Complete homogeneous symmetric polynomial

Template:No footnotes

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 Template:Math in Template:Math variables Template:Math, written Template:Math for Template:Math, is the sum of all monomials of total degree Template:Math in the variables. Formally,

h_{k} (X_{1}, X_{2}, \dots, X_{n}) = \sum_{1 \leq i_{1} \leq i_{2} \leq \dots \leq i_{k} \leq n} X_{i_{1}} X_{i_{2}} \dots X_{i_{k}} .

The formula can also be written as:

h_{k} (X_{1}, X_{2}, \dots, X_{n}) = \sum_{\binom{l_{1} + l_{2} + \dots + l_{n} = k}{l_{i} \geq 0}} X_{1}^{l_{1}} X_{2}^{l_{2}} \dots X_{n}^{l_{n}} .

Indeed, Template:Math is just the multiplicity of Template:Math in the sequence Template:Math.

The first few of these polynomials are

\begin{matrix} h_{0} (X_{1}, X_{2}, \dots, X_{n}) & = 1, \\ [10 p x] h_{1} (X_{1}, X_{2}, \dots, X_{n}) & = \sum_{1 \leq j \leq n} X_{j}, \\ h_{2} (X_{1}, X_{2}, \dots, X_{n}) & = \sum_{1 \leq j \leq k \leq n} X_{j} X_{k}, \\ h_{3} (X_{1}, X_{2}, \dots, X_{n}) & = \sum_{1 \leq j \leq k \leq l \leq n} X_{j} X_{k} X_{l} . \end{matrix}

Thus, for each nonnegative integer Template:Math, there exists exactly one complete homogeneous symmetric polynomial of degree Template:Math in Template:Math variables.

Another way of rewriting the definition is to take summation over all sequences Template:Math, without condition of ordering Template:Math:

h_{k} (X_{1}, X_{2}, \dots, X_{n}) = \sum_{1 \leq i_{1}, i_{2}, \dots, i_{k} \leq n} \frac{m_{1}! m_{2}! \dots m_{n}!}{k!} X_{i_{1}} X_{i_{2}} \dots X_{i_{k}},

here Template:Math is the multiplicity of number Template:Math in the sequence Template:Math.

For example

h_{2} (X_{1}, X_{2}) = \frac{2! 0!}{2!} X_{1}^{2} + \frac{1! 1!}{2!} X_{1} X_{2} + \frac{1! 1!}{2!} X_{2} X_{1} + \frac{0! 2!}{2!} X_{2}^{2} = X_{1}^{2} + X_{1} X_{2} + X_{2}^{2} .

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 Template:Math basic (as explained below) complete homogeneous symmetric polynomials for the first three positive values of Template:Math.

For Template:Math:

h_{1} (X_{1}) = X_{1} .

For Template:Math:

\begin{matrix} h_{1} (X_{1}, X_{2}) & = X_{1} + X_{2} \\ h_{2} (X_{1}, X_{2}) & = X_{1}^{2} + X_{1} X_{2} + X_{2}^{2} . \end{matrix}

For Template:Math:

\begin{matrix} h_{1} (X_{1}, X_{2}, X_{3}) & = X_{1} + X_{2} + X_{3} \\ h_{2} (X_{1}, X_{2}, X_{3}) & = X_{1}^{2} + X_{2}^{2} + X_{3}^{2} + X_{1} X_{2} + X_{1} X_{3} + X_{2} X_{3} \\ h_{3} (X_{1}, X_{2}, X_{3}) & = X_{1}^{3} + X_{2}^{3} + X_{3}^{3} + X_{1}^{2} X_{2} + X_{1}^{2} X_{3} + X_{2}^{2} X_{1} + X_{2}^{2} X_{3} + X_{3}^{2} X_{1} + X_{3}^{2} X_{2} + X_{1} X_{2} X_{3} . \end{matrix}

Properties

Generating function

The complete homogeneous symmetric polynomials are characterized by the following identity of formal power series in Template:Math:

\sum_{k = 0}^{\infty} h_{k} (X_{1}, \dots, X_{n}) t^{k} = \prod_{i = 1}^{n} \sum_{j = 0}^{\infty} (X_{i} t)^{j} = \prod_{i = 1}^{n} \frac{1}{1 - X_{i} t}

(this is called the generating function, or generating series, for the complete homogeneous symmetric polynomials). 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 Template:Math is obtained for exactly one such choice of terms, and comes multiplied by a power of Template:Math equal to the degree of the monomial.

