Some graphical aspects of Frobenius structures

Fauser, Bertfried

doi:10.48550/arxiv.1202.6380

Cited by 3 publications

(11 citation statements)

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“…Nonetheless, in abstract, categorical formulations, a 'measurement basis' is accorded a separate significance, and is regarded as a foundational axiomatic datum equated to the presence of Frobenius algebraic structures [79,41]. Indeed, the basic 'splitting operator' δ , in our multilinear tensor formulation in §2.1, δ(e i ) = i e i ⊗ e i , is precisely the Frobenius comultiplication of [41] (see also [42]). In that case, there is the accompanying multiplication, µ(e i ⊗ e j ) = e i δ i,j .…”

Section: Discussionmentioning

confidence: 99%

“…In summary, we have seen that there is deep algebraic structure bound up with the tensorial construction of the theoretical; phylogenetic branching models. From the Lie algebraic point of view, the emergence of the original coproduct ∇ L with its additional 'quadratic' cross term, is unusual, in that the usual coproduct 42 is the minimal (or primitive) choice ∆(L) = L ⊗ 1 + 1 ⊗ L. We emphasize that ∇ is also a linear operator, but whose cross terms are defined with reference to the form of the generators in the standard, distinguished basis. It is unknown to what extent the homomorphism property extends to other phlyogenetic Lie-Markov models, for example those within the S 2 S 2 class (see figure 5).…”

Section: Coproducts and Phylogenetic Bialgebrasmentioning

confidence: 99%

“…We denote each edge e by a multi-index string comprising the labels of its descendant subtree 44 . Now for 42 While primitive elements p of a Hopf algebra have coproduct ∆(p) = p ⊗ 1 + 1 ⊗ p , grouplike elements g have coproduct ∆(g) = g ⊗ g . The structure of ∇ is heuristically explicable if it is borne in mind that the exponentials of rate generators (Markov matrices, M ) can generically be expressed as 1 + λQ for some scalar λ and rate generator Q (not necessarily ln M ).…”

Section: Phylogenetic Tensors and The Star Lemmamentioning

confidence: 99%

“…Here we consider L to be the leaf set with |L| leaves 49. This theorem is equivalent to the well-known 'spider lemma' in the context of graphical structures and Frobenius algebras[41,42]; for further comments see § §2.3.3,2.3.4 below and the concluding remarks.…”

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confidence: 99%

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Systematics and symmetry in molecular phylogenetic modelling: perspectives from physics

Jarvis¹,

Sumner²

2019

J. Phys. A: Math. Theor.

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The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics which have been intensively used in recent years in biology as the cornerstone of attempts to infer and reconstruct the ancestral relationships between species. We outline the development of theoretical phylogenetics, from the earliest studies based on morphological characters, through to the use of molecular data in a wide variety of forms. We bring the lens of mathematical physics to bear on the formulation of theoretical models, focussing on the applicability of many methods from the toolkit of that tradition -techniques of groups and representations to guide model specification and to exploit the multilinear setting of the models in the presence of underlying symmetries; extensions to coalgebraic properties of the generators associated to rate matrices underlying the models, and possibilities to marry these with the graphical structures (trees and networks) which form the search space for inferring evolutionary trees.Particular aspects which we wish to present to a readership accustomed to thinking from physics, include relating model classes to structural data on relevant matrix Lie algebras, as well as using manipulations with group characters (especially the operation of plethysm, for computing tensor powers) to enumerate various natural polynomial invariants, which can be enormously helpful in tying down robust, low-parameter quantities for use in inference (some of which have only come to light through our perspective). Above all, we wish to emphasize the many features of multipartite entanglement which are shared between descriptions of quantum states on the physics side, and the multi-way tensor probability arrays arising in phylogenetics. In some instances, well-known objects such as the Cayley hyperdeterminant (the 'tangle') can be directly imported into the formalism -in this case, for models with binary character traits, and for providing information about triplets of taxa. In other cases new objects appear, such as the remarkable 'squangle' invariants for quartet tree discrimination, which for DNA data are of quintic degree, with their own unique interpretation in the phylogenetic modelling context. All this hints strongly at the natural and universal presence of entanglement as a phenomenon which reaches across disciplines. We hope that this broad perspective may in turn furnish new insights of use in physics.

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Section: Discussionmentioning

confidence: 99%

Section: Coproducts and Phylogenetic Bialgebrasmentioning

confidence: 99%

Section: Phylogenetic Tensors and The Star Lemmamentioning

confidence: 99%

mentioning

confidence: 99%

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Systematics and symmetry in molecular phylogenetic modelling: perspectives from physics

Jarvis¹,

Sumner²

2019

J. Phys. A: Math. Theor.

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“…Frobenius algebras (FA) have been extensively studied in representation theory and have recently experienced a renewed interest in the context of topological quantum field theory [19], quantum foundations [20] and even natural language processing [21]. See [22] for a survey on graphical aspects. A cause of this multidisciplinary interest is their intuitive topological calculus.…”

Section: Flexibility Of Frobenius Algebrasmentioning

confidence: 99%

When Only Topology Matters

Carette

2021

Preprint

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Graphical languages are symmetric monoidal categories presented by generators and equations. The string diagrams notation allows to transform numerous axioms into low dimension topological rules we are comfortable with as three dimensional space citizens. This aspect is often referred to by the Only Topology Matters paradigm (OTM). However OTM remains quite informal and its exact meaning in terms of rewriting rules is ambiguous. In this paper we define three precise aspects of the OTM paradigm, namely flexsymmetry, flexcyclicity and flexibility of Frobenius algebras. We investigate how this new framework can simplify the presentation of known graphical languages based on Frobenius algebras.

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Hopf algebras, distributive (Laplace) pairings and hash products: a unified approach to tensor product decompositions of group characters

Fauser¹,

Jarvis²,

King³

2014

J. Phys. A: Math. Theor.

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We show for bicommutative graded connected Hopf algebras that a certain distributive (Laplace) subgroup of the convolution monoid of 2-cochains parameterizes certain well behaved Hopf algebra deformations. Using the Laplace group, or its Frobenius subgroup, we define higher derived hash products, and develop a general theory to study their main properties. Applying our results to the (universal) bicommutative graded connected Hopf algebra of symmetric functions, we show that classical tensor product and character decompositions, such as those for the general linear group, mixed co-and contravariant or rational characters, orthogonal and symplectic group characters, Thibon and reduced symmetric group characters, are special cases of higher derived hash products. In the Appendix we discuss a relation to formal group laws.

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Some graphical aspects of Frobenius structures

Cited by 3 publications

References 53 publications

Systematics and symmetry in molecular phylogenetic modelling: perspectives from physics

Systematics and symmetry in molecular phylogenetic modelling: perspectives from physics

When Only Topology Matters

Hopf algebras, distributive (Laplace) pairings and hash products: a unified approach to tensor product decompositions of group characters

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