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OPTIMAL LINE PACKINGS FROM FINITE GROUP ACTIONS

Part of: Circuits, networks Representation theory of groups Discrete geometry Permutation groups Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  20 January 2020

JOSEPH W. IVERSON ,
JOHN JASPER  and
DUSTIN G. MIXON
JOSEPH W. IVERSON
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA Department of Mathematics, Iowa State University, Ames, IA 50011, USA; jwi@iastate.edu
JOHN JASPER
Affiliation:
Department of Mathematics and Statistics, South Dakota State University, Brookings,SD 57007, USA; john.jasper@sdstate.edu
DUSTIN G. MIXON
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA; mixon.23@osu.edu

Abstract

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We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.

MSC classification

Primary: 20B99: None of the above, but in this section 42C15: General harmonic expansions, frames 52C99: None of the above, but in this section
Secondary: 20C15: Ordinary representations and characters 94C30: Applications of design theory
Type
Discrete Mathematics
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e6
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

References

Appleby, D. M., ‘Symmetric informationally complete-positive operator valued measures and the extended Clifford group’, J. Math. Phys. 46(5)052107, 29 (2005).CrossRefGoogle Scholar
Appleby, M., Bengtsson, I., Dumitru, I. and Flammia, S., ‘Dimension towers of SICs. I. Aligned SICs and embedded tight frames’, J. Math. Phys. 58(11)112201, 19 (2017).CrossRefGoogle Scholar
Appleby, M., Chien, T.-Y., Flammia, S. and Waldron, S., ‘Constructing exact symmetric informationally complete measurements from numerical solutions’, J. Phys. A 51(16)165302, 40 (2018).CrossRefGoogle Scholar
Bamberg, J., Betten, A., Cara, P., De Beule, J., Lavrauw, M. and Neunhöffer, M., FinInG–Finite Incidence Geometry, (Version 1.3.3), 2016.Google Scholar
Bannai, E. and Ito, T., Algebraic Combinatorics I: Association Schemes, (Benjamin/Cummings, Menlo Park, CA, 1984).Google Scholar
Benson, C., Jenkins, J. and Ratcliff, G., ‘On Gelfand pairs associated with solvable Lie groups’, Trans. Am. Math. Soc. 321(1) (1990), 85–116.Google Scholar
Benson, C. and Ratcliff, G., ‘Gelfand pairs associated with finite Heisenberg groups’, inRepresentations, Wavelets, and Frames, Appl. Numer. Harmon. Anal. (Birkhäuser, Boston, MA, 2008), 13–31.CrossRefGoogle Scholar
Benson, C. and Ratcliff, G., ‘Spherical functions for the action of a finite unitary group on a finite Heisenberg group’, inNew developments in Lie Theory and Geometry, Contemp. Math., 491 (American Mathematical Society, Providence, RI, 2009), 151–166.CrossRefGoogle Scholar
Bodmann, B. G. and Haas, J., ‘Achieving the orthoplex bound and constructing weighted complex projective 2-designs with singer sets’, Linear Algebra Appl. 511 (2016), 54–71.CrossRefGoogle Scholar
Casazza, P. G. and Haas, J. I., ‘On the rigidity of geometric and spectral properties of Grassmannian frames’. Preprint, 2016, arXiv:1605.02012.Google Scholar
Casazza, P. G., Kutyniok, G. and Philipp, F., ‘Introduction to finite frame theory’, inFinite Frames, Appl. Numer. Harmon. Anal. (Birkhäuser/Springer, New York, 2013), 1–53.CrossRefGoogle Scholar
Ceccherini-Silberstein, T., Scarabotti, F. and Tolli, F., Harmonic Analysis on Finite Groups, Cambridge Studies in Advanced Mathematics, 108 (Cambridge University Press, Cambridge, 2008).CrossRefGoogle Scholar
Christensen, O., An Introduction to Frames and Riesz Bases, (Springer, Boston, 2003).CrossRefGoogle Scholar
Conway, J. H., Hardin, R. H. and Sloane, N. J. A., ‘Packing lines, planes, etc.: packings in Grassmannian spaces’, Experiment. Math. 5(2) (1996), 139–159.CrossRefGoogle Scholar
Ding, C. and Feng, T., ‘A generic construction of complex codebooks meeting the Welch bound’, IEEE Trans. Inf. Theory 53(11) (2007), 4245–4250.CrossRefGoogle Scholar
Dutta, K. and Prasad, A., ‘Degenerations and orbits in finite abelian groups’, J. Combin. Theory Ser. A 118(6) (2011), 1685–1694.CrossRefGoogle Scholar
Dutta, K. and Prasad, A., ‘Combinatorics of finite abelian groups and Weil representations’, Pacific J. Math. 275(2) (2015), 295–324.CrossRefGoogle Scholar