The formula above can be seen as a special case of the MacMahon master theorem. The right hand side can be interpreted as $1 / \det (1 - t M)$ where $t \in ℝ$ and $M = diag (X_{1}, \dots, X_{N})$ . On the left hand side, one can identify the complete homogeneous symmetric polynomials as special cases of the multinomial coefficient that appears in the MacMahon expression.

Performing some standard computations, we can also write the generating function as $\sum_{k = 0}^{\infty} h_{k} (X_{1}, \dots, X_{n}) t^{k} = \exp (\sum_{j = 1}^{\infty} (X_{1}^{j} + \dots + X_{n}^{j}) \frac{t^{j}}{j})$ which is the power series expansion of the plethystic exponential of $(X_{1} + \dots + X_{n}) t$ (and note that $p_{j} : = X_{1}^{j} + \dots + X_{n}^{j}$ is precisely the j-th power sum symmetric polynomial).

Relation with the elementary symmetric polynomials

There is a fundamental relation between the elementary symmetric polynomials and the complete homogeneous ones:

\sum_{i = 0}^{m} (- 1)^{i} e_{i} (X_{1}, \dots, X_{n}) h_{m - i} (X_{1}, \dots, X_{n}) = 0,

which is valid for all Template:Math, and any number of variables Template:Math. The easiest way to see that it holds is from an identity of formal power series in Template:Math for the elementary symmetric polynomials, analogous to the one given above for the complete homogeneous ones, which can also be written in terms of plethystic exponentials as:

\sum_{k = 0}^{\infty} e_{k} (X_{1}, \dots, X_{n}) (- t)^{k} = \prod_{i = 1}^{n} (1 - X_{i} t) = P E [- (X_{1} + \dots + X_{n}) t]

(this is actually an identity of polynomials in Template:Math, because after Template:Math the elementary symmetric polynomials become zero). Multiplying this by the generating function for the complete homogeneous symmetric polynomials, one obtains the constant series 1 (equivalently, plethystic exponentials satisfy the usual properties of an exponential), and the relation between the elementary and complete homogeneous polynomials follows from comparing coefficients of Template:Math. A somewhat more direct way to understand that relation is to consider the contributions in the summation involving a fixed monomial Template:Math of degree Template:Math. For any subset Template:Math of the variables appearing with nonzero exponent in the monomial, there is a contribution involving the product Template:Math of those variables as term from Template:Math, where Template:Math, and the monomial Template:Math from Template:Math; this contribution has coefficient Template:Math. The relation then follows from the fact that

\sum_{s = 0}^{l} (\binom{l}{s}) (- 1)^{s} = (1 - 1)^{l} = 0 for l > 0,

by the binomial formula, where Template:Math denotes the number of distinct variables occurring (with nonzero exponent) in Template:Math. Since Template:Math and Template:Math 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:

\begin{matrix} h_{1} (X_{1}, \dots, X_{n}) & = e_{1} (X_{1}, \dots, X_{n}), \\ h_{2} (X_{1}, \dots, X_{n}) & = h_{1} (X_{1}, \dots, X_{n}) e_{1} (X_{1}, \dots, X_{n}) - e_{2} (X_{1}, \dots, X_{n}), \\ h_{3} (X_{1}, \dots, X_{n}) & = h_{2} (X_{1}, \dots, X_{n}) e_{1} (X_{1}, \dots, X_{n}) - h_{1} (X_{1}, \dots, X_{n}) e_{2} (X_{1}, \dots, X_{n}) + e_{3} (X_{1}, \dots, X_{n}), \end{matrix}

