{"ID":23510311,"CreatedAt":"2026-09-18T02:21:44.056544415Z","UpdatedAt":"2026-09-18T02:21:44.056544415Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20567","arxiv_id":"2609.20567","title":"Rogers--Ramanujan identities from the geometry of $X^a=Y^b$","abstract":"We prove the conjecture of Huang, Jiang, and Oblomkov (HJO) giving a geometric extension of the Rogers--Ramanujan and Andrews--Gordon identities for every torus-knot singularity $X^a=Y^b$ with coprime $1\u003ca\u003cb.$ For a prime power $q$, let $\\mathcal{NC}_n^{a,b}(\\mathbb F_q)$ denote the set of pairs of commuting nilpotent $n\\times n$ matrices $(A,B)$ over $\\mathbb F_q$ satisfying $A^a=B^b$. We establish the threefold equality between their normalized counts, the HJO $q$-series $Z_{a,b}$, and the explicit infinite product $P_{a,b}$: \\[ \\underbrace{\\vphantom{\\Bigg|} \\prod_{m\\geq1}(1-q^{-m}) \\Biggl(\\sum_{n=0}^{\\infty} \\frac{\\lvert\\mathcal{NC}_n^{a,b}(\\mathbb F_q)\\rvert} {\\lvert\\operatorname{GL}_n(\\mathbb F_q)\\rvert}\\Biggr) }_{\\text{point count}} = \\underbrace{\\vphantom{\\Bigg|}Z_{a,b}(q^{-1}) }_{\\text{\\(q\\)-series}} = \\underbrace{\\vphantom{\\Bigg|}P_{a,b}(q^{-1}) }_{\\text{infinite product}}. \\] Our main result is a stronger finite identity: the rank $N$ HJO sum equals $(q;q)_N$ times the generating function for balanced cylindric partitions with entries bounded by $N$. Taking $N\\to\\infty$ yields the HJO conjecture. The proof combines the compositional rational shuffle theorem of Bergeron--Garsia--Leven--Xin and Mellit with a multiplicativity theorem for slope operators and a determinantal model for bounded cylindric partitions, linked by a common $q$-difference equation. The finite identity and the HJO conjecture have been formalized in Lean by AxiomProver, conditional on two stated literature inputs.","short_abstract":"We prove the conjecture of Huang, Jiang, and Oblomkov (HJO) giving a geometric extension of the Rogers--Ramanujan and Andrews--Gordon identities for every torus-knot singularity $X^a=Y^b$ with coprime $1\u003ca\u003cb.$ For a prime power $q$, let $\\mathcal{NC}_n^{a,b}(\\mathbb F_q)$ denote the set of pairs of commuting nilpotent...","url_abs":"https://arxiv.org/abs/2609.20567","url_pdf":"https://arxiv.org/pdf/2609.20567v1","authors":"[\"Yifeng Huang\",\"Kenny Lau\",\"Ken Ono\"]","published":"2026-09-17T15:28:54Z","proceeding":"math.NT","tasks":"[\"math.NT\",\"math.AG\",\"math.CO\",\"math.RT\"]","methods":"[]","has_code":false}