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| Main Authors: | , , |
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| Format: | Preprint |
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2011
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1102.1714 |
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| _version_ | 1866911775373393920 |
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| author | Shaw, Ronald Gordon, Neil Havlicek, Hans |
| author_facet | Shaw, Ronald Gordon, Neil Havlicek, Hans |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1102_1714 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | Aspects of the Segre variety S_{1,1,1}(2) Shaw, Ronald Gordon, Neil Havlicek, Hans Combinatorics 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). |
| title | Aspects of the Segre variety S_{1,1,1}(2) |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1102.1714 |