Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T08:54:00.300Z Has data issue: false hasContentIssue false
  • English
  • Français

ADVENTURES IN INVARIANT THEORY

Part of: Algebraic combinatorics Mathematical biology in general Forms and linear algebraic groups

Published online by Cambridge University Press:  15 December 2014

P. D. JARVIS  and
J. G. SUMNER
P. D. JARVIS*
Affiliation:
School of Mathematics and Physics, University of Tasmania, Private Bag 37 GPO, Hobart Tas 7001, Australia email [email protected], [email protected]
J. G. SUMNER
Affiliation:
School of Mathematics and Physics, University of Tasmania, Private Bag 37 GPO, Hobart Tas 7001, Australia email [email protected], [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

group characterinvariantplethysmSchur functionentanglementqubittanglesquanglequartetgeneral Markov model

MSC classification

Primary: 05E05: Symmetric functions and generalizations
Secondary: 11E57: Classical groups 81P68: Quantum computation 92B05: General biology and biomathematics
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 2 , October 2014 , pp. 105 - 115
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Allman, E. S., Jarvis, P. D., Rhodes, J. A. and Sumner, J. G., “Tensor rank, invariants, inequalities, and applications”, SIAM. J. Matrix Anal. Appl. 34 (2013) 1014–1045; doi:10.1137/120899066.Google Scholar
Allman, E. S. and Rhodes, J. A., “Phylogenetic ideals and varieties for the general Markov model”, Adv. Appl. Math. 20 (2007) 127–148; doi:10.1016/j.aam.2006.10.002.Google Scholar
Barry, D. and Hartigan, J. A., “Asynchronous distance between homologous DNA sequences”, Biometrics 43 (1987) 261–276; doi:10.2307/2531811.Google Scholar
Buneman, P., “The recovery of trees from measures of dissimilarity”, in: Mathematics in the archaeological and historical sciences (Edinburgh University Press, Edinburgh, 1971) 387–395.Google Scholar
Cavender, J. A. and Felsenstein, J., “Invariants of phylogenies in a simple case with discrete states”, J. Classification 4 (1987) 57–71; doi:10.1007/BF01890075.Google Scholar
Coffman, V., Kundu, J. and Wootters, W. K., “Distributed entanglement”, Phys. Rev. A 61 (2000); doi:10.1103/PhysRevA.61.052306.Google Scholar
Eltschka, C. and Siewert, J., “Quantifying entanglement resources”, J. Phys. A Math. Theor. 47 (2014); doi:10.1088/1751-8113/47/42/424005.CrossRefGoogle Scholar
Fauser, B. and Jarvis, P. D., “A Hopf laboratory for symmetric functions”, J. Phys. A: Math. Gen. 37 (2004) 1633–1663; doi:10.1088/0305-4470/37/5/012.Google Scholar
Fauser, B., Jarvis, P. D., King, R. C. and Wybourne, and B. G., “New branching rules induced by plethysm”, J. Phys. A: Math. Gen. 39 (2006) 2611–2655; doi:10.1088/0305-4470/39/11/006.Google Scholar
Goodman, R. and Wallach, N. R., Representations and invariants of the classical groups, Volume 68 of Enyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 1998).Google Scholar
Grassl, M., Rötteler, M. and Beth, T., “Computing local invariants of quantum-bit systems”, Phys. Rev. A 58 (1998) 1833–1839; doi:10.1103/PhysRevA.58.1833.Google Scholar
Hall, B. C., Quantum theory for mathematicians, Volume 267 of Graduate Texts in Mathematics (Springer, New York, 2013).Google Scholar
Holland, B. R., Jarvis, P. D. and Sumner, J. G., “Low-parameter phylogenetic inference under the general Markov model”, Syst. Biol. 62 (2013) 78–92; doi:10.1093/sysbio/sys072.Google Scholar
Horodecki, R., Horodecki, P., Horodecki, M. and Horodecki, K., “Quantum entanglement”, Rev. Mod. Phys. 81 (2009) 865–942; doi:10.1103/RevModPhys.81.865.Google Scholar
Jarvis, P. D., “The mixed two qutrit system: local unitary invariants, entanglement monotones and the SLOCC group $SL(3,\mathbb{C})$”, J. Phys. A: Math. Gen. 47 (2014) 215–302; doi:10.1088/1751-8113/47/21/215302.CrossRefGoogle Scholar
Jarvis, P. D. and Sumner, J. G., “Matrix group structure and Markov invariants in the strand symmetric phylogenetic substitution model”, Preprint, 15 pp., arXiv:1307.5574.Google Scholar
