, for ''n'' = 2, the relevant complete homogeneous symmetric polynomials are {{math|1=''h''<sub>1</sub>(''X''<sub>1</sub>, ''X''<sub>2</sub>) = ''X''<sub>1</sub> + ''X''<sub>2</sub>}} and {{math|1=''h''<sub>2</sub>(''X''<sub>1</sub>, ''X''<sub>2</sub>) = ''X''<sub>1</sub><sup>2</sup> + ''X''<sub>1</sub>''X''<sub>2</sub> + ''X''<sub>2</sub><sup>2</sup>}}. The first polynomial in the list of examples above can then be written as
:<math>X_1^3+ X_2^3-7 = -2h_1(X_1,X_2)^3+3h_1(X_1,X_2)h_2(X_1,X_2)-7.</math>
As in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond ''h''<sub>''n''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>), allowing them to be expressed in terms of the ones up to that point; again the resulting identities become invalid when the number of variables is increased.

An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be expressed as the identities
:<math>\sum_{i=0}^k(-1)^i e_i(X_1,\ldots,X_n)h_{k-i}(X_1,\ldots,X_n) = 0</math>, for all ''k''&nbsp;&gt;&nbsp;0, and any number of variables&nbsp;''n''.
Since ''e''<sub>0</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) and ''h''<sub>0</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) are both equal to&nbsp;1, one can isolate either the first or the last term of these summations; the former gives a set of equations that allows one to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse. This implicitly shows that any symmetric polynomial can be expressed in terms of the ''h''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) with 1&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''n'': one first expresses the symmetric polynomial in terms of the elementary symmetric polynomials, and then expresses those in terms of the mentioned complete homogeneous ones.

=== Schur polynomials ===
{{Main|Schur polynomial}}
Another class of symmetric polynomials is that of the Schur polynomials, which are of fundamental importance in the applications of symmetric polynomials to [[representation theory]]. They are however not as easy to describe as the other kinds of special symmetric polynomials; see the main article for details.