Triangular Products of Group Representations and Their Applications:
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Boston, MA
Birkhäuser Boston
1981
|
| Series: | Progress in Mathematics
17 |
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | The construction considered in these notes is based on a very simple idea. Let (A, G ) and (B, G ) be two group representations, for definiteness faithful and finite 1 2 dimensional, over an arbitrary field. We shall say that a faithful representation (V, G) is an extension of (A, G ) by (B, G ) if there is a G-submodule W of V such that 1 2 the naturally arising representations (W, G) and (V/W, G) are isomorphic, modulo their kernels, to (A, G ) and (B, G ) respectively. 1 2 Question. Among all the extensions of (A, G ) by (B, G ), does there exist 1 2 such a "universal" extension which contains an isomorphic copy of any other one? The answer is in the affirmative. Really, let dim A = m and dim B = n, then the groups G and G may be considered as matrix groups of degrees m and n 1 2 respectively. If (V, G) is an extension of (A, G ) by (B, G ) then, under certain 1 2 choice of a basis in V, all elements of G are represented by (m + n) x (m + n) mat rices of the form (*) ~1-~ ~-J lh I g2 I |
| Physical Description: | 1 Online-Ressource (X, 132 p) |
| ISBN: | 9781468467215 9781468467239 |
| DOI: | 10.1007/978-1-4684-6721-5 |
Staff View
MARC
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| 500 | |a The construction considered in these notes is based on a very simple idea. Let (A, G ) and (B, G ) be two group representations, for definiteness faithful and finite 1 2 dimensional, over an arbitrary field. We shall say that a faithful representation (V, G) is an extension of (A, G ) by (B, G ) if there is a G-submodule W of V such that 1 2 the naturally arising representations (W, G) and (V/W, G) are isomorphic, modulo their kernels, to (A, G ) and (B, G ) respectively. 1 2 Question. Among all the extensions of (A, G ) by (B, G ), does there exist 1 2 such a "universal" extension which contains an isomorphic copy of any other one? The answer is in the affirmative. Really, let dim A = m and dim B = n, then the groups G and G may be considered as matrix groups of degrees m and n 1 2 respectively. If (V, G) is an extension of (A, G ) by (B, G ) then, under certain 1 2 choice of a basis in V, all elements of G are represented by (m + n) x (m + n) mat rices of the form (*) ~1-~ ~-J lh I g2 I | ||
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Record in the Search Index
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-sort | 3512.2 |
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| format | Electronic eBook |
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| id | DE-604.BV042421104 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:08Z |
| institution | BVB |
| isbn | 9781468467215 9781468467239 |
| language | English |
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| physical | 1 Online-Ressource (X, 132 p) |
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| publishDate | 1981 |
| publishDateSearch | 1981 |
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| publisher | Birkhäuser Boston |
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| series2 | Progress in Mathematics |
| spelling | Vovsi, Samuel M. Verfasser (DE-588)106708066X aut Triangular Products of Group Representations and Their Applications by Samuel M. Vovsi Boston, MA Birkhäuser Boston 1981 1 Online-Ressource (X, 132 p) txt rdacontent c rdamedia cr rdacarrier Progress in Mathematics 17 The construction considered in these notes is based on a very simple idea. Let (A, G ) and (B, G ) be two group representations, for definiteness faithful and finite 1 2 dimensional, over an arbitrary field. We shall say that a faithful representation (V, G) is an extension of (A, G ) by (B, G ) if there is a G-submodule W of V such that 1 2 the naturally arising representations (W, G) and (V/W, G) are isomorphic, modulo their kernels, to (A, G ) and (B, G ) respectively. 1 2 Question. Among all the extensions of (A, G ) by (B, G ), does there exist 1 2 such a "universal" extension which contains an isomorphic copy of any other one? The answer is in the affirmative. Really, let dim A = m and dim B = n, then the groups G and G may be considered as matrix groups of degrees m and n 1 2 respectively. If (V, G) is an extension of (A, G ) by (B, G ) then, under certain 1 2 choice of a basis in V, all elements of G are represented by (m + n) x (m + n) mat rices of the form (*) ~1-~ ~-J lh I g2 I Mathematics Group theory Group Theory and Generalizations Mathematik Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 s Gruppentheorie (DE-588)4072157-7 s 1\p DE-604 https://doi.org/10.1007/978-1-4684-6721-5 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Vovsi, Samuel M. Triangular Products of Group Representations and Their Applications Mathematics Group theory Group Theory and Generalizations Mathematik Darstellungstheorie (DE-588)4148816-7 gnd Gruppentheorie (DE-588)4072157-7 gnd |
| subject_GND | (DE-588)4148816-7 (DE-588)4072157-7 |
| title | Triangular Products of Group Representations and Their Applications |
| title_auth | Triangular Products of Group Representations and Their Applications |
| title_exact_search | Triangular Products of Group Representations and Their Applications |
| title_full | Triangular Products of Group Representations and Their Applications by Samuel M. Vovsi |
| title_fullStr | Triangular Products of Group Representations and Their Applications by Samuel M. Vovsi |
| title_full_unstemmed | Triangular Products of Group Representations and Their Applications by Samuel M. Vovsi |
| title_short | Triangular Products of Group Representations and Their Applications |
| title_sort | triangular products of group representations and their applications |
| topic | Mathematics Group theory Group Theory and Generalizations Mathematik Darstellungstheorie (DE-588)4148816-7 gnd Gruppentheorie (DE-588)4072157-7 gnd |
| topic_facet | Mathematics Group theory Group Theory and Generalizations Mathematik Darstellungstheorie Gruppentheorie |
| url | https://doi.org/10.1007/978-1-4684-6721-5 |
| work_keys_str_mv | AT vovsisamuelm triangularproductsofgrouprepresentationsandtheirapplications |