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<a<b. For a prime power q, let 𝒩𝒞_n^a,b(𝔽_q) denote the set of pairs of commuting nilpotent n× n matrices (A,B) over 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: underbracevphantom| ∏_m≥1(1−q^−m) Biggl(∑_n=0^∞ lvert𝒩𝒞_n^a,b(𝔽_q)rvert/lvertGL_n(𝔽_q)rvertBiggr) _{point count} = underbracevphantom|Z_a,b(q⁻¹) _(q)−series = underbracevphantom|P_a,b(q⁻¹) _{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→∞ 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 unconditionally in Lean by AxiomProver.

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Cite

BibTeX RIS

BibTeX

@misc{huang2026rogersramanujan,
  title = {{Rogers--Ramanujan identities from the geometry of $X^a=Y^b$}},
  author = {Yifeng Huang and Kenny Lau and Ken Ono},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.20567},
  eprint = {2609.20567},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20567},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - Rogers--Ramanujan identities from the geometry of X^a=Y^b
AU  - Yifeng Huang
AU  - Kenny Lau
AU  - Ken Ono
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.20567
UR  - https://arxiv.org/abs/2609.20567
AB  - 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<a<b. For a prime power q, let 𝒩𝒞_n^a,b(𝔽_q) denote the set of pairs of commuting nilpotent n× n matrices (A,B) over 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: underbracevphantom| ∏_m≥1(1−q^−m) Biggl(∑_n=0^∞ lvert𝒩𝒞_n^a,b(𝔽_q)rvert/lvertGL_n(𝔽_q)rvertBiggr) _point count = underbracevphantom|Z_a,b(q^−1) _(q)−series = underbracevphantom|P_a,b(q^−1) _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→∞ 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 unconditionally in Lean by AxiomProver.
ER  -

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