The Quintic Grassmannian $$G_{1,4,2}$$ in PG(9, 2)
Abstract The 155 points of the Grassmannian $$G_{1,4,2}$$ of lines of PG (4, 2) = $$\mathbb{P}V\left( {5,2} \right)$$ are those points $$x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$$ which satisfy a certain quintic equation Q(x) = 0....
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Shaw, R. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2004 |
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| Anmerkung: |
© Kluwer Academic Publishers 2004 |
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| Übergeordnetes Werk: |
Enthalten in: Designs, codes and cryptography - Kluwer Academic Publishers, 1991, 32(2004), 1-3 vom: Mai, Seite 381-396 |
|---|---|
| Übergeordnetes Werk: |
volume:32 ; year:2004 ; number:1-3 ; month:05 ; pages:381-396 |
| Links: |
|---|
| DOI / URN: |
10.1023/B:DESI.0000029236.10701.61 |
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| Katalog-ID: |
OLC2027284486 |
|---|
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| 520 | |a Abstract The 155 points of the Grassmannian $$G_{1,4,2}$$ of lines of PG (4, 2) = $$\mathbb{P}V\left( {5,2} \right)$$ are those points $$x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X$$ \subset $$ PG (9, 2) will be termed odd or even according as X intersects $$G_{1,4,2}$$ in an odd or even number of points. Let $$Q^\ddag \left( {x_1 ,...,x_5 } \right)$$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X$$ \subset $$ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$. Because $$Q^\ddag$$ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X$$ \subseteq$$X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. An example of an even 4-flat X such that $$\left( {X^\# } \right)^\$$# = X is provided by any 4-flat X which is external to $$G_{1,4,2}$$. However, it appears that the two possibilities just illustrated, namely X# = X for an odd 4-flat and $$\left( {X^\# } \right)^\$$# = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X# = PG (9, 2) and of even 4-flats for which $${X^{\# \# \# } }$$ = X. | ||
| 700 | 1 | |a Gordon, N. A. |4 aut | |
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10.1023/B:DESI.0000029236.10701.61 doi (DE-627)OLC2027284486 (DE-He213)B:DESI.0000029236.10701.61-p DE-627 ger DE-627 rakwb eng 004 VZ 17,1 ssgn Shaw, R. verfasserin aut The Quintic Grassmannian $$G_{1,4,2}$$ in PG(9, 2) 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Kluwer Academic Publishers 2004 Abstract The 155 points of the Grassmannian $$G_{1,4,2}$$ of lines of PG (4, 2) = $$\mathbb{P}V\left( {5,2} \right)$$ are those points $$x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X$$ \subset $$ PG (9, 2) will be termed odd or even according as X intersects $$G_{1,4,2}$$ in an odd or even number of points. Let $$Q^\ddag \left( {x_1 ,...,x_5 } \right)$$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X$$ \subset $$ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$. Because $$Q^\ddag$$ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X$$ \subseteq$$X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. An example of an even 4-flat X such that $$\left( {X^\# } \right)^\$$# = X is provided by any 4-flat X which is external to $$G_{1,4,2}$$. However, it appears that the two possibilities just illustrated, namely X# = X for an odd 4-flat and $$\left( {X^\# } \right)^\$$# = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X# = PG (9, 2) and of even 4-flats for which $${X^{\# \# \# } }$$ = X. Gordon, N. A. aut Enthalten in Designs, codes and cryptography Kluwer Academic Publishers, 1991 32(2004), 1-3 vom: Mai, Seite 381-396 (DE-627)130994197 (DE-600)1082042-5 (DE-576)029154375 0925-1022 nnns volume:32 year:2004 number:1-3 month:05 pages:381-396 https://doi.org/10.1023/B:DESI.0000029236.10701.61 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_40 GBV_ILN_62 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2244 GBV_ILN_4126 AR 32 2004 1-3 05 381-396 |
