A CLASS OF BINARY SIGNALING ALPHABETS 229 



corresponding to the fact that 



(1011) (0110) = (1101) 



If the +1,-1 symbols are regarded as shorthand for diagonal matrices, 

 so that for example 



-11 -1 -1 



then group multiplication corresponds to matrix multiplication. 



(While much of what follows here can be established in an elementary 

 way for the simple group at hand, it is convenient to fall back upon the 

 established general theory of group representations for several proposi- 

 tions. 



The substitution (13) converts a modular representation table (col- 

 umn / included) into a square array of +l's and — I's. Each column (or 

 row) of this array is clearly an irreducible representation of Ck ■ Since Ck 

 is Abelian it has precisely 2 irreducible representations each of degree 

 one. These are furnished by the converted modular table. This table also 

 furnishes then the characters of the irreducible representations of Ck 

 and we refer to it henceforth as a character table. 



Let x"(^) be the entry of the character table in the row labelled A and 

 column labelled a. The orthogonality relationship for characters gives 



E x'{A)/{A) = 2'8., 



ACCk 



Z x%A)x"(B) = 2'b 



<xCCk 



AB 



where 8 is the usual Kronecker symbol. In particular 



E xiA)x\A) = Z AA) = 0, ^^I 



ACCk ACCk 



Since each x (A) is +1 or — 1, these must occur in eciual numbers in any 

 column ^ 9^ I. This implies that each column except / of the modular 

 representation table contains 2 ~ ones, a fact used earlier. 



Every matrix representation of Ck can be reduced to its irreducible 

 components. If the trace of the matrix representing the element A in an 

 arbitrary matrix representation of Ck is x{A), then this representation 

 contains the irreducible representation having label ^ in the character 

 table dp times where 



