Elementary symmetric polynomial

Template:Short description Template:No footnotes In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial Template:Math is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree Template:Math in Template:Math variables for each positive integer Template:Math, and it is formed by adding together all distinct products of Template:Math distinct variables.

Definition

The elementary symmetric polynomials in Template:Math variables Template:Math, written Template:Math for Template:Math, are defined by

<math>\begin{align}

 e_1 (X_1, X_2, \dots, X_n) &= \sum_{1 \leq a \leq n} X_a,\\
 e_2 (X_1, X_2, \dots, X_n) &= \sum_{1 \leq a < b \leq n} X_a X_b,\\
 e_3 (X_1, X_2, \dots, X_n) &= \sum_{1 \leq a < b < c \leq n} X_a X_b X_c,\\

\end{align}</math> and so forth, ending with

<math> e_n (X_1, X_2, \dots,X_n) = X_1 X_2 \cdots X_n.</math>

In general, for Template:Math we define

<math> e_k (X_1 , \ldots , X_n )=\sum_{1\le a_1 < a_2 < \cdots < a_k \le n} X_{a_1} X_{a_2} \dotsm X_{a_k},</math>

Also, Template:Math if Template:Math.

Sometimes, Template:Math is included among the elementary symmetric polynomials, but excluding it allows generally simpler formulation of results and properties.

Thus, for each positive integer Template:Mvar less than or equal to Template:Mvar there exists exactly one elementary symmetric polynomial of degree Template:Mvar in Template:Mvar variables. To form the one that has degree Template:Mvar, we take the sum of all products of Template:Mvar-subsets of the Template:Mvar variables. (By contrast, if one performs the same operation using multisets of variables, that is, taking variables with repetition, one arrives at the complete homogeneous symmetric polynomials.)

Given an integer partition (that is, a finite non-increasing sequence of positive integers) Template:Math, one defines the symmetric polynomial Template:Math, also called an elementary symmetric polynomial, by

<math>e_\lambda (X_1, \dots,X_n) = e_{\lambda_1}(X_1, \dots, X_n) \cdot e_{\lambda_2}(X_1, \dots, X_n) \cdots e_{\lambda_m}(X_1, \dots, X_n)</math>.

Sometimes the notation Template:Math is used instead of Template:Math.

Recursive definition

The following definition is equivalent to the above and might be useful for computer implementations:

<math>\begin{align}

 e_1 (X_1, \dots, X_n) &= \sum_{1 \leq j \leq n} X_j,\\
 e_k (X_1, \dots, X_n) &= \sum_{1 \leq j \leq n - k + 1} X_j e_{k - 1} (X_{j + 1}, \dots, X_n) \\

\end{align}</math>

Examples

The following lists the Template:Math elementary symmetric polynomials for the first four positive values of Template:Math.

For Template:Math:

<math>e_1(X_1) = X_1.</math>

For Template:Math:

<math>\begin{align}

e_1(X_1,X_2)  &= X_1 + X_2,\\  
e_2(X_1,X_2) &= X_1X_2.\,\\

\end{align}</math>

For Template:Math:

<math>\begin{align}

e_1(X_1,X_2,X_3) &= X_1 + X_2 + X_3,\\ 
e_2(X_1,X_2,X_3) &= X_1X_2 + X_1X_3 + X_2X_3,\\
e_3(X_1,X_2,X_3) &= X_1X_2X_3.\,\\

\end{align}</math>

For Template:Math:

<math>\begin{align}

e_1(X_1,X_2,X_3,X_4) &= X_1 + X_2 + X_3 + X_4,\\
e_2(X_1,X_2,X_3,X_4) &= X_1X_2 + X_1X_3 + X_1X_4 + X_2X_3 + X_2X_4 + X_3X_4,\\
e_3(X_1,X_2,X_3,X_4) &= X_1X_2X_3 + X_1X_2X_4 + X_1X_3X_4 + X_2X_3X_4,\\
e_4(X_1,X_2,X_3,X_4) &= X_1X_2X_3X_4.\,\\

\end{align}</math>

Properties

The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity

<math>\prod_{j=1}^n ( \lambda - X_j)=\lambda^n - e_1(X_1,\ldots,X_n)\lambda^{n-1} + e_2(X_1,\ldots,X_n)\lambda^{n-2} + \cdots +(-1)^n e_n(X_1,\ldots,X_n).</math>

That is, when we substitute numerical values for the variables Template:Math, we obtain the monic univariate polynomial (with variable Template:Math) whose roots are the values substituted for Template:Math and whose coefficients are – up to their sign – the elementary symmetric polynomials. These relations between the roots and the coefficients of a polynomial are called Vieta's formulas.

