Isotypic Decompositions Optional

Just as any natural number can be written uniquely as a product of primes, 

We start with a convenient definition. Just as prime powers play a particular role in number theory, Cartesian sums of copies of one irreducible representation play a particular role in representation theory. 

Definition 6.4 Suppose (G, W, p) is a group representation and k is a natural number. Set 

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

Our first proposition in this section establishes a useful isomorphism. 

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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. 

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