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 $f_1\le i-1$, $f_{L-1}\le i^{\prime }-1$ and $f_j+f_{j+1}\le k$. 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 $({\mathrm{A}}_1^{(1)}{)}_k\times ({\mathrm{A}}_1^{(1)}{)}_{\ell }/({\mathrm{A}}_1^{(1)}{)}_{k+\ell }$ of fractional level $\ell$.

31 pages, Latex, 9 Postscript figures

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• Bailey flows and Bose-Fermi identities for the conformal coset models $(A_1^{(1)}{)}_N\times (A_1^{(1)}{)}_{N^{\prime }}/(A_1^{(1)}{)}_{N+N^{\prime }}$
• Polynomial Triangles Revisited
• $q$-Trinomial identities
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• Parafermionic derivation of Andrews-type multiple sums
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• Proof of the Flohr-Grabow-Koehn conjectures for characters of logarithmic conformal field theory
• Integrable $\widehat{{\mathfrak{sl}}_2}$-modules as infinite tensor products
• Fermi-gas interpretation of the RSOS path representation of the superconformal unitary minimal models
• Q-characters of the tensor products
• On simplifications of certain $q$-multisums