# 50.17 OrthogonalComponents

`OrthogonalComponents( tbl, chars, m )`

If chi is a (nonlinear) character with indicator +1, a splitting of the tensor power chi^m is given by the so-called Murnaghan functions (see~Mur58). These components in general have fewer irreducible constituents than the symmetrizations with the symmetric group of degree m (see Symmetrisations).

`OrthogonalComponents` returns the set of orthogonal symmetrisations of the characters of the character table tbl in the list chars, up to the power m, where the integer m is one of { 2, 3, 4, 5, 6 }.

Note: It is not checked if all characters in chars do really have indicator +1; if there are characters with indicator 0 or -1, the result might contain virtual characters, see also SymplecticComponents.

The Murnaghan functions are implemented as in~Fra82.

```    gap> t:= CharTable( "A8" );; chi:= t.irreducibles[2];
[ 7, -1, 3, 4, 1, -1, 1, 2, 0, -1, 0, 0, -1, -1 ]
gap> OrthogonalComponents( t, [ chi ], 4 );
[ [ 21, -3, 1, 6, 0, 1, -1, 1, -2, 0, 0, 0, 1, 1 ],
[ 27, 3, 7, 9, 0, -1, 1, 2, 1, 0, -1, -1, -1, -1 ],
[ 105, 1, 5, 15, -3, 1, -1, 0, -1, 1, 0, 0, 0, 0 ],
[ 35, 3, -5, 5, 2, -1, -1, 0, 1, 0, 0, 0, 0, 0 ],
[ 77, -3, 13, 17, 2, 1, 1, 2, 1, 0, 0, 0, 2, 2 ],
[ 189, -3, -11, 9, 0, 1, 1, -1, 1, 0, 0, 0, -1, -1 ],
[ 330, -6, 10, 30, 0, -2, -2, 0, -2, 0, 1, 1, 0, 0 ],
[ 168, 8, 8, 6, -3, 0, 0, -2, 2, -1, 0, 0, 1, 1 ],
[ 35, 3, -5, 5, 2, -1, -1, 0, 1, 0, 0, 0, 0, 0 ],
[ 182, 6, 22, 29, 2, 2, 2, 2, 1, 0, 0, 0, -1, -1 ] ]```

GAP 3.4.4
April 1997