alggroupcenter

Function: alggroupcenter
Section: algebras
C-Name: alggroupcenter
Prototype: GDGD&
Help: alggroupcenter(gal,{p=0},{&cc}): constructs the center of the group
 algebra of gal over Q (resp. Fp), and sets cc to the conjugacy classes of gal.
Doc: initializes the center~$Z(K[G])$ of the group algebra~$K[G]$ over~$K=\Q$
 ($p = 0$ or omitted) or~$\F_p$ where~$G$ is the underlying group of the
 \kbd{galoisinit} structure~\var{gal}. The input~\var{gal} is also allowed to
 be a \typ{VEC} of permutations that is closed under products.
 Sets~\var{cc} to a \typ{VEC}~$[\var{elts},\var{conjclass},\var{rep},\var{flag}]$
 where~\var{elts} is a sorted \typ{VEC} containing the list of elements
 of~$G$, \var{conjclass} is a \typ{VECSMALL} of the same length as~\var{elts}
 containing the index of the conjugacy class of the corresponding element (an
 integer between $1$ and the number of conjugacy classes), and~\var{rep} is a
 \typ{VECSMALL} of length the number of conjugacy classes giving for each
 conjugacy class the index in~\var{elts} of a representative of this conjugacy
 class. Finally \var{flag} is $1$ if and only if the permutation
 representation of $G$ is transitive, in which case the $i$-th element
 of \var{elts} is characterized by $g[1] = i$; this is always the case
 when \var{gal} is a \kbd{galoisinit} structure. The basis of~$Z(K[G])$ as
 output consists of the indicator functions of the conjugacy classes in the
 ordering given by~\var{cc}. Example:
 \bprog
 ? K = nfsplitting(x^4+x+1);
 ? gal = galoisinit(K); \\ S4
 ? al = alggroupcenter(gal,,&cc);
 ? algiscommutative(al)
 %4 = 1
 ? #cc[3] \\ number of conjugacy classes of S4
 %5 = 5
 ? gal = [Vecsmall([1,2,3]),Vecsmall([1,3,2])]; \\ C2
 ? al = alggroupcenter(gal,,&cc);
 ? cc[3]
 %8 = Vecsmall([1, 2])
 ? cc[4]
 %9 = 0
 @eprog