e irreps together. Suppose T is a representation of G acting on a space V and T ′ is a representation of the same G acting on a space V ′. Then a linear map φ : V → V ′ is said morphism of representations or intertwining map if for all g ∈ G and all v ∈ V : φ (Tg (v)) = T ′ g (φ (v)), i.e., V Tg−→ V φ ↓ ↓ φ V ′ T ′g−→ V ′ commutes. If φ is also invertible, then φ is said to be an isomorphism of representations, and the representations are said to be isomorphic or equivalent. The remarkable fact is that each group has a fixed set of inequivalent irreps, so the distinct irreps are a characteristic of the group itself, not of a particular representation. The uniqueness and canonical nature of the partition obtained in the operator case by equality of eigenvalues is now obtained using equivalence of irreps and their underlying irreducible subspaces. All the irreducible subspacesWi for irreps equivalent to an irrep L in any such direct sum V = ∑l i=1⊕Wi are grouped together (by direct sum) to obtain the invariant carrierWL for a primary representation- where a representation is primary if all its irreducible subrepresentations are equivalent and the underlying carrier space is also called primary. Note that some of the inequivalent irreps of the group G may not be involved in a particular representation. The decomposition of V as the direct sum ∑ L⊕WL of the invariant primary subspaces for the primary representations is unique. "It is the invariant subspaces [WL] which are the analogues of the eigenspaces of a single operator." [30, p. 244] In terms of representations rather than their carrier subspaces, it is the unique "canonical decomposition into primary representations." [30, p. 244] Thus we have the top-down construction of the vector space partition V = ∑ ⊕Vi of eigenspaces Vi given by an operator (or vector-space attribute) and the bottom-up construction of the vectorspace partition V = ∑ ⊕WL of the carriers WL for primary representations given by a vector