The "representation" in "representation theory" that the wikipedia article refers to is a term of art, referring to a class of one-to-one maps between groups (sets endowed with a binary operation with specific properties) and square matrices, in such a way that matrix multiplication preserves the properties of this binary operation between the original group elements being mapped to matrices. There is no mention of this sort of algebraic operation whatsoever in the paper you have linked, which, as far as I can tell, uses the word "representation" only in a nontechnical descriptive sense. I believe you might be misunderstanding the nature of this breakthrough.