Enumerating coloured partitions in 2 and 3 dimensions

Davison, Ben; Ongaro, Jared; Szendrői, Balázs

doi:10.1017/s0305004119000252

Cited by 7 publications

(3 citation statements)

References 27 publications

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“…One can simply multiply (−1) n 0 +n 1 for the terms q n 0 0 q n 1 1 in Z c to recover 10 the correct signs in Z crystal . Alternatively, one may consider the twisted PE introduced in [60]. We find that in general given Z c = PE[g], the twisted PE of g is precisely Z crystal .…”

Section: Jhep06(2022)016mentioning

confidence: 99%

Crystal melting, BPS quivers and plethystics

Bao

2022

J. High Energ. Phys.

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We study the refined and unrefined crystal/BPS partition functions of D6-D2-D0 brane bound states for all toric Calabi-Yau threefolds without compact 4-cycles and some non-toric examples. They can be written as products of (generalized) MacMahon functions. We check our expressions and use them as vacuum characters to study the gluings. We then consider the wall crossings and discuss possible crystal descriptions for different chambers. We also express the partition functions in terms of plethystic exponentials. For ℂ3 and tripled affine quivers, we find their connections to nilpotent Kac polynomials. Similarly, the partition functions of D4-D2-D0 brane bound states can be obtained by replacing the (generalized) MacMahon functions with the inverse of (generalized) Euler functions.

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Section: Jhep06(2022)016mentioning

confidence: 99%

Crystal melting, BPS quivers and plethystics

Bao

2022

J. High Energ. Phys.

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“…and the cohomological DT theory of the Jacobi algebras in (1.12) turns out to be the same. The cohomological BPS invariants for the noncommutative conifold can be deduced from [30] and purity (proved as in [12,Thm.4.7]):…”

Section: 11)mentioning

confidence: 99%

A boson-fermion correspondence in cohomological Donaldson–Thomas theory

Davison¹

2022

Glasgow Math. J.

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We introduce and study a fermionisation procedure for the cohomological Hall algebra $\mathcal{H}_{\Pi_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson–Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel–Moore homology of the stack of representations of the $\unicode{x03BC}$ -deformed preprojective algebra introduced by Crawley–Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras and my earlier results on the Borel–Moore homology of the stack of representations of the undeformed preprojective algebra.

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“…We also remark that in Corollary 1 there are 7 cases, while there are only 5 in Theorem 1. This is because part (5) in Theorem 1 encompasses cases ( 5) and (7) in Corollary 1, and part (4) in Theorem 1 encompasses cases ( 4) and (6) in Corollary 1.…”

Section: Introductionmentioning

confidence: 98%

On Kostant's weight $q$-multiplicity formula for $\mathfrak{sl}_{4}(\mathbb{C})$

Garcia¹,

Harris²,

Loving³

et al. 2020

Preprint

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The q-analog of Kostant's weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant's partition function. This formula, when evaluated at q = 1, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra sl4(C) and give closed formulas for the q-analog of Kostant's weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the q-analog of Kostant's partition function by counting restricted colored integer partitions. These formulas, when evaluated at q = 1, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant's weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of sl4(C), which are associated to the Weyl alternation sets. This work answers a question posed in 2019

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Enumerating coloured partitions in 2 and 3 dimensions

Cited by 7 publications

References 27 publications

Crystal melting, BPS quivers and plethystics

Crystal melting, BPS quivers and plethystics

A boson-fermion correspondence in cohomological Donaldson–Thomas theory

On Kostant's weight $q$-multiplicity formula for $\mathfrak{sl}_{4}(\mathbb{C})$

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