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

\begin{matrix} e_{1} (X_{1}, \dots, X_{n}) & = h_{1} (X_{1}, \dots, X_{n}), \\ e_{2} (X_{1}, \dots, X_{n}) & = h_{1} (X_{1}, \dots, X_{n}) e_{1} (X_{1}, \dots, X_{n}) - h_{2} (X_{1}, \dots, X_{n}), \\ e_{3} (X_{1}, \dots, X_{n}) & = h_{1} (X_{1}, \dots, X_{n}) e_{2} (X_{1}, \dots, X_{n}) - h_{2} (X_{1}, \dots, X_{n}) e_{1} (X_{1}, \dots, X_{n}) + h_{3} (X_{1}, \dots, X_{n}), \end{matrix}

and so forth, that allows doing the inverse. The first Template:Math 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 Template:Math elementary and first Template:Math complete homogeneous symmetric functions.

The set of complete homogeneous symmetric polynomials of degree 1 to Template:Math 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

ℤ [h_{1} (X_{1}, \dots, X_{n}), \dots, h_{n} (X_{1}, \dots, X_{n})] .

This can be formulated by saying that

h_{1} (X_{1}, \dots, X_{n}), \dots, h_{n} (X_{1}, \dots, X_{n})

form a transcendence basis of the ring of symmetric polynomials in Template:Math with integral coefficients (as is also true for the elementary symmetric polynomials). The same is true with the ring $ℤ$ 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:

\begin{matrix} h_{n} (1, 2, \dots, k) & = {\begin{matrix} n + k \\ k \end{matrix}} \\ e_{n} (1, 2, \dots, k) & = [\binom{k + 1}{k + 1 - n}] \end{matrix}

Relation with the monomial symmetric polynomials

The polynomial Template:Math is also the sum of all distinct monomial symmetric polynomials of degree Template:Math in Template:Math, for instance

\begin{matrix} 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}^{2} X_{2} + X_{1}^{2} X_{3} + X_{1} X_{2}^{2} + X_{1} X_{3}^{2} + X_{2}^{2} X_{3} + X_{2} X_{3}^{2}) + (X_{1} X_{2} X_{3}) . \end{matrix}

Relation with power sums

Newton's identities for homogeneous symmetric polynomials give the simple recursive formula

k h_{k} = \sum_{i = 1}^{k} h_{k - i} p_{i},

where $h_{k} = h_{k} (X_{1}, \dots, X_{n})$ and p_k is the k-th power sum symmetric polynomial: $p_{k} (X_{1}, \dots, X_{n}) = \sum_{i = 1}^{n} x_{i}^{k} = X_{1}^{k} + \dots + X_{n}^{k}$ , as above.

For small $k$ we have

\begin{matrix} h_{1} & = p_{1}, \\ 2 h_{2} & = h_{1} p_{1} + p_{2}, \\ 3 h_{3} & = h_{2} p_{1} + h_{1} p_{2} + p_{3} . \end{matrix}

Relation with symmetric tensors

Consider an Template:Math-dimensional vector space Template:Math and a linear operator Template:Math with eigenvalues Template:Math. Denote by Template:Math its Template:Mathth symmetric tensor power and Template:Math the induced operator Template:Math.

Proposition:

{Trace}_{{Sym}^{k} (V)} (M^{Sym (k)}) = h_{k} (X_{1}, X_{2}, \dots, X_{n}) .

The proof is easy: consider an eigenbasis Template:Math for Template:Math. The basis in Template:Math can be indexed by sequences Template:Math, indeed, consider the symmetrizations of

e_{i_{1}} \otimes e_{i_{2}} \otimes \dots \otimes e_{i_{k}}

.

All such vectors are eigenvectors for Template:Math with eigenvalues

X_{i_{1}} X_{i_{2}} \dots X_{i_{k}},

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 Template:Math.