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  • P. D. JARVIS (a1) and J. G. SUMNER (a1)


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.

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[1]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) 10141045; doi:10.1137/120899066.
[2]Allman, E. S. and Rhodes, J. A., “Phylogenetic ideals and varieties for the general Markov model”, Adv. Appl. Math. 20 (2007) 127148; doi:10.1016/j.aam.2006.10.002.
[3]Barry, D. and Hartigan, J. A., “Asynchronous distance between homologous DNA sequences”, Biometrics 43 (1987) 261276; doi:10.2307/2531811.
[4]Buneman, P., “The recovery of trees from measures of dissimilarity”, in: Mathematics in the archaeological and historical sciences (Edinburgh University Press, Edinburgh, 1971) 387395.
[5]Cavender, J. A. and Felsenstein, J., “Invariants of phylogenies in a simple case with discrete states”, J. Classification 4 (1987) 5771; doi:10.1007/BF01890075.
[6]Coffman, V., Kundu, J. and Wootters, W. K., “Distributed entanglement”, Phys. Rev. A 61 (2000); doi:10.1103/PhysRevA.61.052306.
[7]Eltschka, C. and Siewert, J., “Quantifying entanglement resources”, J. Phys. A Math. Theor. 47 (2014); doi:10.1088/1751-8113/47/42/424005.
[8]Fauser, B. and Jarvis, P. D., “A Hopf laboratory for symmetric functions”, J. Phys. A: Math. Gen. 37 (2004) 16331663; doi:10.1088/0305-4470/37/5/012.
[9]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) 26112655; doi:10.1088/0305-4470/39/11/006.
[10]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).
[11]Grassl, M., Rötteler, M. and Beth, T., “Computing local invariants of quantum-bit systems”, Phys. Rev. A 58 (1998) 18331839; doi:10.1103/PhysRevA.58.1833.
[12]Hall, B. C., Quantum theory for mathematicians, Volume 267 of Graduate Texts in Mathematics (Springer, New York, 2013).
[13]Holland, B. R., Jarvis, P. D. and Sumner, J. G., “Low-parameter phylogenetic inference under the general Markov model”, Syst. Biol. 62 (2013) 7892; doi:10.1093/sysbio/sys072.
[14]Horodecki, R., Horodecki, P., Horodecki, M. and Horodecki, K., “Quantum entanglement”, Rev. Mod. Phys. 81 (2009) 865942; doi:10.1103/RevModPhys.81.865.
[15]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) 215302; doi:10.1088/1751-8113/47/21/215302.
[16]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.
[17]Jarvis, P. D. and Sumner, J. G., “Markov invariants for phylogenetic rate matrices derived from embedded submodels”, Trans. Comp. Biol. Bioinform. 9 (2012) 828836; doi:10.1109/TCBB.2012.24.
[18]Johnson, J. E., “Markov-type Lie groups in $\text{GL}(n,\mathbb{R})$”, J. Math. Phys. 26 (1985) 252257; doi:10.1109/TCBB.2012.24.
[19]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.
[20]Lake, J. A., “A rate-independent technique for analysis of nucleic acid sequences: evolutionary parsimony”, Mol. Biol. Evol. 4(2) (1987) 167191.
[21]Lake, J. A., “Reconstructing evolutionary trees from DNA and protein sequences: Paralinear distances”, Proc. Natl. Acad. Sci., USA 91 (1994) 14551459; doi:10.1073/pnas.91.4.1455.
[22]Littlewood, D. E., The theory of group characters (Clarendon Press, Oxford, 1940).
[23]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) 605612;
[24]Makhlin, Y., “Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations”, Quantum Inf. Process. 1 (2002) 243252; doi:10.1023/A:1022144002391.
[25]Molien, T., “Über die Invarianten der linearen Substitutionsgruppen”, Sitzungsber. König. Preuss. Akad. Wiss. (1897) 11521156;
[26]Mourad, B., “On a Lie-theoretic approach to generalised doubly stochastic matrices and applications”, Linear Multilinear Algebra 52 (2004) 99113; doi:10.1080/0308108031000140687.
[27]Semple, C. and Steel, M., Phylogenetics (Oxford University Press, Oxford, 2003).
[28]Sumner, J. G., “Entanglement, invariants, and phylogenetics”, Ph. D. Thesis, University of Tasmania, 2006.
[29]Sumner, J. G., Charleston, M. A., Jermiin, L. S. and Jarvis, P. D., “Markov invariants, plethysms, and phylogenetics”, J. Theor. Biol. 253 (2008) 601615; doi:10.1016/j.jtbi.2008.04.001.
[30]Sumner, J. G. and Jarvis, P. D., “Entanglement invariants and phylogenetic branching”, J. Math. Biol. 51 (2005) 1836 (erratum); 53 (2006) 490; doi:10.1007/s00285-004-0309-z.
[31]Sumner, J. G. and Jarvis, P. D., “Using the tangle: A consistent construction of phylogenetic distance matrices for quartets”, Math. Biosci. 204 (2006) 4967; doi:10.1016/j.mbs.2006.05.008.
[32]Sumner, J. G. and Jarvis, P. D., “Markov invariants and the isotropy subgroup of a quartet tree”, J. Theor. Biol. 258 (2009) 302310; doi:10.1016/j.jtbi.2009.01.021.
[33]Sumner, J. G., Jarvis, P. D., Allman, E. S. and Rhodes, J. A., “Phylogenetic invariants from group characters alone”, in preparation, 2014.
[34]Sumner, J., Fernández-Sánchez, J. and Jarvis, P., “Lie Markov models”, J. Theor. Biol. 298 (2012) 1631; doi:10.1016/j.jtbi.2011.12.017.
[35]Sumner, J., Fernández-Sánchez, J., Woodhams, M. and Jarvis, P., “Lie Markov models with purine/pyrimidine symmetry”, J. Math. Biol. (2014) 147; doi:10.1007/s00285-014-0773-z.
[36]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.
[37]Vidal, G., “Entanglement monotones”, J. Modern Opt. 47 (2000) 355376; doi:10.1080/095003400148268.
[38]Weyl, H., The classical groups: their invariants and representations (Princeton University Press, Princeton, NJ, 1939).
[39]Wybourne, B. G. et al. , SCHUR group theory software, an interactive program for calculating properties of Lie groups and symmetric functions,
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  • P. D. JARVIS (a1) and J. G. SUMNER (a1)


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