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:
- 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).
- 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.