Uniformly generated submodules of permutation modules: Over fields of characteristic 0

This paper is motivated by a link between algebraic proof complexity and the representation theory of the finite symmetric groups. Our perspective leads to a new avenue of investigation in the representation theory of Sn. Most of our technical results concern the structure of “uniformly” generated submodules of permutation modules. For example, we consider sequences image of submodules of the permutation modules M(n−k,1k) and prove that if the sequence Wn is given in a uniform (in n) way – which we make precise – the dimension p(n) of Wn (as a vector space) is a single polynomial with rational coefficients, for all but finitely many “singular” values of n. Furthermore, we show that dim(Wn)