Isotypic decomposition

This definition makes the isotypic decomposition given in Proposition 6.11 easier to write down. 

Now we prove the existence of the isotypic decomposition for finite-dimensional representations. Just as any natural number has a prime factorization, every finite-dimensional representation of a compact group has an isotypic decomposition. This decomposition tells us what irreducible representations appear as subrepresentations and what their multiplicities are. Note that Proposition 6.11 guarantees uniqueness as well, since the selection of irreducible representations and exponents are uniquely determined. 

This Cartesian sum representation is called the isotypic decomposition of V. The list of representations Wj and their multiplicities Cj is called the isotype of V. 

Proposition 6.10 implies that there is no injective homomorphisms from W into V for any natural number k > k. Thus there can be no subrepresentation (other than Wq ) isomorphic to a power of Wo in the isotypic decomposition of This completes the inductive step. ... 

Decomposition of a mixture of nitrates is often a convenient method for the preparation of mixed transition metal oxides. The success of the method depends upon the fact that typically, the decomposition of the nitrates results in an intimate oxide mixture that readily reacts at higher temperatures to form a homogeneous, complex oxide. Oftentimes nitrate mixtures can be formed as precnrsors by dissolving a stoichiometric mixture of the oxides or carbonates in nitric acid and evaporating to dryness. This method is one of the options used to prepare the superconducting YBa2Cn30y and isotypes. 

Li2BeF has a hexagonal unit cell (space group r5) isotypic with phenacite (6). This structure shows no phase transitions between room temperature and T (3, 4, 7), but exposure to traces of moisture at elevated temperatures causes decomposition to LiP and BeO (7, 8). Metastable forms have been reported (4) but not confirmed or identified. 

The first stage of the decomposition appears to be the formation of Si3N4 and silica, which then reacts with silicon from the dissociation of Si3N4 to form SiO. The preparation of the two isotypic monoclinic phases Caio+xSii2-2xM,6 (M = P or As) has been described. The materials may only be obtained in a pure state when... 

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