  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.
