Representations of group rings and groups

Document Type : Ischia Group Theory 2016


National University of Ireland Galway


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 $A\mapsto P^{-1}AP$ (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.


Main Subjects

[1] O. M. Baksalary, D. S. Bernstein and G. Trenkler, On the equality between rank and trace of an idempotent matrix, Appl. Math. Comput., 217 (2010) 4076–4080.
[2] R. E. Blahut, Algebraic Codes for Data Transmission, CUP, 2003.
[3] C. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Amer. Math. Soc.,Chelsea, 1966.
[4] T. Hurley, Group rings and rings of matrices, Inter. J. Pure & Appl. Math., 31 (2006) 319–335.
[5] B. Hurley and T. Hurley, Paraunitary matrices and group rings, Int. J. of Group Theory, 3 (2014) 31-56.
[6] I. Martin Isaacs, Character Theory of Finite Groups, Dover, 2011.
[7] C. Polcino Milies and S. Sehgal, An introduction to Group Rings, Klumer, 2002.