Subrepresentation semirings and an analog of 6j-symbols

Abstract

Let V be a complex representation of the compact group G. The subrepresentation semiring associated to V is the set of subrepresentations of the algebra of linear endomorphisms of V with operations induced by the matrix operations. The study of these semirings has been motivated by recent advances in materials science, in which the search for microstructure-independent exact relations for physical properties of composites has been reduced to the study of these semirings for the rotation group SO(3). In this case, the structure constants for subrepresentation semirings can be described explicitly in terms of the 6j-symbols familiar from the quantum theory of angular momentum. In this paper, we investigate subrepresentation semirings for the class of quasisimply reducible groups defined by Mackey [“Multiplicity free representations of finite groups,” Pac. J. Math. 8, 503 (1958)]. We introduce a new class of symbols called twisted 6j-symbols for these groups, and we explicitly calculate the structure constants for subrepresentation semirings in terms of these symbols. Moreover, we show that these symbols satisfy analog of the standard properties of classical 6j-symbols.