Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you (quickly) figure out what that basis is?

Note that this may be much easier than finding the shortest vector in a general lattice, since (for instance) you know ahead of time exactly how long the shortest vector actually is.

Motivation: Suppose you wanted to use Brauer's induction theorem to compute the character table of a big group. After computing enough induced characters, you can check (by calculating a suitable determinant) that you have a set of representations generating the Grothendieck group. You know ahead of time that the irreducible representations form an orthonormal basis - so it's natural to ask whether you can use this information to quickly figure out what the characters actually are.