The AndrewsGordon Identities and qMultinomial Coefficients
Abstract
We prove polynomial bosonfermion identities for the generating function of the number of partitions of $n$ of the form $n=\sum_{j=1}^{L1} j f_j$, with $f_1\leq i1$, $f_{L1} \leq i'1$ 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 onedimensional latticegas of fermionic particles. In the limit $L\to\infty$, our identities reproduce the analytic form of Gordon's generalization of the RogersRamanujan 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 $\ell$.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 1997
 DOI:
 10.1007/s002200050058
 arXiv:
 arXiv:qalg/9601012
 Bibcode:
 1997CMaPh.184..203W
 Keywords:

 Mathematics  Quantum Algebra;
 High Energy Physics  Theory
 EPrint:
 31 pages, Latex, 9 Postscript figures