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10.1023/B:DESI.0000029236.10701.61 doi (DE-627)OLC2027284486 (DE-He213)B:DESI.0000029236.10701.61-p DE-627 ger DE-627 rakwb eng 004 VZ 17,1 ssgn Shaw, R. verfasserin aut The Quintic Grassmannian $$G_{1,4,2}$$ in PG(9, 2) 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Kluwer Academic Publishers 2004 Abstract The 155 points of the Grassmannian $$G_{1,4,2}$$ of lines of PG (4, 2) = $$\mathbb{P}V\left( {5,2} \right)$$ are those points $$x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X$$ \subset $$ PG (9, 2) will be termed odd or even according as X intersects $$G_{1,4,2}$$ in an odd or even number of points. Let $$Q^\ddag \left( {x_1 ,...,x_5 } \right)$$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X$$ \subset $$ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$. Because $$Q^\ddag$$ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X$$ \subseteq$$X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. An example of an even 4-flat X such that $$\left( {X^\# } \right)^\$$# = X is provided by any 4-flat X which is external to $$G_{1,4,2}$$. However, it appears that the two possibilities just illustrated, namely X# = X for an odd 4-flat and $$\left( {X^\# } \right)^\$$# = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X# = PG (9, 2) and of even 4-flats for which $${X^{\# \# \# } }$$ = X. Gordon, N. A. aut Enthalten in Designs, codes and cryptography Kluwer Academic Publishers, 1991 32(2004), 1-3 vom: Mai, Seite 381-396 (DE-627)130994197 (DE-600)1082042-5 (DE-576)029154375 0925-1022 nnns volume:32 year:2004 number:1-3 month:05 pages:381-396 https://doi.org/10.1023/B:DESI.0000029236.10701.61 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_40 GBV_ILN_62 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2244 GBV_ILN_4126 AR 32 2004 1-3 05 381-396 |
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10.1023/B:DESI.0000029236.10701.61 doi (DE-627)OLC2027284486 (DE-He213)B:DESI.0000029236.10701.61-p DE-627 ger DE-627 rakwb eng 004 VZ 17,1 ssgn Shaw, R. verfasserin aut The Quintic Grassmannian $$G_{1,4,2}$$ in PG(9, 2) 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Kluwer Academic Publishers 2004 Abstract The 155 points of the Grassmannian $$G_{1,4,2}$$ of lines of PG (4, 2) = $$\mathbb{P}V\left( {5,2} \right)$$ are those points $$x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X$$ \subset $$ PG (9, 2) will be termed odd or even according as X intersects $$G_{1,4,2}$$ in an odd or even number of points. Let $$Q^\ddag \left( {x_1 ,...,x_5 } \right)$$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X$$ \subset $$ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$. Because $$Q^\ddag$$ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X$$ \subseteq$$X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. An example of an even 4-flat X such that $$\left( {X^\# } \right)^\$$# = X is provided by any 4-flat X which is external to $$G_{1,4,2}$$. However, it appears that the two possibilities just illustrated, namely X# = X for an odd 4-flat and $$\left( {X^\# } \right)^\$$# = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X# = PG (9, 2) and of even 4-flats for which $${X^{\# \# \# } }$$ = X. Gordon, N. A. aut Enthalten in Designs, codes and cryptography Kluwer Academic Publishers, 1991 32(2004), 1-3 vom: Mai, Seite 381-396 (DE-627)130994197 (DE-600)1082042-5 (DE-576)029154375 0925-1022 nnns volume:32 year:2004 number:1-3 month:05 pages:381-396 https://doi.org/10.1023/B:DESI.0000029236.10701.61 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_40 GBV_ILN_62 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2244 GBV_ILN_4126 AR 32 2004 1-3 05 381-396 |
