The Huang-Jiang-Oblomkov Conjecture at a = 3
Prior state unknown→proved
Proves the a = 3 layer; the conjecture is layered in a and remains open for larger a.
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number-theory / q-series; arithmetic geometry
The Huang-Jiang-Oblomkov Conjecture at a = 3
Huang, Jiang and Oblomkov conjectured that the Eulerian $q$-series counting commuting pairs of nilpotent matrices with $X^a = Y^b$ equals an explicit theta-and-eta product, making the point count essentially modular. The conjecture is layered in $a$; the $a = 2$ layer is classical, including Rogers-Ramanujan and Andrews-Gordon. Nothing was known for $a = 3$. That layer is now proved in full, yielding a new infinite family of Rogers-Ramanujan identities and a geometric origin for Warnaar's products.
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Huang, Jiang and Oblomkov conjectured that the Eulerian $q$-series counting commuting pairs of nilpotent matrices with $X^a = Y^b$ equals an explicit theta-and-eta product, making the point count essentially modular. The conjecture is layered in $a$; the $a = 2$ layer is classical, including Rogers-Ramanujan and Andrews-Gordon. Nothing was known for $a = 3$. That layer is now proved in full, yielding a new infinite family of Rogers-Ramanujan identities and a geometric origin for Warnaar's products.
Proves the a = 3 layer; the conjecture is layered in a and remains open for larger a.
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