The characteristic polynomial of a square matrix is an example of application of Vieta's formulas. The roots of this polynomial are the eigenvalues of the matrix. When we substitute these eigenvalues into the elementary symmetric polynomials, we obtain – up to their sign – the coefficients of the characteristic polynomial, which are invariants of the matrix. In particular, the trace (the sum of the elements of the diagonal) is the value of Template:Math, and thus the sum of the eigenvalues. Similarly, the determinant is – up to the sign – the constant term of the characteristic polynomial, i.e. the value of Template:Math. Thus the determinant of a square matrix is the product of the eigenvalues.

The set of elementary symmetric polynomials in Template:Math variables generates the ring of symmetric polynomials in Template:Math variables. More specifically, the ring of symmetric polynomials with integer coefficients equals the integral polynomial ring Template:Math. (See below for a more general statement and proof.) This fact is one of the foundations of invariant theory. For another system of symmetric polynomials with the same property see Complete homogeneous symmetric polynomials, and for a system with a similar, but slightly weaker, property see Power sum symmetric polynomial.

Fundamental theorem of symmetric polynomials

For any commutative ring Template:Math, denote the ring of symmetric polynomials in the variables Template:Math with coefficients in Template:Math by Template:Math. This is a polynomial ring in the n elementary symmetric polynomials Template:Math for Template:Math.

This means that every symmetric polynomial Template:Math has a unique representation

<math> P(X_1,\ldots, X_n)=Q\big(e_1(X_1 , \ldots ,X_n), \ldots, e_n(X_1 , \ldots ,X_n)\big) </math>

for some polynomial Template:Math. Another way of saying the same thing is that the ring homomorphism that sends Template:Math to Template:Math for Template:Math defines an isomorphism between Template:Math and Template:Math.

Proof sketch

The theorem may be proved for symmetric homogeneous polynomials by a double induction with respect to the number of variables Template:Math and, for fixed Template:Math, with respect to the degree of the homogeneous polynomial. The general case then follows by splitting an arbitrary symmetric polynomial into its homogeneous components (which are again symmetric).

In the case Template:Math the result is trivial because every polynomial in one variable is automatically symmetric.

Assume now that the theorem has been proved for all polynomials for Template:Math variables and all symmetric polynomials in Template:Math variables with degree Template:Math. Every homogeneous symmetric polynomial Template:Math in Template:Math can be decomposed as a sum of homogeneous symmetric polynomials

<math> P(X_1,\ldots,X_n)= P_{\text{lacunary}} (X_1,\ldots,X_n) + X_1 \cdots X_n \cdot Q(X_1,\ldots,X_n). </math>

Here the "lacunary part" Template:Math is defined as the sum of all monomials in Template:Math which contain only a proper subset of the Template:Math variables Template:Math, i.e., where at least one variable Template:Math is missing.

Because Template:Math is symmetric, the lacunary part is determined by its terms containing only the variables Template:Math, i.e., which do not contain Template:Math. More precisely: If Template:Math and Template:Math are two homogeneous symmetric polynomials in Template:Math having the same degree, and if the coefficient of Template:Math before each monomial which contains only the variables Template:Math equals the corresponding coefficient of Template:Math, then Template:Math and Template:Math have equal lacunary parts. (This is because every monomial which can appear in a lacunary part must lack at least one variable, and thus can be transformed by a permutation of the variables into a monomial which contains only the variables Template:Math.)

But the terms of Template:Math which contain only the variables Template:Math are precisely the terms that survive the operation of setting Template:Math to 0, so their sum equals Template:Math, which is a symmetric polynomial in the variables Template:Math that we shall denote by Template:Math. By the inductive hypothesis, this polynomial can be written as

<math> \tilde{P}(X_1, \ldots, X_{n-1})=\tilde{Q}(\sigma_{1,n-1}, \ldots, \sigma_{n-1,n-1})</math>

for some Template:Math. Here the doubly indexed Template:Math denote the elementary symmetric polynomials in Template:Math variables.

