@article{delsarte_spherical_1977, title = {Spherical codes and designs}, volume = {6}, journal = {Geometriae Dedicata}, author = {Delsarte, P and Goethals, J M and Seidel, J J}, year = {1977}, keywords = {ALCO GAP, Alco paper}, pages = {363--388},
file = {Delsarte et al. - Spherical codes and designs.pdf:C\:\\Users\\benna\\Zotero\\storage\\X523HBK5\\Delsarte et al. - Spherical codes and designs.pdf:application/pdf},
}
@book{colbourn_triple_1999, title = {Triple systems}, publisher = {Oxford University Press}, author = {Colbourn, Charles J. and Rosa, Alexander}, year = {1999}, keywords = {ALCO GAP},
file = {Snapshot:C\:\\Users\\benna\\Zotero\\storage\\9X65KJ7D\\books.html:text/html},
}
@book{mccrimmon_taste_2004, address = {New York},
series = {Universitext}, title = {A {Taste} of {Jordan} {Algebras}},
isbn = {978-0-387-95447-9},
url = {https://www.springer.com/gp/book/9780387954479},
urldate = {2020-06-28}, publisher = {Springer-Verlag}, author = {McCrimmon, Kevin}, year = {2004},
doi = {10.1007/b97489}, keywords = {ALCO GAP},
file = {McCrimmon - 2004 - A Taste of Jordan Algebras.pdf:C\:\\Users\\benna\\Zotero\\storage\\FF3WELAZ\\McCrimmon - 2004 - A Taste of Jordan Algebras.pdf:application/pdf;Snapshot:C\:\\Users\\benna\\Zotero\\storage\\KBL5UUDJ\\9780387954479.html:text/html},
}
@book{springer_octonions_2000, address = {Berlin Heidelberg},
series = {Springer {Monographs} in {Mathematics}}, title = {Octonions, {Jordan} {Algebras} and {Exceptional} {Groups}},
isbn = {978-3-540-66337-9},
url = {https://www.springer.com/gp/book/9783540663379}, abstract = {The 1963 Göttingen notes of T. A. Springer are well-known in the field but have been unavailable for some time. This book is a translation of those notes, completely updated and revised. The part of the book dealing with the algebraic structures is on a fairly elementary level, presupposing basic results from algebra. In the group-theoretical part use is made of some results from the theory of linear algebraic groups. The book will be useful to mathematicians interested in octonion algebras and Albert algebras, or in exceptional groups. It is suitable for use in a graduate course in algebra.},
urldate = {2020-06-28}, publisher = {Springer-Verlag}, author = {Springer, Tonny A. and Veldkamp, Ferdinand D.}, year = {2000},
doi = {10.1007/978-3-662-12622-6}, keywords = {Alco paper, ALCO GAP},
file = {Snapshot:C\:\\Users\\benna\\Zotero\\storage\\5MSRYAXH\\9783540663379.html:text/html;Springer and Veldkamp - 2000 - Octonions, Jordan Algebras and Exceptional Groups.pdf:C\:\\Users\\benna\\Zotero\\storage\\NQP2MAYK\\Springer and Veldkamp - 2000 - Octonions, Jordan Algebras and Exceptional Groups.pdf:application/pdf},
}
@book{wilson_finite_2009, address = {London},
series = {Graduate {Texts} in {Mathematics}}, title = {The {Finite} {Simple} {Groups}},
isbn = {978-1-84800-987-5},
url = {https://www.springer.com/gp/book/9781848009875}, abstract = {The finite simple groups are the building blocks from which all the finite groups are made and as such they are objects of fundamental importance throughout mathematics. The classification of the finite simple groups was one of the great mathematical achievements of the twentieth century, yet these groups remain difficult to study which hinders applications of the classification. This textbook brings the finite simple groups to life by giving concrete constructions of most of them, sufficient