LECTURES 5{6. SYMMETRIC POLYNOMIALS IN SEVERAL VARIABLES S. YAKOVENKO 1. Notation In the situation where we have three variables, the traditional labels for them are the letters x; y; z. However, we will use a different notation, stress- ing the full democracy for all variables. Throughout this lecture, n will always stand for the number of different variables, and the variables them- selves will be labeled as x1; : : : ; xn. The letter x will be used to denote the tuple of all variables, so that we will deal with polynomials denoted by p(x); q(x) from the ring R[x] which is a shorthand for R[x1; : : : ; xn], etc. α1 α2 αn A monomial is a product of the form x1 x2 ··· xn , where αi > 0 are the corresponding nonnegative powers; αi = 0 means that xi is ab- sent in the above product. We will, following the above logic, denote by n α = (α1; : : : ; αn) 2 Z+ the tuple of these powers and abbreviate the product above as xα. n If α; β 2 Z+ are two such tuples (vectors), then obviously the sum α + β makes perfect sense as a vector (componentwise) sum. The rule of product means that α β α+β n x · x = x 8α; β 2 Z+: (1) The vector α can also appear as a (multiple) index, so that a general poly- P α nomial in n variables can be written as a (finite) sum p(x) = α cαx . To stress the fact that the polynomial has degree d, we will use the convention that the norm (\length") of α is the sum of its entries: α = (α1; : : : ; αn) jαj = α1 + ··· + αn: Note that jαj > 0, and jαj = 0 if and only if α = (0;:::; 0). Using these conventions, we can write a general polynomial in n variables of degree 6 n with real coefficients as the sum X α p(x) = cαx ; cα 2 R: jαj6d 2. Symmetric polynomials Definition 1. A polynomial p 2 R[x] is called symmetric, if it remains unchanged by any permutation of variables. 1 2 S. YAKOVENKO To make this intuitive definition slightly more formal, consider the full permutation group on n symbols: by definition, each permutation is one-to- one transformation of the set f1; 2; : : : ; ng. Permutations of this type form a group with the operation \composition", denoted usually by Sn (this group is often called also the symmetric group). A permutation g 2 S acts on the symbol i 2 f1; 2; : : : ; ng and maps it into g(i) 2 f1; 2; : : : ; ng so that g(i) 6= g(j) if i 6= j. The standard way to write explicitly a permutation is by a two-row matrix with columns containing i and g(i) for all labels. For instance, a cyclical permutation on n symbols is the permutation 1 2 ··· n − 1 n 2 3 ··· n 1 Permutation on symbols can be extended to act also on any objects la- beled by these symbols. For instance, a permutation g can be extended n n as a transformation of the space g∗ : R ! R into itself sending the tuple (point) (x1; : : : ; xn) into the point g∗x = (xg(1); xg(2); : : : ; xg(n)). In the same way the symmetric group acts on monomials by the following rules: ∗ ∗ ∗ g (xi) = xg(i); g (xixj) = xg(i)xg(j); g (xixjxk) = xg(i)xg(j)xg(k); ··· i; j; k; ··· = 1; : : : ; n: In the abbreviated form this action can be described as follows, ∗ α α n g (x ) = (g∗x) ; g 2 Sn; α 2 Z+: Finally, after explaining how a \permutation of the variables" g acts on an arbitrary polynomial, we extend it by linearity: X α ∗ X α p = cαx =) g p = cα(g∗x) ; g 2 Sn: α α Lemma 2. The action g∗ respects both the addition/substraction and the multiplication operations on polynomials: for any two polynomials p; q 2 ∗ ∗ ∗ ∗ ∗ ∗ R[x], we have g (p ± q) = g p ± g q, g (p · q) = (g p) · (g q). The action preserves the degree: deg g∗p = deg p. Proof. Obvious. Now we can give a formal definition of a symmetric polynomial. ∗ Definition 3. A polynomial p 2 R[x] is called symmetric, if g p = p for all permutations g 2 Sn. Example 4. The polynomials σ1(x) = x1 + ··· + xn and σn(x) = x1; : : : ; xn Pn k are obviously symmetric. Besides that, all sums of powers sk(x) = 1 xi , k = 2; n; : : : , are obviously symmetric. Our global task is to describe all symmetric polynomials in n > 3 and find for them representation analogous to that in two variables. 