Aspects of the Segre variety S1,1,1(2)

Shaw, Ron; Gordon, Neil; Havlicek, Hans

Abstract

We consider various aspects of the Segre variety S := S_{1,1,1}(2) in PG(7,2), whose stabilizer group G_S < GL(8, 2) has the structure N {\rtimes} Sym(3), where N := GL(2,2)\times GL(2,2)\times GL(2,2). In particular we prove that S determines a distinguished Z_3-subgroup Z < GL(8, 2) such that AZA^{-1} = Z, for all A in G_S, and in consequence S determines a G_S-invariant spread of 85 lines in PG(7,2). Furthermore we see that Segre varieties S_{1,1,1}(2) in PG(7,2) come along in triplets {S,S',S"} which share the same distinguished Z_3-subgroup Z < GL(8,2). We conclude by determining all fifteen G_S-invariant polynomial functions on PG(7,2) which have degree < 8, and their relation to the five G_S-orbits of points in PG(7,2).

Citation

Shaw, R., Gordon, N., & Havlicek, H. (2012). Aspects of the Segre variety S1,1,1(2). Designs, codes, and cryptography, 62(2), 225-239. https://doi.org/10.1007/s10623-011-9508-y

• Journal Article Type: Article
• Online Publication Date: Apr 23, 2011
• Publication Date: 2012-02
• Deposit Date: Nov 13, 2014
• Journal: Designs Codes And Cryptography
• Print ISSN: 0925-1022
• Electronic ISSN: 1573-7586
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 62
• Issue: 2
• Pages: 225-239
• DOI: https://doi.org/10.1007/s10623-011-9508-y
• Keywords: Applied Mathematics; Computer Science Applications
• Public URL: https://hull-repository.worktribe.com/output/463696