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Symmetric Functions

Readings

• Schur functor

Morphisms of the simplex

• Power symmetric function {$p_k(x_1, \dots , x_n)$}: one morphism from each simplex {$[k-1]$} to every vertex of {$[n-1]$}.
• Elementary symmetric function {$p_k(x_1, \dots , x_n)$}: one morphism for every subsimplex.
• Homogeneous symmetric function {$p_k(x_1, \dots , x_n)$}: one morphism for every monotone function (way of building up).
• Monomial symmetric function?

Compare these types of morphisms with those of various categories:

• Inclusion defines morphism for the category of the open sets of a topological space. This space is important for defining a topological presheaf {$C^{op}\rightarrow \mathrm{Top}$}.

Thus we can think of elementary symmetric functions as defining a "topological" perspective on the simplex.