The last example looked at Hecke algebras with parameter q=1 and R
a field of characteristic~3 (so e=3); that is, the group algebra of
the symmetric group over a field of characteristic 3. More, generally,
the command Specht(p, p) can be used to consider the group
algebras of the symmetric groups over fields of characteristic p
(ie. e=p, and R a field of characteristic~p).
To consider Hecke algebras defined over arbitrary fields Specht must
also be supplied with a valuation map val as an argument. The
function val is a map from some PID into the natural numbers; at
present it is needed only by functions which rely (at least
implicitly), upon the q--analogue of Schaper's theorem. In general,
val depends upon q and the characteristic of R; full details can
be found in [JM2].
Over fields of characteristic zero, and in the symmetric group case,
the function val is automatically defined by Specht. When R is a
field of characteristic zero, val([i]_q) is 1 if e divides
i and~0 otherwise (this is the valuation map associated to the
prime ideal in C[v] generated by the e--th cyclotomic
polynomial). When <q>=1 and R is a field of characteristic p,
val is the usual p--adic valuation map.
As another example, if <q>=4 and R is a field of characteristic 5 (so <e>=2), then the valuation map sends the integer x to nu_5([4]_x) where [4]_x is interpreted as an integer and nu_5 is the usual 5--adic valuation. To consider this Hecke algebra one could proceed as follows:
gap> val:=function(x) local v;
> x:=Sum([0..x-1],v->4^v); # x-${>}$[x]\_q
> v:=0; while x mod 5=0 do x:=x/5; v:=v+1; od;
> return v;
> end;;
gap> H:=Specht(2,5,val,"e2q4");
Specht(e=2, p=5, S(), P(), D(), HeckeRing="e2q4")
Notice the string ``e2q4'' which was also passed to Specht in
this example. Although it is not strictly necessary, it is a good idea
when using a ``non--standard'' valuation map val to specify the
value of H.HeckeRing=HeckeRing. This string is used for internal
bookkeeping by Specht; in particular, it is used to determine
filenames when reading and saving decomposition matrices. If a
``standard'' valuation map is used then HeckeRing is set to the
string ``e{<}e{>}p{<}p{>}''; otherwise it defaults to
``unknown''. The function SaveDecompositionMatrix will not save
any decomposition matrix for any Hecke algebra H with
H.HeckeRing=``unknown''.
GAP 3.4.4