• DOI:10.1007/s002200050058
• Corpus ID: 16034446

The Andrews–Gordon Identities and q-Multinomial Coefficients

@article{Warnaar1997TheAI,
  title={The Andrews–Gordon Identities and q-Multinomial Coefficients
},
  author={S. Ole Warnaar},
  journal={Communications in Mathematical Physics},
  year={1997},
  volume={184},
  pages={203-232}
}

• S. Warnaar
• Published 15 January 1996
• Mathematics
• Communications in Mathematical Physics

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 xa 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
$\ell$.