Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes Original Research Article

Let A be a semisimple, n-dimensional, commutative algebra over a field image. Fix a basis B of A, and denote by M(a; B) the transpose of the matrix over image that represents a ε A regularly with respect to B. It is easy to see that the set {M(a; B) a ε A} can be simultaneously diagonalized over many fields (including all perfect fields). We use this fact in order to give an elementary proof that such an algebra over an infinite field is generated by a single element, and to describe the subalgebras of A in terms of certain partitions of the set {1,2,3,…, n}. Several applications of these results are shown: (1) We give a new proof for the theorem stating that every finite-dimensional, separable field extension has a primitive element. (2) We show that every finite group G has a character θ such that every other generalized character of G is a polynomial in θ with rational coefficients. (This is true for Brauer characters as well.) (3) We give a necessary condition for two generalized characters (or Brauer characters) ζ and χ that forces the field of values of ζ to contain that of χ. (4) Many collections of patterned matrices over a field image, such as circulant matrices and some of their generalizations are known to be algebras generated by a single matrix. We observe that each subalgebra of such a collection is also generated by a single matrix. Also, if a and b are two elements of such a collection, we give a necessary and sufficient condition, in terms of the eigenvalue pattern of a and b, for a to be a polynomial in b with coefficients in image. (5) We show that if A is a (generalized) cyclic code, then the eigenvalues of M(a; B) are the so-called Matteson-Solomon coefficients of the codeword a. Other applications to coding, to groups, and to field extensions are discussed as well.