# 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

$\begin{split}\begin{pmatrix} 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}\end{split}$