Zeons, lattices of partitions, and free probability

Schott, René; Staples, G. Stacey

doi:10.31390/cosa.4.3.02

Cited by 3 publications

(5 citation statements)

References 15 publications

(16 reference statements)

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“…Zeons are already appearing in current work, e.g. [8]. Further explorations, extensions, and applications are for the future.…”

Section: Discussionmentioning

confidence: 96%

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REPRESENTATIONS OF sl(2) IN THE BOOLEAN LATTICE, AND THE HAMMING AND JOHNSON SCHEMES

Feinsilver

2012

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Abstract. Starting with the zero-square "zeon algebra", the regular representation gives rise to a Boolean lattice representation of sl(2). We detail the su(2) content of the Boolean lattice, providing the irreducible representations carried by the algebra generated by the subsets of an n-set. The group elements are found, exhibiting the "special functions" in this context. The corresponding Leibniz rule and group law are shown. Krawtchouk polynomials, the Hamming and the Johnson schemes appear naturally. Applications to the Boolean poset and the structure of Hadamard-Sylvester matrices are shown as well.

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“…Zeons are already appearing in current work, e.g. [8]. Further explorations, extensions, and applications are for the future.…”

Section: Discussionmentioning

confidence: 96%

“…On any subrepresentation with principal number N, we have the basis φ j with the action, cf. equation (8), and so on for higher n. Proof. Proceeding by induction, for n = 1, we have ℓ = 0, with the vacuum state e ∅ = 1.…”

Section: 2mentioning

confidence: 91%

REPRESENTATIONS OF sl(2) IN THE BOOLEAN LATTICE, AND THE HAMMING AND JOHNSON SCHEMES

Feinsilver

2012

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…Zeons underlie a significant portion of Liu's work on enumeration of Hamiltonian cycles [9], and their combinatorial properties have been developed in a number of the current author's joint works (e.g., [11,12,13]). Combinatorial properties of zeons are also useful for defining partition-dependent stochastic integrals [14,15]. In other recent work, Neto and dos Anjos [10] establish a number of combinatorial identities using Grassmann-Berezin type integration on zeon algebras.…”

Section: Introduction and Notational Preliminariesmentioning

confidence: 99%

“…I⊆[n]u I ζ I . Combinatorial properties of zeons have been developed in a number of works in recent years[3,4,14,16].…”

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confidence: 99%

Hamiltonian Cycle Enumeration via Fermion-Zeon Convolution

Staples

2017

Int J Theor Phys

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Operators are induced on fermion and zeon algebras by the action of adjacency matrices and combinatorial Laplacians on the vector spaces spanned by the graph's vertices. Properties of the algebras automatically give information about the graph's spanning trees and vertex coverings by cycles & matchings. Combining the properties of operators induced on fermions and zeons gives a fermion-zeon convolution that recovers the number of Hamiltonian cycles in an arbitrary graph. The mathematics underlying the graph-theoretic interpretation of these operators is provided by Kirchhoff's theorem and by the seminal works of Goulden and Jackson and Liu, who established formulas for enumeration of Hamiltonian cycles and paths using determinants and permanents of adjacency matrices.

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“…Further, they have been useful in defining partition-dependent stochastic integrals. In fact, expanding powers of zeon elements is equivalent to summing over partitions [23].…”

Section: Introductionmentioning

confidence: 99%

Zeons, Orthozeons, and Graph Colorings

Staples

Stellhorn

2016

Adv. Appl. Clifford Algebras

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Nilpotent adjacency matrix methods have proven useful for counting self-avoiding walks (paths, trails, cycles, and circuits) in finite graphs. In the current work, these methods are extended for the first time to problems related to graph colorings. Nilpotent-algebraic formulations of graph coloring problems include necessary and sufficient conditions for kcolorability, enumeration (counting) of heterogeneous and homogeneous paths, trails, cycles, and circuits in colored graphs, and a matrix-based greedy coloring algorithm. Introduced here also are the orthozeons and their application to counting monochromatic self-avoiding walks in colored graphs. The algebraic formalism easily lends itself to symbolic computations, and Mathematica-computed examples are presented throughout.

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Zeons, lattices of partitions, and free probability

Cited by 3 publications

References 15 publications

REPRESENTATIONS OF sl(2) IN THE BOOLEAN LATTICE, AND THE HAMMING AND JOHNSON SCHEMES

REPRESENTATIONS OF sl(2) IN THE BOOLEAN LATTICE, AND THE HAMMING AND JOHNSON SCHEMES

Hamiltonian Cycle Enumeration via Fermion-Zeon Convolution

Zeons, Orthozeons, and Graph Colorings

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