Symmetric function

From Academic Kids

In mathematics, the theory of symmetric functions is part of the theory of polynomial equations, and also a substantial chapter of combinatorics. If P(x) is a polynomial with roots

α1, α2, ..., αn,

a symmetric function of the roots of P means

S1, α2, ..., αn)

where S is a function of n variables that is symmetric in the sense of not depending on the order of the n-tuple of arguments.

For example

S(X1, X2, ..., Xn)

might be

X1 + X2 + ... Xn,

or

X1X2...Xn.

In those cases there are direct relations one can draw between the symmetric functions of the roots, and coefficients in the polynomial P. Assume for simplicity that P is monic, so that

P(x) = xn + an-1xn-1 + ... + a0 = (x - α1)(x - α2)...(x - αn),

where the ai are in a field K and the αi are in an algebraic closure, we have the sum of all the αi equal to −an−1 and their product equal to (−1)na0. Further, it is classical algebra (Vičte's formulas) that the intermediate coefficients of P are plus or minus the sums of the products of the roots taken j at a time, for 1 < j < n. The sign is alternately + and −; these formulae are the basis of the traditional theory of equations. In symbols

(−1)njanj = Σ αk(1)...αk(j)

with the summation taken over all index sets

1 ≤ k(1) < ... < k(j) ≤ n.

Those sums σj for j = 1, 2, ..., n are the elementary symmetric functions of the roots; that is, the j-th elementary symmetric function is given by the same formula, but in indeterminates Xi. A basic theorem states that any symmetric polynomial function S of n variables can be expressed as a polynomial in the elementary symmetric functions. In particular, as a consequence applying to the solution of equations, the symmetric polynomials of the roots lie in K.

The polynomial relations underlying that assertion are universal (independent of choice of P); and, if we work with the symmetric polynomials created from a monomial, we can eliminate dependence on K, too, to get formulae with integer coefficients. Putting this more algebraically, we can define a subring Symm(n) of Z[X1, X2, ..., Xn] consisting of the integral symmetric polynomials (those invariant under the action of the symmetric group on indices); and then assert that the formulae for σj, for which we retain the notation, are ring generators of Symm(n). What is more, they are independent generators (no algebraic relations hold), so that Symm(n) is abstractly also a polynomial ring on n generators. A great deal of attention was paid, in older algebra textbooks, to algorithmic procedures expressing the procedural content of this (which has been stated as an existence theorem but has computational content).

The most important single application is to the power sums α1k + α2k + ... + αnk, in terms of the aj. The formulae for doing this are attributed to Isaac Newton. They were encountered in K-theory too, where they underlie the Adams operations.

They also support the theory of the Newton polygon, part of the theory of ramification. In Newton's case the point was to work with aj in a formal power series ring; here passage to the algebraic closure is the theory of Puiseux expansions in fractional powers, and the Newton polygon is a device for computing the required exponents.

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