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Edit: Let me add a few words about the motivations. Once you already know the fundamental theorem of symmetric functions, the above isomorphism may seems tautological and not very interesting. In fact, the existence of an (actually several) very explicit isomorphism(s) from the algebra of symmetric functions to $A$ is nothing but a reformulation of this theorem. On the other hand, the above definition is arguably one of the most natural definition of $A$, and you get the Hopf structure for free.

Somehow, the fundamental theorem tells you that the algebra structure of symmetric functions is not that interesting. But it turns out that many interesting combinatorial identities can be deduced from the Hopf structure, and especially from the fact that it's self dual. Hence the pull pack of the coproduct and antipode to the algebra of symmetric functions itself has many interesting combinatorial applications. The same is true for the other Hopf algebra structure, since it identifies combinatorial identities between symmetric functions, and the computation of the composition inverse of a formal power serie.

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In fact there are several Hopf algebra structures on this algebra, mainly because it's one of the many occurence of the free commutative graded algebra with exactly one generator in each degree $A=\mathbb C[h_1,\dots,h_n,\dots]$, where $h_i$ is of degree $i$. Here the $h_i$'s can be either the elementary or the complete symmetric polynomials. Of course that's the grading which makes things interesting, otherwise it would just be the usual polynomial algebra on infinitely many variables.

Indeed, let $G$ be the group of formal power series with constant terme equal to 1. Then the algebra $O(G)$ of polynomial function on $G$ is generated by the linear maps $\lambda_k$ defined by $$\langle \lambda_k,1+\sum a_nX^n \rangle=a_k/k!$$

Letting $\lambda_k$ having degree $k$, then the map $h_k \rightarrow \lambda_k$ is an isomorphism of graded algebra. But being an algebra of function on a group, $O(G)$ has a natural Hopf algebra structure given by

$$\Delta(f)(a \otimes b)=f(ab)$$ and $$S(f)(a)=f(a^{-1})$$

If you think as the $h_i$'s as being the elementary symmetric polynomials, then this coproduct is the same as the coproduct of Dan's answer (if I'm not wrong). This is not just an abstract isomorphism, however, but if I remenber well it is reminiscent of the fact that coefficients of a polynomial are elementary symmetric functions of its roots.

But if you take instead the group of formal power series of the form $$X+\sum_{n\geq 1} a_nX^{n+1}$$ , whose multiplication is given by the composition of formal power series, then you get again the same graded algebra but the above formula leads to a non-commutative coproduct (leading to the so-called Faa di Bruno Hopf algebra).

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In fact there are several Hopf algebra structures on this algebra, mainly because it's one of the many occurence of the free commutative graded algebra with exactly one generator in each degree $A=\mathbb C[h_1,\dots,h_n,\dots]$, where $h_i$ is of degree $i$. Here the $h_i$'s can be either the elementary or the complete symmetric polynomials. Of course that's the grading which makes things interesting, otherwise it would just be the usual polynomial algebra on infinitely many variables.

Indeed, let $G$ be the group of formal power series with constant terme equal to 1. Then the algebra $O(G)$ of polynomial function on $G$ is generated by the linear maps $\lambda_k$ defined by $$\langle \lambda_k,1+\sum a_nX^n \rangle=a_k/k!$$

Letting $\lambda_k$ having degree $k$, then the map $h_k \rightarrow \lambda_k$ is an isomorphism of graded algebra. But being an algebra of function on a group, $O(G)$ has a natural Hopf algebra structure given by

$$\Delta(f)(a \otimes b)=f(ab)$$ and $$S(f)(a)=f(a^{-1})$$

If you think as the $h_i$'s as being the elementary symmetric polynomials, then this coproduct is the same as the coproduct of Dan's answer (if I'm not wrong). This is not just an abstract isomorphism, however, but if I remenber well it is reminiscent of the fact that coefficients of a polynomial are elementary symmetric functions of its roots.

But if you take instead the group of formal power series with zero constant termof the form $$X+\sum_{n\geq 1} a_nX^{n+1}$$ , whose multiplication is given by the composition of formal power series, then you get again the same graded algebra but the above formula leads to a non-commutative coproduct (leading to the so-called Faa di Bruno Hopf algebra).

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