# 49.2 Examples of Generic Character Tables

1. The generic table of the cyclic group:

For the cyclic group C_q = < x > of order q, there is one type of classes. The class parameters are integers k in {0,ldots,q-1}, the class of the parameter k consists of the group element x^k. Group order and centralizer orders are the identity function q mapsto q, independent of the parameter k.

The representative order function maps (q,k) to frac{q}{gcd(q,k)}, the order of x^k in C_q; the p-th powermap is the function (q,k,p) mapsto [1,(kpbmod q)].

There is one type of characters with parameters l in {0,ldots,q-1}; for e_q a primitive complex q-th root of unity, the character values are chi_l(x^k) = e_q^{kl}.

The library file contains the following generic table:

```    rec(name:="Cyclic",
specializedname:=(q->ConcatenationString("C",String(q))),
order:=(n->n),
text:="generic character table for cyclic groups",
centralizers:=[function(n,k) return n;end],
classparam:=[(n->[0..n-1])],
charparam:=[(n->[0..n-1])],
powermap:=[function(n,k,pow) return [1,k*pow mod n];end],
orders:=[function(n,k) return n/Gcd(n,k);end],
irreducibles:=[[function(n,k,l) return E(n)^(k*l);end]],
domain:=(n->IsInt(n) and n>0),
libinfo:=rec(firstname:="Cyclic",othernames:=[]),
isGenericTable:=true);```

2. The generic table of the general linear group rm{GL}(2,q):

We have four types t_1, t_2, t_3, t_4 of classes according to the

rational canonical form of the elements::

t_1 scalar matrices,
t_2 nonscalar diagonal matrices,
t_3 companion matrices of (X-rho)^2 for elements rho in F_q^{ast} and
t_4 companion matrices of irreducible polynomials of degree 2 over F_q.

The sets of class parameters of the types are in bijection with
F_q^{ast} for t_1 and t_3, {{rho,tau}; rho, tau in F_q^{ast}, rhonot=tau} for t_2 and {{epsilon,epsilon^q}; epsilon in F_{q^2}setminus F_q} for t_4.

The centralizer order functions are q mapsto (q^2-1)(q^2-q) for type t_1, q mapsto (q-1)^2 for type t_2, q mapsto q(q-1) for type t_3 and q mapsto q^2-1 for type t_4.

The representative order function of t_1 maps (q,rho) to the order of rho in F_q, that of t_2 maps (q,{rho,tau}) to the least common multiple of the orders of rho and tau.

The file contains something similar to this table:

```    rec(name:="GL2",
specializedname:=(q->ConcatenationString("GL(2,",String(q),")")),
order:= ( q -> (q^2-1)*(q^2-q) ),
text:= "generic character table of GL(2,q),\
see Robert Steinberg: The Representations of Gl(3,q), Gl(4,q),\
PGL(3,q) and PGL(4,q), Canad. J. Math. 3 (1951)",
classparam:= [ ( q -> [0..q-2] ), ( q -> [0..q-2] ),
( q -> Combinations( [0..q-2], 2 ) ),
( q -> Filtered( [1..q^2-2], x -> not (x mod (q+1) = 0)
and (x mod (q^2-1)) < (x*q mod (q^2-1)) ))],
charparam:= [ ( q -> [0..q-2] ), ( q -> [0..q-2] ),
( q -> Combinations( [0..q-2], 2 ) ),
( q -> Filtered( [1..q^2-2], x -> not (x mod (q+1) = 0)
and (x mod (q^2-1)) < (x*q mod (q^2-1)) ))],
centralizers := [ function( q, k ) return (q^2-1) * (q^2-q); end,
function( q, k ) return q^2-q; end,
function( q, k ) return (q-1)^2; end,
function( q, k ) return q^2-1; end],
orders:= [ function( q, k ) return (q-1)/Gcd( q-1, k ); end,
..., ..., ... ],
classtext:= [ ..., ..., ..., ... ],
powermap:=
[ function( q, k, pow ) return [1, (k*pow) mod (q-1)]; end,
..., ..., ... ],
irreducibles := [[ function( q, k, l ) return E(q-1)^(2*k*l); end,
function( q, k, l ) return E(q-1)^(2*k*l); end,
...,
function( q, k, l ) return E(q-1)^(k*l); end    ],
[ ..., ..., ..., ... ],
[ ..., ..., ..., ... ],
[ ..., ..., ..., ... ]],
domain := ( q->IsInt(q) and q>1 and Length(Set(FactorsInt(q)))=1 ),
isGenericTable := true )```

GAP 3.4.4
April 1997