  We relate two apparently different bases in the representations of affine Lie
algebras of type A: one arising from statistical mechanics, the other from
gauge theory. We show that the two are governed by the same combinatorics and
therefore can be viewed as identical. In particular, we are able to give an
alternative and much simpler geometric proof of a result of E. Date, M. Jimbo,
A. Kuniba, T. Miwa and M. Okado on the construction of bases of affine Lie
algebra representations. At the same time, we give a simple parametrization of
the irreducible components of Nakajima quiver varieties associated to infinite
and cyclic quivers. We also define new varieties whose irreducible components
are in one-to-one correspondence with bases of the highest weight
representations of affine gl_{n+1}.
