The Andrews–Gordon Identities and q-Multinomial Coefficients

Abstract:

We prove polynomial boson-fermion identities for the generating function of the number of partitions of n of the form $n=\sum_{j=1}^{L-1} j f_j$, with $f_1\leq i-1$, $f_{L-1} \leq i'-1$ and $f_j+f_{j+1}\leq k$. The bosonic side of the identities involves q-deformations of the coefficients of x ^a in the expansion of $(1+x+\cdots+ x^k)^L$. A combinatorial interpretation for these q-multinomial coefficients is given using Durfee dissection partitions. The fermionic side of the polynomial identities arises as the partition function of a one-dimensional lattice-gas of fermionic particles.

In the limit $L\to\infty$, our identities reproduce the analytic form of Gordon's generalization of the Rogers–Ramanujan identities, as found by Andrews. Using the $q \to 1/q$ duality, identities are obtained for branching functions corresponding to cosets of type $({\rm A}^{(1)}_1)_k \times ({\rm A}^{(1)}_1)_{\ell} / ({\rm A}^{(1)}_1)_{k+\ell}$ of fractional level $\ell$.