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The concept of system of imprimitivity is used in mathematics, particularly in algebra and analysis, both within the context of the theory of group representations. It was used by George Mackey as the basis for his theory of induced unitary representations of locally compact groups. The simplest case, and the context in which the idea was first noticed, is that of finite groups (see primitive permutation group). Consider a group ''G'' and subgroups ''H'' and ''K'', with ''K'' contained in ''H''. Then the left cosets of ''H'' in ''G'' are each the union of left cosets of ''K''. Not only that, but translation (on one side) by any element ''g'' of ''G'' respects this decomposition. The connection with induced representations is that the permutation representation on cosets is the special case of induced representation, in which a representation is induced from a trivial representation. The structure, combinatorial in this case, respected by translation shows that either ''K'' is a maximal subgroup of ''G'', or there is a system of imprimitivity (roughly, a lack of full 'mixing'). In order to generalise this to other cases, the concept is re-expressed: first in terms of functions on ''G'' constant on ''K''-cosets, and then in terms of projection operators (for example the averaging over ''K''-cosets of elements of the group algebra). Mackey also used the idea for his explication of quantization theory based on preservation of relativity groups acting on configuration space. This generalized work of Eugene Wigner and others and is often considered to be one of the pioneering ideas in canonical quantization. == Illustrative Example == To motivate the general definitions, we first formulate a definition in the case of finite groups and their representations on finite-dimensional vector spaces. Suppose ''G'' is a finite group and ''U'' a representation of ''G'' on a finite-dimensional complex vector space ''H''. The action of ''G'' on elements of ''H'' induces an action of ''G'' on the vector subspaces ''W'' of ''H'' in an obvious way: : Suppose ''X'' is a set of subspaces of ''H'' such that # the elements of ''X'' are permuted by the action of ''G'' on subspaces and # ''H'' is the (internal) algebraic direct sum of the elements of ''X'', i.e., : Then (''U'',''X'') is a system of imprimitivity for ''G''. Two assertions must hold in the definition above: # the spaces ''W'' for ''W'' ∈ ''X'' must span ''H'', and # the spaces ''W'' ∈ ''X'' must be linearly independent, that is, : holds only when all the coefficients ''c''''W'' are zero. If the action of ''G'' on the elements of ''X'' is transitive, then we say this is a transitive system of imprimitivity. Suppose ''G'' is a finite group, ''G''0 a subgroup of ''G''. A representation ''U'' of ''G'' is induced from a representation ''V'' of ''G''0 if and only if there exist the following: * a transitive system of imprimitivity (''U'', ''X'') and * a subspace ''W''0 ∈ ''X'' such that ''G''0 is the fixed point subgroup of ''W'' under the action of ''G'', i.e. : and ''V'' is equivalent to the representation of ''G''0 on ''W''0 given by ''U''''h'' | ''W''0 for ''h'' ∈ ''G''0. Note that by this definition, ''induced by'' is a relation between representations. We would like to show that there is actually a mapping on representations which corresponds to this relation. For finite groups one can easily show that a well-defined inducing construction exists on equivalence of representations by considering the character of a representation ''U'' defined by : In fact if a representation ''U'' of ''G'' is induced from a representation ''V'' of ''G''0, then : Thus the character function χ''U'' (and therefore ''U'' itself) is completely determined by χ''V''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「system of imprimitivity」の詳細全文を読む スポンサード リンク
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