dorsal/arxiv
View SchemaThe Andrews-Gordon identities and $q$-multinomial coefficients
| Authors | S. O. Warnaar |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9601012 |
| URL | https://arxiv.org/abs/q-alg/9601012 |
| DOI | 10.1007/s002200050058 |
| Journal | Commun.Math.Phys. 184 (1997) 203-232 |
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 $\ell$.
{
"annotation_id": "d53f0b9d-303a-4af9-aef9-b1c56cd5fcd2",
"date_created": "2026-03-02T18:01:27.550000Z",
"date_modified": "2026-03-02T18:01:27.550000Z",
"file_hash": "63f77f6082dee6be258daa267800566096f14813f3499092e49a9606130cb23f",
"private": false,
"record": {
"abstract": "We prove polynomial boson-fermion identities for the generating function of\nthe number of partitions of $n$ of the form $n=\\sum_{j=1}^{L-1} j f_j$, with\n$f_1\\leq i-1$, $f_{L-1} \\leq i\u0027-1$ and $f_j+f_{j+1}\\leq k$. The bosonic side of\nthe identities involves $q$-deformations of the coefficients of $x^a$ in the\nexpansion of $(1+x+\\cdots+ x^k)^L$. A combinatorial interpretation for these\n$q$-multinomial coefficients is given using Durfee dissection partitions. The\nfermionic side of the polynomial identities arises as the partition function of\na one-dimensional lattice-gas of fermionic particles. In the limit\n$L\\to\\infty$, our identities reproduce the analytic form of Gordon\u0027s\ngeneralization of the Rogers--Ramanujan identities, as found by Andrews. Using\nthe $q \\to 1/q$ duality, identities are obtained for branching functions\ncorresponding to cosets of type $({\\rm A}^{(1)}_1)_k \\times ({\\rm\nA}^{(1)}_1)_{\\ell} / ({\\rm A}^{(1)}_1)_{k+\\ell}$ of fractional level $\\ell$.",
"arxiv_id": "q-alg/9601012",
"authors": [
"S. O. Warnaar"
],
"categories": [
"q-alg",
"hep-th",
"math.QA"
],
"doi": "10.1007/s002200050058",
"journal_ref": "Commun.Math.Phys. 184 (1997) 203-232",
"title": "The Andrews-Gordon identities and $q$-multinomial coefficients",
"url": "https://arxiv.org/abs/q-alg/9601012"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "fe13555d-492b-44ab-b07e-880fe13bedf9",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}