Symmetrisations( tbl, chars, Sn )
Symmetrisations( tbl, chars, n )
returns the list of nonzero symmetrisations of the characters chars, regarded as characters of the character table tbl, with the ordinary characters of the symmetric group of degree n; alternatively, the table of the symmetric group can be entered as Sn.
The symmetrisation chi^{[lambda]} of the character chi of tbl with the character lambda of the symmetric group S_n of degree n is defined by
[ chi^[lambda](g) = frac1n! sum_rho in S_n lambda(rho) prod_k=1^n chi(g^k)^a_k(rho), ]
where a_k(rho) is the number of cycles of length k in rho.
For special symmetrisations, see SymmetricParts, AntiSymmetricParts, MinusCharacter and OrthogonalComponents, SymplecticComponents.
gap> t:= CharTable( "A5" );;
gap> chars:= Sublist( t.irreducibles, [ 1 .. 3 ] );;
gap> Symmetrisations( t, chars, 3 );
[ [ 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0 ], [ 1, 1, 1, 1, 1 ],
[ 1, 1, 1, 1, 1 ], [ 8, 0, -1, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ],
[ 10, -2, 1, 0, 0 ], [ 1, 1, 1, 1, 1 ],
[ 8, 0, -1, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ], [ 10, -2, 1, 0, 0 ] ]
Note that the returned list may contain zero characters, and duplicate characters are not deleted.
GAP 3.4.4