The $\mathrm{A}_2$ Andrews-Gordon identities and cylindric partitions

Warnaar, S. Ole

doi:10.48550/arxiv.2111.07550

Cited by 6 publications

(17 citation statements)

References 44 publications

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“…Warnaar proves this identity in his paper [26]. Let us review his proof as it will help us in the case for all t.…”

Section: Representation-theoretic Interpretationmentioning

confidence: 82%

“…At all moduli, the sum-product versions (obtained after setting z = 1 and using character formulas to get products) of our seed conjectures can be seen to be appropriate truncations of A (1) 2 "infinite level" sum-product conjectures, which we provide in Section 3. A subset of these conjectures was already proved by Warnaar in [26]. In a way, infinite level, i.e., ℓ = ∞ means that the affine singular vector x θ (−1) ℓ+1 gets pushed down to infinity and thus vanishes.…”

Section: 2)mentioning

confidence: 93%

“…Recently, Corteel, Dousse and Uncu [8] gave the complete set of identities in the manifestly non-negative form for level 5 (i.e., mod 8 identities). Even more recently, Warnaar [26] added more identities at every level not divisible by 3. Still, the problem of completing the set of Andrews-Schilling-Warnaar identities remains, and the present paper is a step in the direction of answering this problem.…”

Section: 2)mentioning

confidence: 99%

“…1 provided by Warnaar [26] to include principal characters of generalized Verma modules induced from all finite dimensional sl 2 -modules. In purely combinatorial terms, we prove infinite level analogues of all Andrews-Gordon (or Andrews-Bressoud) identities:…”

Section: 2)mentioning

confidence: 99%

“…We also have the following connection to representation theory. [26]). Let L(λ) be a standard module of A (1) r and let χ(L(λ)) denote the principally specialized character, i.e., χ(L(λ)) = e −λ ch L(λ) e −α 0 ,...,e −αr →q .…”

Section: Cylindric Partitions and Corteel-welsh Recursionmentioning

confidence: 99%

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Completing the $\mathrm{A}_2$ Andrews-Schilling-Warnaar identities

Kanade¹,

Russell²

2022

Preprint

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We study the Andrews-Schilling-Warnaar sum-sides for the principal characters of standard (i.e., integrable, highest weight) modules of A(1) 2 . These characters have been studied recently by various subsets of Corteel, Dousse, Foda, Uncu, Warnaar and Welsh. We prove complete sets of identities for moduli 5 through 8 and 10, in Andrews-Schilling-Warnaar form. The cases of moduli 6 and 10 are new. Our methods depend on the Corteel-Welsh recursions governing the cylindric partitions and on certain relations satisfied by the Andrews-Schilling-Warnaar sum-sides. We speculate on the role of the latter in the proofs of higher modulus identities. Further, we provide a complete set of conjectures for modulus 9. In fact, we show that at any given modulus, a complete set of conjectures may be deduced using a subset of "seed" conjectures. These seed conjectures are obtained by appropriately truncating conjectures for the "infinite" level. Additionally, for moduli 3k, we use an identity of Weierstraß to deduce new sum-product identities starting from the results of Andrews-Schilling-Warnaar.

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“…Warnaar proves this identity in his paper [26]. Let us review his proof as it will help us in the case for all t.…”

Section: Representation-theoretic Interpretationmentioning

confidence: 82%

Section: 2)mentioning

confidence: 93%

Section: 2)mentioning

confidence: 99%

Section: 2)mentioning

confidence: 99%

Section: Cylindric Partitions and Corteel-welsh Recursionmentioning

confidence: 99%

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Completing the $\mathrm{A}_2$ Andrews-Schilling-Warnaar identities

Kanade¹,

Russell²

2022

Preprint

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Weighted cylindric partitions

Bridges

Uncu

2022

J Algebr Comb

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Recently Corteel and Welsh outlined a technique for finding new sum-product identities by using functional relations between generating functions for cylindric partitions and a theorem of Borodin. Here, we extend this framework to include very general product-sides coming from work of Han and Xiong. In doing so, we are led to consider structures such as weighted cylindric partitions, symmetric cylindric partitions and weighted skew double-shifted plane partitions. We prove some new identities and obtain new proofs of known identities, including the Göllnitz–Gordon and Little Göllnitz identities as well as some beautiful Schmidt-type identities of Andrews and Paule.

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Bilateral identities of the Rogers–Ramanujan type

Schlosser

2023

Trans. Amer. Math. Soc. Ser. B

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We derive by analytic means a number of bilateral identities of the Rogers–Ramanujan type. Our results include bilateral extensions of the Rogers–Ramanujan and the Göllnitz–Gordon identities, and of related identities by Ramanujan, Jackson, and Slater. We give corresponding results for multisums including multilateral extensions of the Andrews–Gordon identities, of the Andrews–Bressoud generalization of the Göllnitz–Gordon identities, of Bressoud’s even modulus identities, and other identities. Our closed form bilateral and multilateral summations appear to be the very first of their kind.

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The $\mathrm{A}_2$ Andrews-Gordon identities and cylindric partitions

Cited by 6 publications

References 44 publications

Completing the $\mathrm{A}_2$ Andrews-Schilling-Warnaar identities

Completing the $\mathrm{A}_2$ Andrews-Schilling-Warnaar identities

Weighted cylindric partitions

Bilateral identities of the Rogers–Ramanujan type

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