to illuminate their structure and permit real calculations both in the groups themselves and in the underlying geometrical or algebraic structures. This is the first time that all the finite simple groups have been treated together in this way and the bookpoints out their connections, for example between exceptional behaviour of generic groups and the existence of sporadic groups, and discusses a number of new approaches to some of the groups. Many exercises of varying difficulty are provided. The Finite Simple Groups is aimed at advanced undergraduate and graduate students in algebra as well as professional mathematicians and scientists who use groups and want to apply the knowledge which the classification has given us. The main prerequisite is an undergraduate course in group theory up to the level of Sylow’s theorems.},
urldate = {2020-06-28}, publisher = {Springer-Verlag}, author = {Wilson, Robert A.}, year = {2009},
doi = {10.1007/978-1-84800-988-2}, keywords = {Alco paper, ALCO GAP},
file = {Snapshot:C\:\\Users\\benna\\Zotero\\storage\\FGGWCRJX\\9781848009875.html:text/html;Wilson - 2009 - The Finite Simple Groups.pdf:C\:\\Users\\benna\\Zotero\\storage\\P2AC3QMP\\Wilson - 2009 - The Finite Simple Groups.pdf:application/pdf},
}
@book{cameron_designs_1991, address = {Cambridge}, title = {Designs, graphs, codes and their links}, volume = {3}, publisher = {Cambridge University Press}, author = {Cameron, Peter Jephson and Van Lint, Jacobus Hendricus}, year = {1991}, keywords = {ALCO GAP},
file = {Snapshot:C\:\\Users\\benna\\Zotero\\storage\\Z9BRHXE7\\320010.html:text/html},
}
@book{faraut_analysis_1994, title = {Analysis on {Symmetric} {Cones}},
isbn = {978-0-19-853477-8}, publisher = {Clarendon Press}, author = {Faraut, Jacques and Koranyi, Adam}, year = {1994}, keywords = {History / Europe / General, Alco paper, ALCO GAP},
}
@article{hoggar_t-designs_1992, title = {t-{Designs} with general angle set}, volume = {13},
issn = {0195-6698},
url = {http://www.sciencedirect.com/science/article/pii/S0195669805800320},
doi = {10.1016/S0195-6698(05)80032-0}, number = {4},
urldate = {2020-08-07}, journal = {European Journal of Combinatorics}, author = {Hoggar, Stuart G.}, year = {1992}, keywords = {Alco paper, ALCO GAP}, pages = {257--271},
file = {ScienceDirect Full Text PDF:C\:\\Users\\benna\\Zotero\\storage\\KMA8WTIF\\Hoggar - 1992 - t-Designs with general angle set.pdf:application/pdf;ScienceDirect Snapshot:C\:\\Users\\benna\\Zotero\\storage\\BBNR39RS\\S0195669805800320.html:text/html},
}
@article{elkies_exceptional_1996, title = {The exceptional cone and the {Leech} lattice}, volume = {1996}, number = {14}, journal = {International Mathematics Research Notices}, author = {Elkies, Noam D. and Gross, Benedict H.}, year = {1996},
note = {Publisher: Citeseer}, keywords = {Must Review, ALCO GAP}, pages = {665--698},
file = {Full Text:C\:\\Users\\benna\\Zotero\\storage\\FWCSPUIT\\Elkies and Gross - 1996 - The exceptional cone and the Leech lattice.pdf:application/pdf},
}
@article{hoggar_t-designs_1982, title = {t-{Designs} in projective spaces}, volume = {3}, number = {3}, journal = {European Journal of Combinatorics}, author = {Hoggar, Stuart G.}, year = {1982}, keywords = {Projective t-designs, Alco paper, ALCO GAP}, pages = {233--254},
file = {Hoggar - 1982 - t-Designs in projective spaces.pdf:C\:\\Users\\benna\\Zotero\\storage\\V72F42UH\\Hoggar - 1982 - t-Designs in projective spaces.pdf:application/pdf},
