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Polynomial endomorphisms of the Cuntz algebras arising from permutations . II — Branching laws of endomorphisms —

For a subgroup H of a group G, an irreducible decomposition of π|H for an irreducible representation π of G is one of main study in representation theory([12]). When such decomposition holds, the decomposition formula is called a branching law. This is reformulated as the branching which is brought by the inclusion map ι from H to G, that is, ι∗(π) ≡ π ◦ ι gives a map from RepG to RepH. In general, any homomorphism φ from a group G1 to other G2 arises a transformation φ∗ from RepG2 to RepG1. Specially, when G = G1 = G2, ρ ∈ EndG gives a transformation on RepG and we can consider the branching law by ρ∗ in this situation. On the other hand, the branching law by embeddings of C∗-algebras is studied in [1, 2]. The difference of two theories of quantum string field is explained by the difference of representations arising from two embeddings of a pseudo Cuntz algebra in [3]. In the last paper [11], we introduce a class of endomorphisms of the Cuntz algebra ON arising from permutations and show the complete reducibility of the action of them on permutative representations. For a representation π and an endomorphism ρ, if there is a decomposition π ◦ρ = π1⊕· · ·⊕πM , we call this formula by the branching law of ρ at π. Because this branching is brought by ρ, the branching law of ρ shows a property of ρ. In application of branching laws for geometry, such method to study morphisms is already considered(§ 4.A. in [12]). In