# ADVENTURES IN INVARIANT THEORY

@article{Jarvis2014ADVENTURESII, title={ADVENTURES IN INVARIANT THEORY}, author={Peter D. Jarvis and Jeremy G. Sumner}, journal={The ANZIAM Journal}, year={2014}, volume={56}, pages={105 - 115} }

Abstract We provide an introduction to enumerating and constructing invariants of group representations via character methods. The problem is contextualized via two case studies, arising from our recent work: entanglement invariants for characterizing the structure of state spaces for composite quantum systems; and Markov invariants, a robust alternative to parameter-estimation intensive methods of statistical inference in molecular phylogenetics.

## 7 Citations

### Dimensional reduction for phylogenetic tree models

- Biology
- 2016

A general method of dimensional reduction for phylogenetic tree models that identifies an invariant subspace which depends only bilinearly on the model parameters; in contrast to the usual multi-linear dependence in the full model space.

### Developing a statistically powerful measure for quartet tree inference using phylogenetic identities and Markov invariants

- BiologyJournal of mathematical biology
- 2017

A statistically bias-corrected version of the Markov invariants approach is developed which satisfies all three desirable statistical properties of any invariant-based phylogenetic method, and several new phylogenetic inference methods are developed and explored.

### Dimensional Reduction for the General Markov Model on Phylogenetic Trees

- BiologyBulletin of mathematical biology
- 2017

A method of dimensional reduction for the general Markov model of sequence evolution on a phylogenetic tree is presented and a key feature is the identification of an invariant subspace which depends only bilinearly on the model parameters, in contrast to the usual multi-linear dependence in the full space.

### Developing a statistically powerful measure for phylogenetic tree inference using phylogenetic identities and Markov invariants

- Biology
- 2016

A statistical bias-corrected version of the Markov invariants approach is developed which satisfies all three desirable statistical properties any phylogenetic method should satisfy, and several new phylogenetic inference methods are developed and explored.

### Matrix group structure and Markov invariants in the strand symmetric phylogenetic substitution model

- MathematicsJournal of mathematical biology
- 2016

This work applies representation theoretic techniques and provides the means to classify and enumerate the associated Markov invariants for any (fixed) number of phylogenetic taxa L and polynomial degree D of interest, and proves there are precisely quadratic and cubic MarkOV invariants.

### The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group SL(3,C)?>

- Mathematics
- 2014

We consider local unitary invariants and entanglement monotones for the mixed two qutrit system. Character methods for the local SU(3) × SU(3) transformation group are used to establish the count of…

### Systematics and symmetry in molecular phylogenetic modelling: perspectives from physics

- BiologyJournal of Physics A: Mathematical and Theoretical
- 2019

The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics, as the foundations on which the practical implementations are based, and 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.

## References

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- MathematicsJournal of theoretical biology
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### Entanglement invariants and phylogenetic branching

- BiologyJournal of mathematical biology
- 2005

A powerful approach to entanglement measures in quantum physics and its relevance to phylogenetic analysis are summarized and distance measures are found to give distance measures that are equivalent to, and expand upon, those already known in phylogenetics.

### Entanglement, Invariants, and Phylogenetics

- Mathematics
- 2006

This thesis develops and expands upon known techniques of mathematical physics relevant to the analysis of the popular Markov model of phylogenetic trees required in biology to reconstruct the evolutionary relationships of taxonomic units from biomolecular sequence data and takes a group theoretical approach to the problem.

### Quantifying entanglement resources

- Physics
- 2014

We present an overview of the quantitative theory of single-copy entanglement in finite-dimensional quantum systems. In particular we emphasize the point of view that different entanglement measures…

### Markov Invariants for Phylogenetic Rate Matrices Derived from Embedded Submodels

- MathematicsIEEE/ACM Transactions on Computational Biology and Bioinformatics
- 2012

A prescription for identifying and counting Markov invariants for such "symmetric embedded” models, and enumerations of these for the first few cases with a small number of character states are provided.

### Entanglement monotones

- Mathematics
- 2000

Abstract In the context of quantifying entanglement we study those functions of a multipartite state which do not increase under the set of local transformations. A mathematical characterization of…

### A Hopf laboratory for symmetric functions

- Mathematics
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An analysis of symmetric function theory is given from the perspective of the underlying Hopf and bi-algebraic structures. These are presented explicitly in terms of standard symmetric function…

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- Mathematics
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In this renowned volume, Hermann Weyl discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from…

### Matrix group structure and Markov invariants in the strand symmetric phylogenetic substitution model

- MathematicsJournal of mathematical biology
- 2016

This work applies representation theoretic techniques and provides the means to classify and enumerate the associated Markov invariants for any (fixed) number of phylogenetic taxa L and polynomial degree D of interest, and proves there are precisely quadratic and cubic MarkOV invariants.