Representations of group rings and groups
Publish place: International Journal of Group Theory، Vol: 7، Issue: 2
Publish Year: 1397
نوع سند: مقاله ژورنالی
زبان: English
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JR_THEGR-7-2_006
تاریخ نمایه سازی: 14 اردیبهشت 1400
Abstract:
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 $U^{-۱}AU= diag(T_۱,T_۲,\ldots, T_r)$ 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 $A\mapsto P^{-۱}AP$ (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.
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Authors
Ted Hurley
National University of Ireland Galway
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