Abstract
We consider the fusion algebras arising in e.g. Wess-Zumino-Witten conformal field theories, affine Kac-Moody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix
S
, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the
A
r
fusion algebra at level
k
. We prove that for many choices of rank
r
and level
k
, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most
r
and
k
. We also find new, systematic sources of zeros in the modular matrix
S
. In addition, we obtain a formula relating the entries of
S
at fixed points, to entries of
S
at smaller ranks and levels. Finally, we identify the number fields generated over the rationals by the entries of
S
, and by the fusion (Verlinde) eigenvalues.