}
@book{colbourn_handbook_2007, title = {Handbook of {Combinatorial} {Designs}},
isbn = {978-1-4200-1054-1}, publisher = {CRC Press}, author = {Colbourn, Charles J. and Dinitz, Jeffrey H.}, year = {2007}, keywords = {Computers / Operating Systems / General, Computers / Programming / Algorithms, Mathematics / Combinatorics, Mathematics / Discrete Mathematics, Technology \& Engineering / Automation, Alco paper, ALCO GAP},
file = {Colbourn and Dinitz - 2007 - Handbook of Combinatorial Designs.pdf:C\:\\Users\\benna\\Zotero\\storage\\6LU7KDKY\\Colbourn and Dinitz - 2007 - Handbook of Combinatorial Designs.pdf:application/pdf},
}
@article{baez_octonions_2002, title = {The octonions}, volume = {39}, number = {2}, journal = {Bulletin of the American Mathematical Society}, author = {Baez, John}, year = {2002}, keywords = {Alco paper, ALCO GAP}, pages = {145--205},
file = {Full Text:C\:\\Users\\benna\\Zotero\\storage\\BML67R26\\Baez - 2002 - The octonions.pdf:application/pdf;Snapshot:C\:\\Users\\benna\\Zotero\\storage\\7YJCDJIQ\\home.html:text/html},
}
@book{conway_quaternions_2003, title = {On {Quaternions} and {Octonions}: {Their} {Geometry}, {Arithmetic}, and {Symmetry}},
shorttitle = {On quaternions and octonions}, publisher = {CRC Press}, author = {Conway, John H. and Smith, Derek A.}, year = {2003}, keywords = {Alco paper, ALCO GAP},
file = {Full Text:C\:\\Users\\benna\\Zotero\\storage\\N9CUXPBL\\conway_smith.html:text/html;Snapshot:C\:\\Users\\benna\\Zotero\\storage\\69STE4ZS\\books.html:text/html},
}
@article{lyubich_tight_2009, title = {On tight projective designs}, volume = {51},
issn = {1573-7586},
url = {https://doi.org/10.1007/s10623-008-9240-4},
doi = {10.1007/s10623-008-9240-4}, number = {1},
urldate = {2020-06-28}, journal = {Designs, Codes and Cryptography}, author = {Lyubich, Yu. I.}, month = apr, year = {2009}, keywords = {Projective t-designs, Alco paper, ALCO GAP}, pages = {21--31},
file = {CP1icosahedron.g:C\:\\Users\\benna\\Zotero\\storage\\6MGDJBP4\\CP1icosahedron.g:text/plain;Springer Full Text PDF:C\:\\Users\\benna\\Zotero\\storage\\H6VT7DV3\\Lyubich - 2009 - On tight projective designs.pdf:application/pdf},
}
@article{wilson_octonions_2009, title = {Octonions and the {Leech} lattice}, volume = {322},
issn = {00218693},
url = {https://linkinghub.elsevier.com/retrieve/pii/S0021869309001458},
doi = {10.1016/j.jalgebra.2009.03.021}, number = {6},
urldate = {2020-06-28}, journal = {Journal of Algebra}, author = {Wilson, Robert A.}, month = sep, year = {2009}, keywords = {Alco paper, ALCO GAP}, pages = {2186--2190},
file = {leech_lattice_coords.g:C\:\\Users\\benna\\Zotero\\storage\\U2J7X2DB\\leech_lattice_coords.g:text/plain;Wilson - 2009 - Octonions and the Leech lattice.pdf:C\:\\Users\\benna\\Zotero\\storage\\9USH5Z2L\\Wilson - 2009 - Octonions and the Leech lattice.pdf:application/pdf;wilsonsoctonionicleechlattice.g:C\:\\Users\\benna\\Zotero\\storage\\6NSN9ZUS\\wilsonsoctonionicleechlattice.g:text/plain},
}
@article{wilson_conways_2011, title = {Conway's group and octonions}, volume = {14},
issn = {1433-5883, 1435-4446},
url = {https://www.degruyter.com/view/j/jgth.2011.14.issue-1/jgt.2010.038/jgt.2010.038.xml},
doi = {10.1515/jgt.2010.038}, number = {1},
urldate = {2020-06-28}, journal = {Journal of Group Theory}, author = {Wilson, Robert A.}, month = jan, year = {2011}, keywords = {Alco paper, ALCO GAP}, pages = {1--8},