5. SYMMETRIC POLYNOMIALS IN SEVERAL VARIABLES 3 3. Construction of symmetric polynomials After constructing the action of the permutation group G = Sn on the polynomials, we have a situation already familiar from Lecture 3. For any degree d the polynomials of degree 6 d have the structure of a (finite- dimension) linear space over R and the group acts on this space by linear transformations. Problem 5. Compute the dimension of the space of homogeneous polyno- mials of degree exactly d in n independent variables. Compute the dimension of polynomials of degree 6 d. The symmetric group Sn is finite, it contains n! distinct permutations. Therefore we can apply the construction of symmetrization to construct sufficiently many symmetric polynomials. We recall it. Definition 6. The symmetrization of a polynomial p 2 R[x] is the polyno- mial 1 X Ep = g∗p: (2) jSnj g2Sn Remark 7. We use here the notation Ep, traditional for the Probability Theory, where it denotes the expectation of a random variable. This is not accidental: the polynomial Ep can be considered as the expectation of the polynomial p by a \random permutation" of its variables (all permutations being considered as equiprobable). The result is a new polynomial. The operator E is a projection: E(Ep) = Ep for any polynomial p. Theorem 8. The polynomial p is symmetric if and only if Ep = p. Proof. For the sake of better understanding, we repeat here the proof from Lecture 3. ∗ If p is symmetric, then g p = p for any g 2 Sn. Since g respects all operations in the ring R[x], the action of any g preserves each term in the sum (2), hence Ep = p. Conversely, if Ep = p, then, applying an arbitrary h 2 Sn to the sum and using the fact that h preserves all operations, we conclude that 1 X 1 X 1 X h∗(Ep) = g∗p = h∗(g∗p) = (hg)∗p: jS j jS j jS j n g n g n g Yet for any fixed h if g runs over the entire group Sn, then so does the composition hg 2 Sn. Thus the result of the application is only permutation of different terms in the sum, which does not affect the result. Thus h∗p = ∗ h (Ep) = p, i.e., p is symmetric. This result gives a way for bulk production of symmetric polynomials: 1 Pn 1 P 1 (1) For any i = 1; : : : ; n, E(xi) = n! 1 (n − 1)!xi = n i xi = n σ1(x). k 1 P k 1 (2) In the same way, E(xi ) = n xi = n sk(x) is the sum of powers. 4 S. YAKOVENKO (3) E(x ··· x ) = Qn x · 1 P 1 = 1 σ (x)s (x). 1 n−1 1 i n i xi n n −1 2 P (4) E(x1x2) = n(n−1) i<j xixj|a symmetric polynomial of degree 2, denoted by σ2(x). The construction can be applied to any monomial xα. The result of the averaging is the sum of monomials with a numeric coefficient. Without loss of generality we may always assume that the vector of powers α consists of non-increasing numbers. Such vector can be graphically represented as a so called Young tableau (table), the collection of columns of height α1 > α2 > ··· ; columns of height zero are not plotted. Example 9. The Young diagrams correspond to monomials 3 2 4 3 2 x1; x1x2x3x4; x1x2x3; x1x2(x3x4x5) x6x7: The total area of the tableau is the degree d of the monomials, the width (number of columns) is less or equal to the number of variables n. The result of application of the averaging operator E(xα) depends only on the Young diagram of the monomial. The numeric coefficient can be obtained by solving the corresponding combinatorial problem. Example 10. The result of averaging of the monomial E( ··· ) n of degree d 6 n is a sum of d monomials with the reciprocal coefficient n 1= d . n Example 11. The result of averaging E( ) is the sum of 2 2 = n(n − 1) 2 monomials of the form xi xj, with the reciprocal coefficient. Indeed, the label for the quadratic term can be chosen by n ways, and the remaining 2 2 label by (n − 1) ways. The terms xi xj and xj xi are different, so no need to divide by 2. Problem 12. Assume that the Young diagram consists of ν1 > 1 columns of maximal height, followed by ν2 > 1 columns of second biggest height, fol- lowed by ν3 columns of third biggest height etc, followed by ν0 > (invisible) zero-height columns. Prove that the number of monomials after averaging will be n! : ν1!ν2! ··· ν0! In the preceding Example we have ν1 = ν2 = 1, ν0 = n − 2, so the answer is n! indeed 1! 1! (n−2)! = n(n − 1). 5. SYMMETRIC POLYNOMIALS IN SEVERAL VARIABLES 5 The operator E is obviously linear (respects the operation + on the ring R[x]).