Complete Homogeneous Symmetric Polynomial - Definition

Definition

The complete homogeneous symmetric polynomial of degree k in variables X1, ..., Xn, written hk for k = 0, 1, 2, ..., is the sum of all monomials of total degree k in the variables. Formally,

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, lp is just multiplicity of p in sequence ik.

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 ik, without condition of ordering :

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 mp is the multiplicity of number p in the sequence ik.

For example

The polynomial ring formed by taking all integral linear combinations of products of the complete homogeneous symmetric polynomials is a commutative ring.

Read more about this topic:  Complete Homogeneous Symmetric Polynomial

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