{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T03:17:39Z","timestamp":1765423059956,"version":"build-2065373602"},"reference-count":59,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,1,20]],"date-time":"2019-01-20T00:00:00Z","timestamp":1547942400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["PHY-1315155."],"award-info":[{"award-number":["PHY-1315155."]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Recently, all 1,358,954,496 values of the gadget between the 36,864 adinkras with four colors, four bosons, and four fermions have been computed. In this paper, we further analyze these results in terms of     B  C 3     , the signed permutation group of three elements, and     B  C 4     , the signed permutation group of four elements. It is shown how all 36,864 adinkras can be generated via     B  C 4      boson \u00d7     B  C 3      color transformations of two quaternion adinkras that satisfy the quaternion algebra. An adinkra inner product has been used for some time, known as the gadget, which is used to distinguish adinkras. We show how 96 equivalence classes of adinkras that are based on the gadget emerge in terms of     B  C 3      and     B  C 4     . We also comment on the importance of the gadget as it relates to separating out dynamics in terms of K\u00e4hler-like potentials. Thus, on the basis of the complete analysis of the supersymmetrical representations achieved in the preparatory first four sections, the final comprehensive achievement of this work is the construction of the universal     B  C 4      non-linear    \u03c3   -model.<\/jats:p>","DOI":"10.3390\/sym11010120","type":"journal-article","created":{"date-parts":[[2019,1,22]],"date-time":"2019-01-22T03:08:22Z","timestamp":1548126502000},"page":"120","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Generating All 36,864 Four-Color Adinkras via Signed Permutations and Organizing into \u2113- and \r\n          \r\n            \r\n\t\t\t\r\n              \r\n                \u2113\r\n                \u02dc\r\n              \r\n\t\t\t  \r\n            \r\n          \r\n        -Equivalence Classes"],"prefix":"10.3390","volume":"11","author":[{"suffix":"Jr.","given":"S. James","family":"Gates","sequence":"first","affiliation":[{"name":"Department of Physics, Brown University, Box 1843, 182 Hope Street, Providence, RI 02912, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kevin","family":"Iga","sequence":"additional","affiliation":[{"name":"Natural Science Division, Pepperdine University, 24255 Pacific Coast Hwy., Malibu, CA 90263, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lucas","family":"Kang","sequence":"additional","affiliation":[{"name":"Department of Physics, Brown University, Box 1843, 182 Hope Street, Providence, RI 02912, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vadim","family":"Korotkikh","sequence":"additional","affiliation":[{"name":"Center for String and Particle Theory-Department of Physics, University of Maryland, 4150 Campus Dr., College Park, MD 20472, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1491-1484","authenticated-orcid":false,"given":"Kory","family":"Stiffler","sequence":"additional","affiliation":[{"name":"Department of Physics, Brown University, Box 1843, 182 Hope Street, Providence, RI 02912, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3980613","DOI":"10.1155\/2016\/3980613","article-title":"Structural Theory and Classification of 2D Adinkras","volume":"2016","author":"Iga","year":"2016","journal-title":"Adv. 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