Uniformly generated submodules of permutation modules Over fields of characteristic 0

AbstractThis 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 {Wn}n∈N 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)