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The standard construction (using Specht modules)of the irreducible representations of symmetric groups of finite degree gives representation where all the matrices take integer-valued entries. My questions:

  1. How does the size of the largest among absolute values of entries (taken over all irreducible representations) grow with n (where n is the degree of the symmetric group, or the size of the set it acts upon naturally).
  2. Are the specific matrices that we obtain using the Specht module construction optimal from the viewpoint of minimizing the largest of the entry sizes?

If the answer to (2) is no, we can formulate (1') which asks the same question with the optimal representations (from the viewpoint of minimizing the maximum entry size).

Easy hand computations show me that for symmetric groups of degree $n \le 4$, we can choose bases for all the irreducible representations with all the matrix entries 1, 0, or -1. I'm assuming, however, that this does not continue to larger n.

RELATED NOTE: If a representation can be realized over the field of fractions of a principal ideal domain, it can also be realized over the principal ideal domain. So, if a representation can be realized over the rational numbers, it can also be realized over the integers.

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1 Answer 1

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.

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