Problem 5

The symmetric group \(S_5\) has an irreducible representation of dimension 6. Construct explicit \(6 \times 6\)-matrices such that the matrix group they generate in \({\rm GL}(6,\mathbb{C})\) is isomorphic to that representation.

Solution

GAP

g:=SymmetricGroup(5);
r:=IrreducibleRepresentations(g);

GAP returns a list of all 7 irreducible representations of \(S_5\); the 6-dimensional one looks as follows:

GroupHomomorphismByImages( SymmetricGroup( [ 1 .. 5 ] ), Group(
 [ [ [ 0, 0, 0, 0, -1, 0 ], [ 0, 0, 1, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 1 ],
       [ 0, 0, -1, 0, 0, 0 ], [ -1, 0, 1, 0, 0, -1 ],
       [ 0, -1, 0, 0, 1, 1 ] ],
   [ [ 0, -1, 0, 0, 0, 0 ], [ -1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, -1, 0 ],
       [ 0, 0, 0, -1, 0, 0 ], [ 0, 0, -1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1 ]
      ] ]), [ (1,2,3,4,5), (1,2) ],
 [ [ [ 0, 0, 0, 0, -1, 0 ], [ 0, 0, 1, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 1 ],
       [ 0, 0, -1, 0, 0, 0 ], [ -1, 0, 1, 0, 0, -1 ],
       [ 0, -1, 0, 0, 1, 1 ] ],
   [ [ 0, -1, 0, 0, 0, 0 ], [ -1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, -1, 0 ],
       [ 0, 0, 0, -1, 0, 0 ], [ 0, 0, -1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1 ]
      ] ] )

So the wanted matrices are

\[\begin{split}\begin{pmatrix} 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 & 0 & -1 \\ 0 & -1 & 0 & 0 & 1 & 1 \end{pmatrix}\end{split}\]

and

\[\begin{split}\begin{pmatrix} 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}\end{split}\]