Combinatorial aspects of finite topological spaces via the representation theory of the symmetric group

Abstract

We investigate the representation of the symmetric group S_n derived from linearizing the action of S_n on the power set, P(X_n), of a set X_n :=\{ 1, . . . , n\} , on the power set of the power set, PP(X_n), of X_n, and, finally, on the set of all topologies on X_n, Top(X_n). Moreover, we prove that the latter two cases give an algebra faithful representation of the symmetric group S_n. We decompose the representation of the symmetric group Sn on CY, into irreducibles, for some particular invariant subsets, Y , of P(X_n) and PP(X_n), in the case n = 2, 3, 4. In the general case, we show that for some typical invariant subsets, Y of PP(X_n) the representation on CY is explicitly a tensor product of representations that already have an explicit decomposition into irreducibles. We reduce the action of S_n on Top(X_n) to an action on the set of reflexive, transitive relation on X_n. We use this presentation to find orbits, O that subsets of Top(Xn), such that CO is an algebra faithful representations, and orbits that are in bijection with orbits of the action of S_n on the set of Young tabloids