Jarvis, P. D. and Sumner, J. G., “Markov invariants for phylogenetic rate matrices derived from embedded submodels”, Trans. Comp. Biol. Bioinform. 9 (2012) 828–836; doi:10.1109/TCBB.2012.24.Google Scholar
Johnson, J. E., “Markov-type Lie groups in $\text{GL}(n,\mathbb{R})$”, J. Math. Phys. 26 (1985) 252–257; doi:10.1109/TCBB.2012.24.Google Scholar
King, R. C., Welsh, T. A. and Jarvis, P. D., “The mixed two-qubit system and the structure of its ring of local invariants”, J. Phys. A: Math. Theor. 40 (2007) 10083; doi:10.1088/1751-8113/40/33/011.Google Scholar
Lake, J. A., “A rate-independent technique for analysis of nucleic acid sequences: evolutionary parsimony”, Mol. Biol. Evol. 4(2) (1987) 167–191.Google Scholar
Lake, J. A., “Reconstructing evolutionary trees from DNA and protein sequences: Paralinear distances”, Proc. Natl. Acad. Sci., USA 91 (1994) 1455–1459; doi:10.1073/pnas.91.4.1455.Google Scholar
Littlewood, D. E., The theory of group characters (Clarendon Press, Oxford, 1940).Google Scholar
Lockhart, P. J., Steel, M. A., Hendy, M. D. and Penny, D., “Recovering evolutionary trees under a more realistic model of sequence evolution”, Mol. Biol. Evol. 11 (1994) 605–612;http://www.ncbi.nlm.nih.gov/pubmed/19391266.Google Scholar
Makhlin, Y., “Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations”, Quantum Inf. Process. 1 (2002) 243–252; doi:10.1023/A:1022144002391.Google Scholar
Molien, T., “Über die Invarianten der linearen Substitutionsgruppen”, Sitzungsber. König. Preuss. Akad. Wiss. (1897) 1152–1156;http://www.sciencedirect.com/science/article/pii/S0195669889800496.Google Scholar
Mourad, B., “On a Lie-theoretic approach to generalised doubly stochastic matrices and applications”, Linear Multilinear Algebra 52 (2004) 99–113; doi:10.1080/0308108031000140687.Google Scholar
Semple, C. and Steel, M., Phylogenetics (Oxford University Press, Oxford, 2003).Google Scholar
Sumner, J. G., “Entanglement, invariants, and phylogenetics”, Ph. D. Thesis, University of Tasmania, 2006.Google Scholar
Sumner, J. G., Charleston, M. A., Jermiin, L. S. and Jarvis, P. D., “Markov invariants, plethysms, and phylogenetics”, J. Theor. Biol. 253 (2008) 601–615; doi:10.1016/j.jtbi.2008.04.001.CrossRefGoogle ScholarPubMed
Sumner, J. G. and Jarvis, P. D., “Entanglement invariants and phylogenetic branching”, J. Math. Biol. 51 (2005) 18–36 (erratum); 53 (2006) 490; doi:10.1007/s00285-004-0309-z.CrossRefGoogle ScholarPubMed
Sumner, J. G. and Jarvis, P. D., “Using the tangle: A consistent construction of phylogenetic distance matrices for quartets”, Math. Biosci. 204 (2006) 49–67; doi:10.1016/j.mbs.2006.05.008.CrossRefGoogle ScholarPubMed
Sumner, J. G. and Jarvis, P. D., “Markov invariants and the isotropy subgroup of a quartet tree”, J. Theor. Biol. 258 (2009) 302–310; doi:10.1016/j.jtbi.2009.01.021.Google Scholar
Sumner, J. G., Jarvis, P. D., Allman, E. S. and Rhodes, J. A., “Phylogenetic invariants from group characters alone”, in preparation, 2014.Google Scholar
Sumner, J., Fernández-Sánchez, J. and Jarvis, P., “Lie Markov models”, J. Theor. Biol. 298 (2012) 16–31; doi:10.1016/j.jtbi.2011.12.017.Google Scholar
Sumner, J., Fernández-Sánchez, J., Woodhams, M. and Jarvis, P., “Lie Markov models with purine/pyrimidine symmetry”, J. Math. Biol. (2014) 1–47; doi:10.1007/s00285-014-0773-z.Google Scholar
Verstraete, F., Dehaene, J., de Moor, B. and Verschelde, H., “Four qubits can be entangled in nine different ways”, Phys. Rev. A 65 (2002) 052112; doi:10.1103/PhysRevA.65.052112.Google Scholar
Vidal, G., “Entanglement monotones”, J. Modern Opt. 47 (2000) 355–376; doi:10.1080/095003400148268.Google Scholar
Weyl, H., The classical groups: their invariants and representations (Princeton University Press, Princeton, NJ, 1939).Google Scholar
Wybourne, B. G. et al. , SCHUR group theory software, an interactive program for calculating properties of Lie groups and symmetric functions, http://schur.sourceforge.net/.Google Scholar