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10.1023/B:DESI.0000029236.10701.61 doi (DE-627)OLC2027284486 (DE-He213)B:DESI.0000029236.10701.61-p DE-627 ger DE-627 rakwb eng 004 VZ 17,1 ssgn Shaw, R. verfasserin aut The Quintic Grassmannian $$G_{1,4,2}$$ in PG(9, 2) 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Kluwer Academic Publishers 2004 Abstract The 155 points of the Grassmannian $$G_{1,4,2}$$ of lines of PG (4, 2) = $$\mathbb{P}V\left( {5,2} \right)$$ are those points $$x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X$$ \subset $$ PG (9, 2) will be termed odd or even according as X intersects $$G_{1,4,2}$$ in an odd or even number of points. Let $$Q^\ddag \left( {x_1 ,...,x_5 } \right)$$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X$$ \subset $$ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$. Because $$Q^\ddag$$ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X$$ \subseteq$$X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. An example of an even 4-flat X such that $$\left( {X^\# } \right)^\$$# = X is provided by any 4-flat X which is external to $$G_{1,4,2}$$. However, it appears that the two possibilities just illustrated, namely X# = X for an odd 4-flat and $$\left( {X^\# } \right)^\$$# = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X# = PG (9, 2) and of even 4-flats for which $${X^{\# \# \# } }$$ = X. Gordon, N. A. aut Enthalten in Designs, codes and cryptography Kluwer Academic Publishers, 1991 32(2004), 1-3 vom: Mai, Seite 381-396 (DE-627)130994197 (DE-600)1082042-5 (DE-576)029154375 0925-1022 nnns volume:32 year:2004 number:1-3 month:05 pages:381-396 https://doi.org/10.1023/B:DESI.0000029236.10701.61 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_40 GBV_ILN_62 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2244 GBV_ILN_4126 AR 32 2004 1-3 05 381-396 |
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10.1023/B:DESI.0000029236.10701.61 doi (DE-627)OLC2027284486 (DE-He213)B:DESI.0000029236.10701.61-p DE-627 ger DE-627 rakwb eng 004 VZ 17,1 ssgn Shaw, R. verfasserin aut The Quintic Grassmannian $$G_{1,4,2}$$ in PG(9, 2) 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Kluwer Academic Publishers 2004 Abstract The 155 points of the Grassmannian $$G_{1,4,2}$$ of lines of PG (4, 2) = $$\mathbb{P}V\left( {5,2} \right)$$ are those points $$x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X$$ \subset $$ PG (9, 2) will be termed odd or even according as X intersects $$G_{1,4,2}$$ in an odd or even number of points. Let $$Q^\ddag \left( {x_1 ,...,x_5 } \right)$$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X$$ \subset $$ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$. Because $$Q^\ddag$$ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X$$ \subseteq$$X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. An example of an even 4-flat X such that $$\left( {X^\# } \right)^\$$# = X is provided by any 4-flat X which is external to $$G_{1,4,2}$$. However, it appears that the two possibilities just illustrated, namely X# = X for an odd 4-flat and $$\left( {X^\# } \right)^\$$# = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X# = PG (9, 2) and of even 4-flats for which $${X^{\# \# \# } }$$ = X. Gordon, N. A. aut Enthalten in Designs, codes and cryptography Kluwer Academic Publishers, 1991 32(2004), 1-3 vom: Mai, Seite 381-396 (DE-627)130994197 (DE-600)1082042-5 (DE-576)029154375 0925-1022 nnns volume:32 year:2004 number:1-3 month:05 pages:381-396 https://doi.org/10.1023/B:DESI.0000029236.10701.61 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_40 GBV_ILN_62 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2244 GBV_ILN_4126 AR 32 2004 1-3 05 381-396 |
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(The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X$$ \subset $$ PG (9, 2) will be termed odd or even according as X intersects $$G_{1,4,2}$$ in an odd or even number of points. Let $$Q^\ddag \left( {x_1 ,...,x_5 } \right)$$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X$$ \subset $$ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$. Because $$Q^\ddag$$ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X$$ \subseteq$$X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. An example of an even 4-flat X such that $$\left( {X^\# } \right)^\$$# = X is provided by any 4-flat X which is external to $$G_{1,4,2}$$. However, it appears that the two possibilities just illustrated, namely X# = X for an odd 4-flat and $$\left( {X^\# } \right)^\$$# = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X# = PG (9, 2) and of even 4-flats for which $${X^{\# \# \# } }$$ = X.</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Gordon, N. 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the quintic grassmannian $$g_{1,4,2}$$ in pg(9, 2) |