Fickus, M., Jasper, J. and Mixon, D. G., ‘Packings in real projective spaces’, SIAM J. Appl. Algebra Geom. 2(3) (2018), 377–409.CrossRefGoogle Scholar
Fickus, M., Jasper, J., Mixon, D. G. and Peterson, J., ‘Tremain equiangular tight frames’, J. Combin. Theory Ser. A 153 (2018), 54–66.CrossRefGoogle Scholar
Fickus, M., Jasper, J., Mixon, D. G. and Peterson, J. D., ‘Hadamard equiangular tight frames’, Appl. Comput. Harmon. Anal. https://doi.org/10.1016/j.acha.2019.08.003, in press.Google Scholar
Fickus, M., Jasper, J., Mixon, D. G., Peterson, J. D. and Watson, C. E., ‘Polyphase equiangular tight frames and abelian generalized quadrangles’, Appl. Comput. Harmon. Anal. 47(3) (2019), 628–661.CrossRefGoogle Scholar
Fickus, M. and Mixon, D. G., ‘Tables of the existence of equiangular tight frames’, Preprint, 2015, arXiv:1504.00253.CrossRefGoogle Scholar
Fickus, M., Mixon, D. G. and Jasper, J., ‘Equiangular tight frames from hyperovals’, IEEE Trans. Inform. Theory 62(9) (2016), 5225–5236.CrossRefGoogle Scholar
Fickus, M., Mixon, D. G. and Tremain, J. C., ‘Steiner equiangular tight frames’, Linear Algebra Appl. 436(5) (2012), 1014–1027.CrossRefGoogle Scholar
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.6, (2016).Google Scholar
Godsil, C., Association schemes. http://www.math.uwaterloo.ca/cgodsil/pdfs/assoc2.pdf, 06-03-2010.Google Scholar
Helgason, S., Groups and Geometric Analysis, Mathematical Surveys and Monographs, 83 (American Mathematical Society, Providence, RI, 2000).CrossRefGoogle Scholar
Herman, A., ‘Algebraic aspects of association schemes and scheme rings’, http://uregina.ca/ hermana/ASSR-Lecture9.pdf, 08-23- 2011.Google Scholar
Hoggar, S. G., ‘Bounds for quaternionic line systems and reflection groups’, Math. Scand. 43(2) (1978), 241–249 (1979).CrossRefGoogle Scholar
Hoggar, S. G., ‘Two quaternionic 4-polytopes’, inThe Geometric Vein (Springer, New York–Berlin, 1981), 219–230.CrossRefGoogle Scholar
Iverson, J. W., ‘Frames generated by compact group actions’, Trans. Amer. Math. Soc. 370(1) (2018), 509–551.CrossRefGoogle Scholar
Iverson, J. W., Jasper, J. and Mixon, D. G., ‘Optimal line packings from nonabelian groups’, Discrete Comput. Geom. https://doi.org/10.1007/s00454-019-00084-z, in press.Google Scholar
Iverson, J. W., Jasper, J. and Mixon, D. G., Action packings. https://github.com/jwiverson/action-packings, 2017.Google Scholar
Levenshtein, V., ‘Bounds on the maximal cardinality of a code with bounded modulus of the inner product’, Soviet Math. Dokl 25 (1982), 526–531.Google Scholar
Newton, I., The Correspondence of Isaac Newton: Volume 3, 1688–1694 (ed. Turnball, H. W.) (Cambridge University Press, New York, 1966).Google Scholar
Sloane, N. J., Packings in Grassmannian spaces, http://neilsloane.com/grass/.Google Scholar
Strohmer, T. and Heath, R. W. Jr, ‘Grassmannian frames with applications to coding and communication’, Appl. Comput. Harmon. Anal. 14(3) (2003), 257–275.CrossRefGoogle Scholar
Tammes, P. M. L., ‘On the origin of number and arrangement of the places of exit on the surface of pollen grains’, Recueil de Trav. Botaniques Néerlandais 27 (1930), 1–84.Google Scholar
Vale, R. and Waldron, S., ‘Tight frames and their symmetries’, Constr. Approx. 21(1) (2005), 83–112.Google Scholar
Vale, R. and Waldron, S., ‘Tight frames generated by finite nonabelian groups’, Numer. Algorithms 48(1-3) (2008), 11–27.CrossRefGoogle Scholar
Vale, R. and Waldron, S., ‘The symmetry group of a finite frame’, Linear Algebra Appl. 433(1) (2010), 248–262.CrossRefGoogle Scholar
Weil, A., ‘Sur certains groupes d’opérateurs unitaires’, Acta Math. 111 (1964), 143–211.CrossRefGoogle Scholar
Welch, L., ‘Lower bounds on the maximum cross correlation of signals’, IEEE Trans. Inform. Theory 20(3) (1974), 397–399.CrossRefGoogle Scholar
Xia, P., Zhou, S. and Giannakis, G. B., ‘Achieving the Welch bound with difference sets’, IEEE Trans. Inf. Theory 51(5) (2005), 1900–1907.CrossRefGoogle Scholar
Zauner, G., ‘Quantendesigns: Grundzüge einer nichtkommutativen Designtheorie’, PhD Thesis, University of Vienna, Vienna, Austria, 1999.Google Scholar
Zhu, H., ‘Quantum State Estimation and Symmetric Informationally Complete POMs’, PhD Thesis, National University of Singapore, 2012.Google Scholar
Zieschang, P.-H., Theory of Association Schemes, Springer Monographs in Mathematics (Springer, Berlin, 2005).Google Scholar