We introduce a concept of global decomposition numbers
for cellular algebras A over a principal ideal domain R. Such
algebras have been defined by Graham and Lehrer in [GL].
In the category of R-finitely generated A-modules, you always
have a set of standard resp. costandard modules, the
former of which
Graham and Lehrer called cell representations
and which we will denote by 
resp. 
(for the indicated connections
with quasi-hereditary algebras we refer to [KX]).
Here 
runs through a poset
which comes along with the definition of A.
Now, if there is an R-algebra structure on a field K, i.e.\
a ring homomorphism from R to K, it is shown in [GL] that
there is a unique maximal 
)-submodule in 
(not necessarily
proper) and that
the corresponding nonzero simple quotients 
are absolutely irreducible
and give all the irreducibles 
-modules. In this situation you can
define decomposition numbers 
as the
integer coefficients of 
in the
Grothendieck group of with respect to the basis element
.
In this paper we are going to define analogues of and
with respect to the algebra A over the
original ground ring R, such that you can get the above described
simple modules and decomposition numbers by specialising.
As examples for
A we will treat the group algebras of
the symmetric groups 
and 
over the ring 
.
In this case, the standard modules are known as
Specht modules and the index set consists of partitions
of n. It turns out that the global decomposition numbers contain
informations on the module category of 
, which
vanishes in all specialisations, i.e. which can not be reconstructed
from the knowledge of ordinary
decomposition numbers for all characteristics.
This work is purely conceptual, the proofs being almost trivial. But anyway, it provides a new view and calculus for decomposition numbers. Since more information survives in this calculus, there might be a chance to understand things better.