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The Quintic Grassmannian $$G_{1,4,2}$$ in PG(9, 2) |
| abstract |
Abstract The 155 points of the Grassmannian $$G_{1,4,2}$$ of lines of PG (4, 2) = $$\mathbb{P}V\left( {5,2} \right)$$ are those points $$x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X$$ \subset $$ PG (9, 2) will be termed odd or even according as X intersects $$G_{1,4,2}$$ in an odd or even number of points. Let $$Q^\ddag \left( {x_1 ,...,x_5 } \right)$$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X$$ \subset $$ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$. Because $$Q^\ddag$$ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X$$ \subseteq$$X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. An example of an even 4-flat X such that $$\left( {X^\# } \right)^\$$# = X is provided by any 4-flat X which is external to $$G_{1,4,2}$$. However, it appears that the two possibilities just illustrated, namely X# = X for an odd 4-flat and $$\left( {X^\# } \right)^\$$# = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X# = PG (9, 2) and of even 4-flats for which $${X^{\# \# \# } }$$ = X. © Kluwer Academic Publishers 2004 |
| abstractGer |
Abstract The 155 points of the Grassmannian $$G_{1,4,2}$$ of lines of PG (4, 2) = $$\mathbb{P}V\left( {5,2} \right)$$ are those points $$x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X$$ \subset $$ PG (9, 2) will be termed odd or even according as X intersects $$G_{1,4,2}$$ in an odd or even number of points. Let $$Q^\ddag \left( {x_1 ,...,x_5 } \right)$$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X$$ \subset $$ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$. Because $$Q^\ddag$$ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X$$ \subseteq$$X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. An example of an even 4-flat X such that $$\left( {X^\# } \right)^\$$# = X is provided by any 4-flat X which is external to $$G_{1,4,2}$$. However, it appears that the two possibilities just illustrated, namely X# = X for an odd 4-flat and $$\left( {X^\# } \right)^\$$# = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X# = PG (9, 2) and of even 4-flats for which $${X^{\# \# \# } }$$ = X. © Kluwer Academic Publishers 2004 |
| abstract_unstemmed |
Abstract The 155 points of the Grassmannian $$G_{1,4,2}$$ of lines of PG (4, 2) = $$\mathbb{P}V\left( {5,2} \right)$$ are those points $$x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X$$ \subset $$ PG (9, 2) will be termed odd or even according as X intersects $$G_{1,4,2}$$ in an odd or even number of points. Let $$Q^\ddag \left( {x_1 ,...,x_5 } \right)$$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X$$ \subset $$ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$. Because $$Q^\ddag$$ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X$$ \subseteq$$X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. An example of an even 4-flat X such that $$\left( {X^\# } \right)^\$$# = X is provided by any 4-flat X which is external to $$G_{1,4,2}$$. However, it appears that the two possibilities just illustrated, namely X# = X for an odd 4-flat and $$\left( {X^\# } \right)^\$$# = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X# = PG (9, 2) and of even 4-flats for which $${X^{\# \# \# } }$$ = X. © Kluwer Academic Publishers 2004 |
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(The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X$$ \subset $$ PG (9, 2) will be termed odd or even according as X intersects $$G_{1,4,2}$$ in an odd or even number of points. Let $$Q^\ddag \left( {x_1 ,...,x_5 } \right)$$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X# of a r-flat X$$ \subset $$ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$. Because $$Q^\ddag$$ is quinquelinear, the associate X# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X$$ \subseteq$$X# while if X is an even 4-flat then X# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X# = X. 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