$\sigma_0=1$ instead of 0. The generating function is defined to be $X'(L)=\sum_{\sigma\in {\mathcal{D}}'_L} q^{E(\sigma)}$. Analogous arguments to those in sections \[subsec:fermi\] and \[subsec:bose\] yield the following two expressions for $X'(L)$ $$X'(L)=\sum_{n=0}^\infty q^{n(n+1)} {\genfrac{[}{]}{0pt}{}{L-n-1}{n}}
 =\sum_{j=-\infty}^\infty (-1)^j q^{\frac{j}{2}(5j+3)}
 {\genfrac{[}{]}{0pt}{}{L}{\lfloor \frac{L-5j-1}{2}\rfloor}}.$$ In the limit $L\to\infty$ this polynomial identity implies .

Crystal bases {#sec:crystal}
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In this section we review the main results about crystal bases and explain how they provide a general setting for the definition of one-dimensional configuration sums.

The quantized universal enveloping algebra $U_q({\mathfrak{g}})$ associated with a symmetrizable Kac–Moody Lie algebra ${\mathfrak{g}}$ was introduced independently by Drinfeld [@D] and Jimbo [@J] in their study of two dimensional solvable lattice models in statistical mechanics. The parameter $q$ corresponds to the temperature of the underlying model. Kashiwara [@K] showed that at zero temperature or $q=0$ the representations of $U_q({\mathfrak{g}})$ have bases, which he coined crystal bases, with a beautiful combinatorial structure and favorable properties such as uniqueness and stability under tensor products. The existence and uniqueness of crystal bases for integrable highest weight modules for an arbitrary symmetrizable Kac–Moody algebra was given in [@K1].

Axiomatic definition of crystals
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Let ${\mathfrak{g}}$ be a symmetrizable Kac-Moody algebra, $P$ the weight lattice, $I$ the index 