ABSTRACT
There are two theorems of fundamental importance. These are known as Schur’s lemmas. These lemmas are useful for the study of irreducible representation (IR). Schur’s lemma 1: If Γi is an irreducible representation of a group and if a matrix P commutes with all the matrices of this irreducible representation, then this matrix P must be a constant matrix, i.e., P = C × E, where C is scalar quantity. It means that if a nonconstant-commuting matrix exists, then the representation is reducible. On the other hand, if none exists, then the representation is irreducible. Schur’s lemma 2: If Γi and Γj are two irreducible representations of a group G = {Ai, i = 1, 2, 3, …, n} with li and lj dimensions, respectively and a matrix M (of the order li and lj) satisfy the following relation.
