schur index in GAP

Question

What is the command for the Schur index in GAP? Let $χ$ be an irreducible (complex) character of a finite group, $G$. The Schur index $m_K(χ)$ of $χ$ over the field $K$ is the smallest positive integer $m$ such that $m_χ$ is afforded by a representation over the field $K(χ)$. The most interesting case is $K=\mathbb{Q}$. Given the character table, or only the particular character one is interested in, one can usually derive bounds for $m(χ)=m_\mathbb{Q}(χ)$.

In MAGMA, this could be done as follows (see here):

T:=CharacterTable(Group);
SchurIndex(T[number of row of T]);

Answer 1

In GAP, this functionality had been recently implemented by Allen Herman in the package Wedderga (Wedderburn Decomposition of Group Algebras), starting from version 4.6. To use it, you need to load the package first with LoadPackage("wedderga");.

The two functions are SchurIndex( A ) and SchurIndexByCharacter( F, G, n ). The first of these returns the Schur index of the simple algebra A. The second returns the Schur index of the simple component of the group ring FG corresponding to the irreducible character Irr(G)[n] of G.

See Wedderga manual in your GAP installation or find in online here for the full documentation.

Just two illustrative examples from the Wedderga manual are given below:

gap> LoadPackage("wedderga");
...
true

gap> G:=SmallGroup(63,1);  
<pc group of size 63 with 3 generators>
gap> R:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 3 generators>
gap> W:=WedderburnDecompositionInfo(R);
[ [ 1, Rationals ], [ 1, CF(3) ], [ 1, CF(9) ], 
  [ 1, NF(7,[ 1, 2, 4 ]), 7, [ 3, 2, 0 ] ], 
  [ 1, NF(21,[ 1, 4, 16 ]), 21, [ 3, 4, 7 ] ] ]
gap> SchurIndex(W[5]);
3

gap> G:=SmallGroup(40,1);              
<pc group of size 40 with 4 generators>
gap> Size(Irr(G));                          
16
gap> SchurIndexByCharacter(GaussianRationals,G,16);
2
gap> SchurIndexByCharacter(CF(5),G,16);         
1