file = {Co1and819.g:C\:\\Users\\benna\\Zotero\\storage\\PVPR3T6U\\Co1and819.g:text/plain;ConwayGroupInF4.g:C\:\\Users\\benna\\Zotero\\storage\\83PJE6JF\\ConwayGroupInF4.g:text/plain;Wilson - 2011 - Conway's group and octonions.pdf:C\:\\Users\\benna\\Zotero\\storage\\Y5QQUJV4\\Wilson - 2011 - Conway's group and octonions.pdf:application/pdf;WilsonsConwayGroup.g:C\:\\Users\\benna\\Zotero\\storage\\CBAXM9DN\\WilsonsConwayGroup.g:text/plain},
}
@book{conway_sphere_2013, title = {Sphere packings, lattices and groups}, volume = {290}, publisher = {Springer Science \& Business Media}, author = {Conway, John Horton and Sloane, Neil James Alexander}, year = {2013}, keywords = {Alco paper, ALCO GAP},
file = {Conway and Sloane - 2013 - Sphere packings, lattices and groups.pdf:C\:\\Users\\benna\\Zotero\\storage\\76N4VYYW\\Conway and Sloane - 2013 - Sphere packings, lattices and groups.pdf:application/pdf;Snapshot:C\:\\Users\\benna\\Zotero\\storage\\AQF5FKIS\\books.html:text/html},
}
@book{abramowitz_handbook_1972, address = {New York}, title = {Handbook of {Mathematical} {Functions}}, publisher = {Dover}, editor = {Abramowitz, Milton and Stegun, Irene A.}, year = {1972}, keywords = {Alco paper, ALCO GAP},
}
@book{faraut_analysis_2000, address = {Boston},
series = {Progress in mathematics}, title = {Analysis and {Geometry} on {Complex} {Homogeneous} {Domains}},
isbn = {978-1-4612-1366-6}, number = {185}, publisher = {Birkhäuser}, author = {Faraut, Jacques and Kaneyuki, Soji and Koranyi, Adam and Lu, Qi-keng and Roos, Guy}, year = {2000}, keywords = {Mathematics / Algebra / General, Mathematics / Complex Analysis, Mathematics / Geometry / Differential, Mathematics / Group Theory, Mathematics / Mathematical Analysis, ALCO GAP},
}
@article{elkies_cubic_2001, title = {Cubic {Rings} and the {Exceptional} {Jordan} {Algebra}}, volume = {109}, number = {2}, journal = {Duke Mathematical Journal}, author = {Elkies, Noam D and Gross, Benedict H}, year = {2001}, keywords = {ALCO GAP}, pages = {383--410},
file = {Elkies and Gross - CUBIC RINGS AND THE EXCEPTIONAL JORDAN ALGEBRA.pdf:C\:\\Users\\benna\\Zotero\\storage\\R5UWZB5X\\Elkies and Gross - CUBIC RINGS AND THE EXCEPTIONAL JORDAN ALGEBRA.pdf:application/pdf},
}
@article{lepowsky_e8-approach_1982, title = {An {E8}-approach to the {Leech} lattice and the {Conway} group}, volume = {77},
issn = {0021-8693},
url = {https://www.sciencedirect.com/science/article/pii/002186938290268X},
doi = {10.1016/0021-8693(82)90268-X}, number = {2},
urldate = {2022-04-02}, journal = {Journal of Algebra}, author = {Lepowsky, James and Meurman, Arne}, month = aug, year = {1982}, keywords = {ALCO GAP}, pages = {484--504},
file = {ScienceDirect Full Text PDF:C\:\\Users\\benna\\Zotero\\storage\\JFWBYN3S\\Lepowsky and Meurman - 1982 - An E8-approach to the Leech lattice and the Conway.pdf:application/pdf;ScienceDirect Snapshot:C\:\\Users\\benna\\Zotero\\storage\\Q2D42SBS\\002186938290268X.html:text/html},
}
@article{nasmith_octonions_2022, title = {Octonions and the two strictly projective tight 5-designs}, volume = {5},
issn = {2589-5486},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.215/},
doi = {10.5802/alco.215}, number = {3},
urldate = {2022-07-12}, journal = {Algebraic Combinatorics}, author = {Nasmith, Benjamin}, year = {2022}, keywords = {ALCO GAP}, pages = {401--411},
