Heston model

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 k in $n$ variables X₁, ..., X_n, written h_k for k = 0, 1, 2, ..., is the sum of all monomials of total degree k in the variables. Formally,

$h_{k}(X_{1},X_{2},\dots ,X_{n})=\sum _{1\leq i_{1}\leq i_{2}\leq \cdots \leq i_{k}\leq n}X_{i_{1}}X_{i_{2}}\cdots X_{i_{k}}.$

The formula can also be written as:

$h_{k}(X_{1},X_{2},\dots ,X_{n})=\sum _{l_{1}+l_{2}+\cdots +l_{n}=k;~~l_{i}\geq 0}X_{1}^{l_{1}}X_{2}^{l_{2}}\cdots X_{n}^{l_{n}}.$

Indeed, l_p is just multiplicity of p in sequence i_k.

The first few of these polynomials are

$h_{0}(X_{1},X_{2},\dots ,X_{n})=1,$

$h_{1}(X_{1},X_{2},\dots ,X_{n})=\sum _{1\leq j\leq n}X_{j},$

$h_{2}(X_{1},X_{2},\dots ,X_{n})=\sum _{1\leq j\leq k\leq n}X_{j}X_{k},$

$h_{3}(X_{1},X_{2},\dots ,X_{n})=\sum _{1\leq j\leq k\leq l\leq n}X_{j}X_{k}X_{l}.$

Thus, for each nonnegative integer $k$, there exists exactly one complete homogeneous symmetric polynomial of degree $k$ in $n$ variables.

Another way of rewriting the definition is to take summation over all sequences i_k, without condition of ordering $i_{p}\leq i_{p+1}$:

$h_{k}(X_{1},X_{2},\dots ,X_{n})=\sum _{1\leq i_{1},i_{2},\cdots ,i_{k}\leq n}{\frac {m_{1}!m_{2}!...m_{n}!}{k!}}X_{i_{1}}X_{i_{2}}\cdots X_{i_{k}},$

here m_p is the multiplicity of number p in the sequence i_k.

For example

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

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 $n$ basic (as explained below) complete homogeneous symmetric polynomials for the first three positive values of n.

For n = 1:

$h_{1}(X_{1})=X_{1}\,.$

For n = 2:

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

For n = 3:

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

Properties

Generating function

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

$\sum _{k=0}^{\infty }h_{k}(X_{1},\ldots ,X_{n})t^{k}=\prod _{i=1}^{n}\sum _{j=0}^{\infty }(X_{i}t)^{j}=\prod _{i=1}^{n}{\frac {1}{1-X_{i}t}}$

(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 term of the product is obtained by multiplying one term chosen from each geometric series, and every monomial in the variables X_i is obtained for exactly one such choice of terms, and comes multiplied by a power of t 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 $1/det(1-tM)$, for the diagonal matrix M with X_i on the diagonal. While at the left hand side one can recognize similar expressions as stands in MacMahon 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:

$\sum _{i=0}^{m}(-1)^{i}e_{i}(X_{1},\ldots ,X_{n})h_{m-i}(X_{1},\ldots ,X_{n})=0,$

which is valid for all m > 0, and any number of variables n. The easiest way to see that it holds is from an identity of formal power series in t for the elementary symmetric polynomials, analogous to the one given above for the complete homogeneous ones:

$\sum _{k=0}^{\infty }e_{k}(X_{1},\ldots ,X_{n})(-t)^{k}=\prod _{i=1}^{n}(1-X_{i}t)$

(this is actually an identity of polynomials in t, because after e_n(X₁,…X_n) the elementary symmetric polynomials become zero). 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 t^m. A somewhat more direct way to understand that relation, is to consider the contributions in the summation involving a fixed monomial X^α of degree m. For any subset S of the variables appearing with nonzero exponent in the monomial, there is a contribution involving the product X_S of those variables as term from e_s(X₁,…,X_n), where s = #S, and the monomial X^α / X_S from h_m−s(X₁,…,X_n); this contribution has coefficient (−1)^s. The relation then follows from the fact that

