# 50.18 SymplecticComponents

`SymplecticComponents( tbl, chars, m )`

If chi is a (nonlinear) character with indicator -1, a splitting of the tensor power chi^m is given in terms of 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).

`SymplecticComponents` returns the set of symplectic 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 }.

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 OrthogonalComponents.

```    gap> t:= CharTable( "U3(3)" );; chi:= t.irreducibles[2];
[ 6, -2, -3, 0, -2, -2, 2, 1, -1, -1, 0, 0, 1, 1 ]
gap> SymplecticComponents( t, [ chi ], 4 );
[ [ 14, -2, 5, -1, 2, 2, 2, 1, 0, 0, 0, 0, -1, -1 ],
[ 21, 5, 3, 0, 1, 1, 1, -1, 0, 0, -1, -1, 1, 1 ],
[ 64, 0, -8, -2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0 ],
[ 14, 6, -4, 2, -2, -2, 2, 0, 0, 0, 0, 0, -2, -2 ],
[ 56, -8, 2, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0 ],
[ 70, -10, 7, 1, 2, 2, 2, -1, 0, 0, 0, 0, -1, -1 ],
[ 189, -3, 0, 0, -3, -3, -3, 0, 0, 0, 1, 1, 0, 0 ],
[ 90, 10, 9, 0, -2, -2, -2, 1, -1, -1, 0, 0, 1, 1 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 126, 14, -9, 0, 2, 2, 2, -1, 0, 0, 0, 0, -1, -1 ] ]```

GAP 3.4.4
April 1997