Representations of group rings and groups
An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. It is shown that for any group ring matrix $A$ of $mathbb{C} G$ there exists a matrix $U$ (independent of $A$) such that $U^{-1}AU= diag(T_1,T_2,ldots, T_r)$ for block matrices $T_i$...
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| Format: | Article |
| Language: | English |
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University of Isfahan
2018-06-01
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| Series: | International Journal of Group Theory |
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| Online Access: | http://ijgt.ui.ac.ir/article_21484_2d64c759c091beeacf98923cde8ed7f6.pdf |
| _version_ | 1818331775617728512 |
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| author | Ted Hurley |
| author_facet | Ted Hurley |
| author_sort | Ted Hurley |
| collection | DOAJ |
| description | An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. It is shown that for any group ring matrix $A$ of $mathbb{C} G$ there exists a matrix $U$ (independent of $A$) such that $U^{-1}AU= diag(T_1,T_2,ldots, T_r)$ for block matrices $T_i$ of fixed size $s_i × s_i$ where $r$ is the number of conjugacy classes of $G$ and $s_i$ are the ranks of the group ring matrices of the primitive idempotents. Using the isomorphism of the group ring to the ring of group ring matrices followed by the mapping $Amapsto P^{-1}AP$ (fixed $P$) gives an isomorphism from the group ring to the ring of such block matrices. Specialising to the group elements gives a faithful representation of the group. Other representations of $G$ may be derived using the blocks in the images of the group elements. For a finite abelian group $Q$ an explicit matrix $P$ is given which diagonalises any group ring matrix of $mathbb{C}Q$. The characters of $Q$ and the character table of $Q$ may be read off directly from the rows of the diagonalising matrix $P$. This is a special case of the general block diagonalisation process but is arrived at independently. The case for cyclic groups is well-known: Circulant matrices are the group ring matrices of the cyclic group and the Fourier matrix diagonalises any circulant matrix. This has applications to signal processing. |
| first_indexed | 2024-12-13T13:25:13Z |
| format | Article |
| id | doaj.art-86eb5e2902b3461b958f118b073efba8 |
| institution | Directory Open Access Journal |
| issn | 2251-7650 2251-7669 |
| language | English |
| last_indexed | 2024-12-13T13:25:13Z |
| publishDate | 2018-06-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | International Journal of Group Theory |
| spelling | doaj.art-86eb5e2902b3461b958f118b073efba82022-12-21T23:44:18ZengUniversity of IsfahanInternational Journal of Group Theory2251-76502251-76692018-06-0172314410.22108/ijgt.2017.2148421484Representations of group rings and groupsTed Hurley0National University of Ireland GalwayAn isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. It is shown that for any group ring matrix $A$ of $mathbb{C} G$ there exists a matrix $U$ (independent of $A$) such that $U^{-1}AU= diag(T_1,T_2,ldots, T_r)$ for block matrices $T_i$ of fixed size $s_i × s_i$ where $r$ is the number of conjugacy classes of $G$ and $s_i$ are the ranks of the group ring matrices of the primitive idempotents. Using the isomorphism of the group ring to the ring of group ring matrices followed by the mapping $Amapsto P^{-1}AP$ (fixed $P$) gives an isomorphism from the group ring to the ring of such block matrices. Specialising to the group elements gives a faithful representation of the group. Other representations of $G$ may be derived using the blocks in the images of the group elements. For a finite abelian group $Q$ an explicit matrix $P$ is given which diagonalises any group ring matrix of $mathbb{C}Q$. The characters of $Q$ and the character table of $Q$ may be read off directly from the rows of the diagonalising matrix $P$. This is a special case of the general block diagonalisation process but is arrived at independently. The case for cyclic groups is well-known: Circulant matrices are the group ring matrices of the cyclic group and the Fourier matrix diagonalises any circulant matrix. This has applications to signal processing.http://ijgt.ui.ac.ir/article_21484_2d64c759c091beeacf98923cde8ed7f6.pdfgroupringRepresentation |
| spellingShingle | Ted Hurley Representations of group rings and groups International Journal of Group Theory group ring Representation |
| title | Representations of group rings and groups |
| title_full | Representations of group rings and groups |
| title_fullStr | Representations of group rings and groups |
| title_full_unstemmed | Representations of group rings and groups |
| title_short | Representations of group rings and groups |
| title_sort | representations of group rings and groups |
| topic | group ring Representation |
| url | http://ijgt.ui.ac.ir/article_21484_2d64c759c091beeacf98923cde8ed7f6.pdf |
| work_keys_str_mv | AT tedhurley representationsofgroupringsandgroups |