Editing System of imprimitivity (section)

The concept of a '''system of imprimitivity''' is used in [[mathematics]], particularly in [[algebra]] and [[Mathematical analysis|analysis]], both within the context of the [[theory]] of [[group representation]]s. It was used by [[George Mackey]] as the basis for his theory of [[induced representation|induced unitary representation]]s of [[locally compact group]]s.

The simplest case, and the context in which the idea was first noticed, is that of [[finite group]]s (see [[primitive permutation group]]). Consider a group ''G'' and subgroups ''H'' and ''K'', with ''K'' contained in ''H''. Then the left [[coset]]s 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 representation]]s 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 operator]]s (for example the averaging over ''K''-cosets of elements of the [[group ring|group algebra]]).

Mackey also used the idea for his explication of quantization theory based on preservation of [[Poincaré group|relativity groups]] acting on [[Configuration space (physics)|configuration space]]. This generalized work of [[Eugene Wigner]] and others and is often considered to be one of the pioneering ideas in [[canonical quantization]].