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Berenstein and Kazhdan define perfect bases as an "unquantized" version of crystal bases. A perfect basis is roughly a basis with a crystal structure such that $E_i\cdot v=\mathbb{C}\cdot \tilde{e}_iv+\cdots$ where the $\cdots$ indicates terms in basis vectors killed by $\tilde{e}_i^{\epsilon_i(v)-2}$ (here $E_i$ is an element of the Lie algebra, and $\tilde{e}_i$ is a Kashiwara operator).

The cool theorem is that any given finite-dimensional representation only has one possible crystal attached to it.

Note that many of the "nicest" crystal bases (in particular, the global crystal basis) are perfect bases when specialized at q=1, this is far from universally true. In particular, taking the tensor product of perfect bases in the naive sense doesn't result in a new perfect basis.

Does anyone know of a way of fixing this, and getting in a canonical perfect basis on the tensor product from perfect bases on the factors?

What I particularly want is a natural bijection from the basis in the tensor product to the product of the original bases, sending the induced crystal structure to the crystal tensor product.

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You've probably thought about this. What about Lusztig's tensor product of based modules? – Bruce Westbury May 24 '10 at 16:38
Don't based modules rely in a very heavy way on the q? The whole point of this perfect basis story is not to rely on the q. – Ben Webster May 25 '10 at 3:29
I was exactly wondering this recently myself! It would be very nice to have this, since the perfect basis theory is as powerful (yet more elementary) than the crystal basis theory, except for this point. – Joel Kamnitzer Apr 24 '11 at 0:37

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