Consider now the polynomial

<math>R(X_1, \ldots, X_{n}):= \tilde{Q}(\sigma_{1,n}, \ldots, \sigma_{n-1,n}) .</math>

Then Template:Math is a symmetric polynomial in Template:Math, of the same degree as Template:Math, which satisfies

<math>R(X_1, \ldots, X_{n-1},0) = \tilde{Q}(\sigma_{1,n-1}, \ldots, \sigma_{n-1,n-1}) = P(X_1, \ldots,X_{n-1},0)</math>

(the first equality holds because setting Template:Math to 0 in Template:Math gives Template:Math, for all Template:Math). In other words, the coefficient of Template:Math before each monomial which contains only the variables Template:Math equals the corresponding coefficient of Template:Math. As we know, this shows that the lacunary part of Template:Math coincides with that of the original polynomial Template:Math. Therefore the difference Template:Math has no lacunary part, and is therefore divisible by the product Template:Math of all variables, which equals the elementary symmetric polynomial Template:Math. Then writing Template:Math, the quotient Template:Math is a homogeneous symmetric polynomial of degree less than Template:Math (in fact degree at most Template:Math) which by the inductive hypothesis can be expressed as a polynomial in the elementary symmetric functions. Combining the representations for Template:Math and Template:Math one finds a polynomial representation for Template:Math.

The uniqueness of the representation can be proved inductively in a similar way. (It is equivalent to the fact that the Template:Math polynomials Template:Math are algebraically independent over the ring Template:Math.) The fact that the polynomial representation is unique implies that Template:Math is isomorphic to Template:Math.

Alternative proof

The following proof is also inductive, but does not involve other polynomials than those symmetric in Template:Math, and also leads to a fairly direct procedure to effectively write a symmetric polynomial as a polynomial in the elementary symmetric ones. Assume the symmetric polynomial to be homogeneous of degree Template:Mvar; different homogeneous components can be decomposed separately. Order the monomials in the variables Template:Mvar lexicographically, where the individual variables are ordered Template:Math, in other words the dominant term of a polynomial is one with the highest occurring power of Template:Math, and among those the one with the highest power of Template:Math, etc. Furthermore parametrize all products of elementary symmetric polynomials that have degree Template:Math (they are in fact homogeneous) as follows by partitions of Template:Math. Order the individual elementary symmetric polynomials Template:Math in the product so that those with larger indices Template:Mvar come first, then build for each such factor a column of Template:Mvar boxes, and arrange those columns from left to right to form a Young diagram containing Template:Mvar boxes in all. The shape of this diagram is a partition of Template:Mvar, and each partition Template:Mvar of Template:Math arises for exactly one product of elementary symmetric polynomials, which we shall denote by Template:Math) (the Template:Math is present only because traditionally this product is associated to the transpose partition of Template:Mvar). The essential ingredient of the proof is the following simple property, which uses multi-index notation for monomials in the variables Template:Math.

Lemma. The leading term of Template:Math is Template:Math.

Proof. The leading term of the product is the product of the leading terms of each factor (this is true whenever one uses a monomial order, like the lexicographic order used here), and the leading term of the factor Template:Math is clearly Template:Math. To count the occurrences of the individual variables in the resulting monomial, fill the column of the Young diagram corresponding to the factor concerned with the numbers Template:Math of the variables, then all boxes in the first row contain 1, those in the second row 2, and so forth, which means the leading term is Template:Math.

Now one proves by induction on the leading monomial in lexicographic order, that any nonzero homogeneous symmetric polynomial Template:Mvar of degree Template:Mvar can be written as polynomial in the elementary symmetric polynomials. Since Template:Mvar is symmetric, its leading monomial has weakly decreasing exponents, so it is some Template:Math with Template:Mvar a partition of Template:Math. Let the coefficient of this term be Template:Mvar, then Template:Math is either zero or a symmetric polynomial with a strictly smaller leading monomial. Writing this difference inductively as a polynomial in the elementary symmetric polynomials, and adding back Template:Math to it, one obtains the sought for polynomial expression for Template:Math.

The fact that this expression is unique, or equivalently that all the products (monomials) Template:Math of elementary symmetric polynomials are linearly independent, is also easily proved. The lemma shows that all these products have different leading monomials, and this suffices: if a nontrivial linear combination of the Template:Math were zero, one focuses on the contribution in the linear combination with nonzero coefficient and with (as polynomial in the variables Template:Math) the largest leading monomial; the leading term of this contribution cannot be cancelled by any other contribution of the linear combination, which gives a contradiction.

See also

• Symmetric polynomial
• Complete homogeneous symmetric polynomial
• Schur polynomial
• Newton's identities
• Newton's inequalities
• Maclaurin's inequality
• MacMahon Master theorem
• Symmetric function
• Representation theory

References

• Template:Cite book
• Template:Cite book

External links

• Template:Cite AV media