file = {Full Text PDF:C\:\\Users\\benna\\Zotero\\storage\\2MWVRQNQ\\Nasmith - 2022 - Octonions and the two strictly projective tight 5-.pdf:application/pdf;Snapshot:C\:\\Users\\benna\\Zotero\\storage\\WY7RNGWV\\alco.215.html:text/html},
}
@phdthesis{nasmith_tight_2023, address = {Kingston ON},
type = {{PhD} {Thesis}}, title = {Tight {Projective} 5-{Designs} and {Exceptional} {Structures}},
url = {https://espace.rmc.ca/jspui/handle/11264/1423}, abstract = {A t-design on a sphere or projective space is a finite subset such that the integral of any degree t polynomial over the sphere or projective space is equal to the average value of that polynomial evaluated at the points of the t-design. Tight t-designs are optimal in that they use the minimum possible number of points to achieve a particular value of t. Although t-designs are abundant, tight t-designs are rare structures in combinatorics that continue to resist a full classification. However, there are precisely four tight projective 5-designs: the vertices of a regular hexagon, the vertices of a regular icosahedron, the lines spanning the short vectors of the Leech lattice, and a set of points in the octonion projective plane forming a generalized hexagon finite geometry. This thesis explores the four tight projective 5-designs and their
connections to various exceptional structures. The regular hexagon provides a starting point from which to recover Lie and Jordan theory. We explore an exceptional sequence of Lie algebras that terminates in the Lie algebra of the standard model of particle physics and provides a three generation representation of standard model fermions. The regular icosahedron is unique among tight t-designs, apart from most polygons on the unit circle, in that it has an irrational angle set. We provide proof of this fact by using Jordan algebras to generalize and correct certain previous attempts to prove that tight t-designs have rational angles. Finally, we explore the relationship between the two remaining tight 5-designs by examining octonion constructions of the Leech lattice and their octonion reflection symmetries. We introduce a common construction technique that yields the two remaining tight 5-designs and explore the role that octonion integers and exceptional Jordan algebra integers can play in this common construction.}, language = {en},
urldate = {2023-08-07}, school = {Royal Military College of Canada}, author = {Nasmith, Benjamin}, year = {2023},
note = {Accepted: 2023-07-27T12:24:01Z}, keywords = {ALCO GAP},
file = {Nasmith - 2023 - Tight Projective 5-Designs and Exceptional Structu.pdf:C\:\\Users\\benna\\Zotero\\storage\\TDDLVUE4\\Nasmith - 2023 - Tight Projective 5-Designs and Exceptional Structu.pdf:application/pdf},
}
@book{dummit_abstract_2004, address = {Hoboken, NJ},
edition = {3rd ed}, title = {Abstract algebra},
isbn = {978-0-471-43334-7}, publisher = {Wiley}, author = {Dummit, David Steven and Foote, Richard M.}, year = {2004}, keywords = {ALCO GAP, Algebra, Abstract},
}
@book{bannai_algebraic_2021, address = {Berlin/Boston},
series = {De {Gruyter}, {Series} in {Discrete} {Mathematics} and {Applications}}, title = {Algebraic {Combinatorics}},
isbn = {978-3-11-062763-3}, number = {5}, publisher = {Walter de Gruyter GmbH}, author = {Bannai, Eiichi and Bannai, Etsuko and Ito, Tatsuro and Tanaka, Rie}, year = {2021},
note = {OCLC: on1243061900}, keywords = {ALCO GAP, Analyse combinatoire, Combinatorial analysis},
}
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