$\sum _{s=0}^{l}{l \choose s}(-1)^{s}=(1-1)^{l}=0\quad {\mbox{for }}l>0,$

by the binomial formula, where l ≤ m denotes the number of distinct variables occurring (with nonzero exponent) in X^α. Since e₀(X₁, …, X_n) and h₀(X₁, …, X_n) 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

${\begin{aligned}h_{1}(X_{1},\ldots ,X_{n})&=e_{1}(X_{1},\ldots ,X_{n}),\\h_{2}(X_{1},\ldots ,X_{n})&=h_{1}(X_{1},\ldots ,X_{n})e_{1}(X_{1},\ldots ,X_{n})-e_{2}(X_{1},\ldots ,X_{n}),\\h_{3}(X_{1},\ldots ,X_{n})&=h_{2}(X_{1},\ldots ,X_{n})e_{1}(X_{1},\ldots ,X_{n})-h_{1}(X_{1},\ldots ,X_{n})e_{2}(X_{1},\ldots ,X_{n})+e_{3}(X_{1},\ldots ,X_{n}),\\\end{aligned}}$

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

${\begin{aligned}e_{1}(X_{1},\ldots ,X_{n})&=h_{1}(X_{1},\ldots ,X_{n}),\\e_{2}(X_{1},\ldots ,X_{n})&=h_{1}(X_{1},\ldots ,X_{n})e_{1}(X_{1},\ldots ,X_{n})-h_{2}(X_{1},\ldots ,X_{n}),\\e_{3}(X_{1},\ldots ,X_{n})&=h_{1}(X_{1},\ldots ,X_{n})e_{2}(X_{1},\ldots ,X_{n})-h_{2}(X_{1},\ldots ,X_{n})e_{1}(X_{1},\ldots ,X_{n})+h_{3}(X_{1},\ldots ,X_{n}),\\\end{aligned}}$

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

The set of complete homogeneous symmetric polynomials of degree 1 to n in n variables generates the ring of symmetric polynomials in n variables. More specifically, the ring of symmetric polynomials with integer coefficients equals the integral polynomial ring $\mathbf {Z} [h_{1}(X_{1},\ldots ,X_{n}),\ldots ,h_{n}(X_{1},\ldots ,X_{n})]$. This can be formulated by saying that $h_{1}(X_{1},\ldots ,X_{n}),\ldots ,h_{n}(X_{1},\ldots ,X_{n})$ form an algebraic basis of the ring of symmetric polynomials in X₁, … X_n 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 monomial symmetric polynomials

The polynomial h_k(X₁, …, X_n) is also the sum of all distinct monomial symmetric polynomials of degree k in X₁, …, X_n, for instance

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

Relation with symmetric tensors

Consider an n-dimensional vector space V and a linear operator $M:V\to V$ with eigenvalues X₁, X₂,...,X_n. Denote by Sym^k(V) its k-th symmetric tensor power and M^sym(k) the induced operator $Sym^{k}(V)\to Sym^{k}(V)$.

Proposition:

$Trace_{Sym^{k}(V)}(M^{sym(k)})=h_{k}(X_{1},X_{2},...,X_{n}).$

The proof is easy: consider an eigenbasis e_i for M. The basis in Sym^k(V) can be indexed by sequences i₁≤i₂≤...≤i_k, indeed, consider the symmetrizations of $e_{i_{1}}\otimes e_{i_{2}}\otimes ...\otimes e_{i_{k}}$. All such vectors are eigenvectors for M^sym(k) with eigenvalues $X_{i_{1}}X_{i_{2}}...X_{i_{k}}$, 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 GL(V).

See also

• Symmetric polynomial
• Elementary symmetric polynomial
• Schur polynomial
• Newton's identities
• MacMahon Master theorem
• Symmetric function
• Representation theory

References

• Macdonald, I.G. (1979), Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. Oxford: Clarendon Press.
• Macdonald, I.G. (1995), Symmetric Functions and Hall Polynomials, second ed. Oxford: Clarendon Press. ISBN 0-19-850450-0 (paperback, 1998).
• Richard P. Stanley (1999), Enumerative Combinatorics, Vol. 2. Camridge: Cambridge University Press. ISBN 0-521-56069-1