I've been looking for representations of groups, that is sets of matrices over GF(2) or GF(n) that are isomorphic to Sn, the symmetric group of order n.

One well known group is the standard representation, which works in any Field. The standard representation is the collection of matrices that have a single 1 in each row and each column and zeros elsewhere. The collection of all such nxn matrices forms a group of size n! which is isomorphic to Sn.

In a sense this set of matrices is created from the following rows.

1000...
0100...
0010...
0001...
...

Each row contains n-1 zeros and a single 1. Every matrix in the standard representation consists of n rows, each one of which comes from this set. The elements of the standard representation are also called permutation matrices. A permutation matrix is itself a permutation of the n rows given above.

I have discovered that there is another representation that works for any field. To the n rows given above, add an n+1st row consisting of all -1 s.

so now we have

-1 -1 -1 -1  ...
 1   0  0  0  ...
 0   1  0  0  ...
 0   0  1  0  ...
 0   0  0  1  ...
...

All permutations of these rows give us n+1 nxn matrices. (We have to leave off the bottom row) These matrices are all non-singular, and form a group which is isomorphic to Sn+1. This also works in GF(2) even though -1=1.

(I don't know if I'm the first to discover this representation or not. I've never seen any reference to it, but it seems odd that something this cool would have escaped everyone's notice but mine.)

Let's call the standard representation Rn and the new representation Qn. Let Tn be the transpose of Qn. Tn is also a faithful representation of Sn+1, sometimes Qn and Tn are conjugate to one another, sometimes they are not.

I'd like to make the following conjectures.

1. Any faithful representation of Sn+1 in nxn matrices over any field is conjugate to Qn or Tn.
2. If k>n+1, there is no faithful representation of Sk in nxn matrices over any field.
     (why? I think such a representation would have to permute n+2 rows. An nxn matrix can represent a
      permutation of n+1 rows, because the omitted row is uniquely determined. For n+2, there is no
      way to uniquely determine the order of the two missing rows.)

3. Any faithful representation of Sn is either
    a. Rn
    b. Qn-1
    c. Tn-1
    d. A direct sum of Rk Qi and/or Tj for k<n, i,j<n-1
    e. conjugate to a,b,c or d.