Complete homogeneous symmetric polynomial
In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete homogeneous symmetric polynomials.
Definition
The complete homogeneous symmetric polynomial of degree in variables, written for, is the sum of all monomials of total degree in the variables. Formally,The formula can also be written as:
Indeed, is just the multiplicity of in the sequence.
The first few of these polynomials are
Thus, for each nonnegative integer, there exists exactly one complete homogeneous symmetric polynomial of degree in variables.
Another way of rewriting the definition is to take summation over all sequences, without condition of ordering :
here is the multiplicity of number in the sequence.
For example
The polynomial ring formed by taking all integral linear combinations of products of the complete homogeneous symmetric polynomials is a commutative ring.
Examples
The following lists the basic complete homogeneous symmetric polynomials for the first three positive values of.For :
For :
For :
Properties
Generating function
The complete homogeneous symmetric polynomials are characterized by the following identity of formal power series in :. Here each fraction in the final expression is the usual way to represent the formal geometric series that is a factor in the middle expression. The identity can be justified by considering how the product of those geometric series is formed: each factor in the product is obtained by multiplying together one term chosen from each geometric series, and every monomial in the variables is obtained for exactly one such choice of terms, and comes multiplied by a power of equal to the degree of the monomial.
The formula above is in certain sense equivalent to MacMahon master theorem. Indeed, the right hand side can be interpreted as, for the diagonal matrix with on the diagonal. On the left hand side, one can recognize expressions like those in MacMahon's master theorem. Diagonalizable matrices are dense in the set of all matrices, and this consideration proves the whole theorem.
Relation with the elementary symmetric polynomials
There is a fundamental relation between the elementary symmetric polynomials and the complete homogeneous ones:which is valid for all, and any number of variables. The easiest way to see that it holds is from an identity of formal power series in for the elementary symmetric polynomials, analogous to the one given above for the complete homogeneous ones:
. Multiplying this by the generating function for the complete homogeneous symmetric polynomials, one obtains the constant series 1, and the relation between the elementary and complete homogeneous polynomials follows from comparing coefficients of. A somewhat more direct way to understand that relation, is to consider the contributions in the summation involving a fixed monomial of degree. For any subset of the variables appearing with nonzero exponent in the monomial, there is a contribution involving the product of those variables as term from, where, and the monomial from ; this contribution has coefficient. The relation then follows from the fact that
by the binomial formula, where denotes the number of distinct variables occurring in. Since and are both equal to 1, one can isolate from the relation either the first or the last terms of the summation. The former gives a sequence of equations:
and so on, that allows to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials; the latter gives a set of equations
and so forth, that allows doing the inverse. The first elementary and complete homogeneous symmetric polynomials play perfectly similar roles in these relations, even though the former polynomials then become zero, whereas the latter do not. This phenomenon can be understood in the setting of the ring of symmetric functions. It has a ring automorphism that interchanges the sequences of the elementary and first complete homogeneous symmetric functions.
The set of complete homogeneous symmetric polynomials of degree 1 to in variables generates the ring of symmetric polynomials in variables. More specifically, the ring of symmetric polynomials with integer coefficients equals the integral polynomial ring
This can be formulated by saying that
form an algebraic basis of the ring of symmetric polynomials in with integral coefficients. The same is true with the ring of integers replaced by any other commutative ring. These statements follow from analogous statements for the elementary symmetric polynomials, due to the indicated possibility of expressing either kind of symmetric polynomials in terms of the other kind.
Relation with the Stirling numbers
The evaluation at integers of complete homogeneous polynomials and elementary symmetric polynomials is related to Stirling numbers:Relation with the monomial symmetric polynomials
The polynomial is also the sum of all distinct monomial symmetric polynomials of degree in, for instanceRelation with symmetric tensors
Consider an -dimensional vector space and a linear operator with eigenvalues. Denote by its th symmetric tensor power and the induced operator.Proposition:
The proof is easy: consider an eigenbasis for. The basis in can be indexed by sequences, indeed, consider the symmetrizations of
All such vectors are eigenvectors for with eigenvalues
hence this proposition is true.
Similarly one can express elementary symmetric polynomials via traces over antisymmetric tensor powers. Both expressions are subsumed in expressions of Schur polynomials as traces over Schur functors, which can be seen as the Weyl character formula for.