Jagged Partitions

Fortin, Jean-François; Jacob, P.; Mathieu, Pierre

doi:10.1007/s11139-005-4848-8

Cited by 56 publications

(81 citation statements)

References 9 publications

(22 reference statements)

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“…The generating function (equivalent of Eq. (1)) for the number of (2, r) partitions λ i − λ i+2 ≥ r has been obtained using the theory of jagged partitions [20] [21]. After some algebra, we find a ground-state energy r N 2…”

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confidence: 99%

Properties of Non-Abelian Fractional Quantum Hall States at Fillingν=k/r

2008

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We compute the physical properties of non-Abelian Fractional Quantum Hall (FQH) states described by Jack polynomials at general filling ν = k r . For r = 2, these states are identical to the Z k Read-Rezayi parafermions, whereas for r > 2 they represent new FQH states. The r = k + 1 states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling 2/5, 3/7, 4/9.... We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and show that the non-Abelian quasihole has a well-defined propagator falling off with the distance. The clustering properties of the Jack polynomials, provide a strong indication that the states with r > 2 can be obtained as correlators of fields of non-unitary conformal field theories, but the CFT-FQH connection fails when invoked to compute physical properties such as thermal Hall coefficient or, more importantly, the quasihole propagator. The quasihole wavefuntion, when written as a coherent state representation of Jack polynomials, has an identical structure for all non-Abelian states at filling ν = k r .

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confidence: 99%

Properties of Non-Abelian Fractional Quantum Hall States at Fillingν=k/r

2008

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“…By employing dissection formulas, Fortin, Jacob and Mathieu [7], Hirschhorn and Sellers [9] independently derived various Ramanujan-type congruences for p(n), such as…”

Section: Introductionmentioning

confidence: 99%

Ramanujan-type congruences for overpartitions modulo 16

et al. 2015

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Abstract. Let p(n) denote the number of overpartitions of n. Hirschhorn and Sellers showed that p(4n + 3) ≡ 0 (mod 8) for n ≥ 0. They also conjectured that p(40n + 35) ≡ 0 (mod 40) for n ≥ 0. Chen and Xia proved this conjecture by using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that p(5n) ≡ (−1) n p(4 · 5n) (mod 5) for n ≥ 0 and p(n) ≡ (−1) n p(4n) (mod 8) for n ≥ 0 by using the relation of the generating function of p(5n) modulo 5 found by Treneer and the 2-adic expansion of the generating function of p(n) due to Mahlburg. As a consequence, we deduce that p(4 k (40n + 35)) ≡ 0 (mod 40) for n, k ≥ 0. Furthermore, applying the Hecke operator on φ(q) 3 and the fact that φ(q) 3 is a Hecke eigenform, we obtain an infinite family of congrences p(4 k · 5ℓ 2 n) ≡ 0 (mod 5), where k ≥ 0 and ℓ is a prime such that ℓ ≡ 3 (mod 5) and −n ℓ = −1. Moreover, we show that p(5 2 n) ≡ p(5 4 n) (mod 5) for n ≥ 0. So we are led to the congruences p 4 k 5 2i+3 (5n ± 1) ≡ 0 (mod 5) for n, k, i ≥ 0. In this way, we obtain various Ramanujan-type congruences for p(n) modulo 5 such as p(45(3n + 1)) ≡ 0 (mod 5) and p(125(5n ± 1)) ≡ 0 (mod 5) for n ≥ 0.

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“…Following [9], a jagged partition refers to an ordered collection of non-negative integers (n 1 , n 2 , . .…”

Section: Introductionmentioning

confidence: 99%