  We consider graded Cartan matrices of the symmetric groups and the
Iwahori-Hecke algebras of type A, which have entries in the ring $\mathbb
Z[v,v^{-1}]$. These matrices may also be interpreted as Gram matrices of the
Shapovalov form on sums of weight spaces of a basic representation of an affine
quantum group. We present a conjecture predicting the invariant factors of
these matrices and give evidence for the conjecture by proving its implications
under a localization and certain specializations of the ring $\mathbb
Z[v,v^{-1}]$. This proves and generalizes a conjecture of Ando-Suzuki-Yamada on
the invariants of these matrices over $\mathbb Q[v,v^{-1}]$ and also
generalizes the first author's recent proof of the K\"ulshammer-Olsson-Robinson
conjecture over $\mathbb Z$.

Representation Theory;Quantum Algebra