For an overview of the properties of $\mathbf{Symm}$, see, e. g. Michiel Hazewinkel's http://arxiv.org/abs/0804.3888 Section 10 (errata: http://www.cip.ifi.lmu.de/~grinberg/algebra/typos1short.pdfhttp://mit.edu/~darij/www/algebra/typos1short.pdf ). The antipode, for example, switches elementary symmetric with complete symmetric functions (up to sign), leaving the power sum functions invariant (again, up to sign). See also his Section 18 about the relation of $\mathbf{Symm}$ to the representation theory of symmetric group.
For an overview of the properties of $\mathbf{Symm}$, see, e. g. Michiel Hazewinkel's http://arxiv.org/abs/0804.3888 Section 10 (errata: http://www.cip.ifi.lmu.de/~grinberg/algebra/typos1short.pdf ). The antipode, for example, switches elementary symmetric with complete symmetric functions (up to sign), leaving the power sum functions invariant (again, up to sign). See also his Section 18 about the relation of $\mathbb{Symm}$ \mathbf{Symm}$to the representation theory of symmetric group. 2 added 137 characters in body Since we call the elements of$\mathbf{Symm}$"functions", let us explain what they can be evaluated at: For any$p\in\mathbf{Symm}$, any$u\in\mathbb N$and any$u$-tuple$\left(x_1,x_2,...,x_u\right)$of elements of a commutative$k$-algebra, we can define the "value" of$p$at$\left(x_1,x_2,...,x_u\right)$(denoted by$p\left(x_1,x_2,...,x_u\right)$) to be the result of (1) $p\left(x_1,x_2,...,x_u,y_1,y_2,...,y_v\right) = \sum\limits_{i \in I} q_i\left(x_1,x_2,...,x_u\right) r_i\left(y_1,y_2,...,y_v\right)$for any two multisets$\left(x_1,x_2,...,x_u\right)$and$\left(y_1,y_2,...,y_v\right)$of elements of any commutative$k$-algebra. How do we actually get such$\Delta$and$\varepsilon$? The easy answer is: By the fundamental theorem on symmetric polynomials,$\mathbf{Symm}$is generated by the as a$k$-algebra by the elementary symmetric polynomials (This does not immediately follow from the fundamental theorem on symmetric polynomials, since the fundamental theorem is usually not formulated for infinitely many variables, but you can either apply the same argument (lexicographic induction) in the infinite-variables case, or use the direct-limit construction of$\mathbf{Symm}$to conclude the infinite-variables case from the finite-variables here.one.) Hence, in order to define a$k$-algebra homomorphism from$\mathbf{Symm}$to another commutative$k$-algebra (be it$\mathbf{Symm}\otimes \mathbf{Symm}$or$k$or anything else), it is enough to specify its values at the$e_j$for$j=1,2,3,...$, and we have total freedom to do in doing so so. Since we want$\mathbf{Symm}$to be a$k$-bialgebra, we must define$\Delta$and$\varepsilon$as$k$-algebra homomorphisms; so let us define$\Delta$by requiring that$\Delta\left(e_j\right) = \sum\limits_{m=0}^j e_m \otimes e_{j-m}$for all$j\geq 1$, where$e_0$is defined to mean$1$, $e_j\left(x_1,x_2,...,x_u,y_1,y_2,...,y_v\right) = \sum\limits_{m=0}^j e_m\left(x_1,x_2,...,x_u\right) e_{j-m}\left(y_1,y_2,...,y_v\right)$for any two multisets$\left(x_1,x_2,...,x_u\right)$and$\left(y_1,y_2,...,y_v\right)$of elements of any commutative$k$-algebra, For an overview of the properties of$\mathbf{Symm}$, see, e. g. Michiel Hazewinkel's http://arxiv.org/abs/0804.3888 Section 10 (errata: http://www.cip.ifi.lmu.de/~grinberg/algebra/typos1short.pdf )and see also his Section 18 about its relation to the representation theory of symmetric groups. The antipode, for example, switches elementary symmetric with complete symmetric functions (up to sign), leaving the power sum functions invariant (again, up to sign). See also his Section 18 about the relation of$\mathbb{Symm}\$ to the representation theory of symmetric group.