gr.group theory - Maximum entry magnitudes for representations of symmetric groups

I've no idea on your general question, but as far as eliminating the easy answer that Specht matrices always have entries -1,0,1, I decided to check the computer. According to sage, the maximum entry's absolute value can be larger than 1. The first example appears to be on 7 points:

sage: chi=SymmetricGroupRepresentations(7);chi([2,2,2,1])([2,3,1,5,4,6,7])
[-1  0  0  0  0  1  0  0  0  0  0  0  0  0]
[-1  1  0  0  0  1 -1  0  0  0  0  0  0  0]
[ 0  0  0  0  0  1  0  0  0  0 -1  0  0  0]
[ 0  1  0  0  0  1  0  0  0  0  0  0 -1  0]
[-1  1  0  0  0  1  0  0  0  0  0 -1  0  0]
[-1  0  0  0  0  0  0  0  0  0  0  0  0  0]
[-1  1  0  0  0  0  0  0  0  0  0  0  0  0]
[ 0  0 -1  0  0  0  0  0  0  0  0  0  0  0]
[ 0  1  0  0 -1  0  0  0  0  0  0  0  0  0]
[-1  1  0 -1  0  0  0  0  0  0  0  0  0  0]
[-1  0  0  0  0  1  0 -1  0  0  0  0  0  0]
[-1  1  0  0  0  1  0  0  0 -1  0  0  0  0]
[-2  1  0  0  0  1  0  0 -1  0  0  0  0  0]
[-1  1  0  0  0  1  0  0  0  0  0  0  0 -1]

The first command gives you Irr(Sym(7)) in the variable chi, then chi(partition) gives you the specific irreducible Specht representation, and chi(partition)(permutation) gives you the matrix for that permutation. Permutations are specified in the combinatorics way by listing the images of [1,2,3,4,5,6,7] in order, also called one-line notation. The specific permutation is (1,2,3)(4,5)(6)(7).

According to magma one has:

> SymmetricRepresentation([2,2,2,1],Sym(7)!(1,2,7)(3,4));
[ 0 -1  0  1 -1 -1  0 -2 -1  0  0 -1  0 -1]
[ 0  1  0 -1  2  1 -1  1  1  0  0  0  0  1]
[ 0 -1  0  1 -1  0  1 -1  0  0  0  0  0  0]
[ 0  1  0  0  1  1  0  1  0  0  0  0  0  1]
[ 0  0  0  0 -1 -1  0 -1 -1  0  0  0  0  0]
[ 0  0  0  0  0  0  0  1  0  0  0  0  0  0]
[ 0  0  0  0  0 -1  0 -1  0  0  0  0  0  0]
[ 0  0  0  0 -1  0  1  0  0  0  0  0  0  0]
[ 0  0  0  0  1  1  0  0  0  0  0  0  0  0]
[ 0  1  0 -1  1  1  0  1  1  0  1  1  0  1]
[ 0 -1  0  1  0  0  0 -1  0  1  0 -1  0  0]
[ 1  1  0  0  0  1  0  1  0  0  0  1  0  1]
[ 0  1  0 -1  1  0 -1  0  0  0  0  0 -1  0]
[ 0 -1 -1  0 -1 -1  0  0  0  0  0  0  0 -1]

At any rate, both violate the bound, though requiring different conjugates.

At least when I was looking at the modular representations induced by Specht modules I noticed there is no convention in the published articles or software on which of the two modules is a Specht module and which is its dual. Over a field of characteristic 0 they are isomorphic, but over finite fields they are just dual. I suspect the entries of the matrices in the representation of the Specht module are not terribly well defined; you'll need to have some combinatorial definition to work from to even decide isomorphism over Z/pZ, much less exact entries.