The concept of a 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.
Example To motivate the general definitions, a definition is first formulated, in the case of finite groups and their representations on finite-dimensional vector spaces. Let G be 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 this way:
U g W = { U g w : w ∈ W } . {\displaystyle U_{g}W=\{U_{g}w:w\in W\}.}
Let X be 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.,
H = ⨁ W ∈ X W . {\displaystyle H=\bigoplus _{W\in X}W.}
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,
∑ W ∈ X c W v W = 0 , v W ∈ W ∖ { 0 } {\displaystyle \sum _{W\in X}c_{W}v_{W}=0,\quad v_{W}\in W\setminus \{0\}}
holds only when all the coefficients cW are zero. If the action of G on the elements of X is transitive, then we say this is a transitive system of imprimitivity. Let G be a finite group and G0 a subgroup of G. A representation U of G is induced from a representation V of G0 if and only if there exist the following:
a transitive system of imprimitivity (U, X) and a subspace W0 ∈ X such that G0 is the stabilizer subgroup of W under the action of G, i.e.
G 0 = { g ∈ G : U g W 0 ⊆ W 0 } . {\displaystyle G_{0}=\{g\in G:U_{g}W_{0}\subseteq W_{0}\}.}
and V is equivalent to the representation of G0 on W0 given by Uh | W0 for h ∈ G0. 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 show that a well-defined inducing construction exists on equivalence of representations by considering the character of a representation U defined by
χ U ( g ) = tr ( U g ) . {\displaystyle \chi _{U}(g)=\operatorname {tr} (U_{g}).}
If a representation U of G is induced from a representation V of G0, then
χ U ( g ) = 1 | G 0 | ∑ { x ∈ G : x − 1 g x ∈ G 0 } χ V ( x − 1 g x ) , ∀ g ∈ G . {\displaystyle \chi _{U}(g)={\frac {1}{|G_{0}|}}\sum _{\{x\in G:{x}^{-1}\,g\,x\in G_{0}\}}\chi _{V}({x}^{-1}\ g\ x),\quad \forall g\in G.}
Thus the character function χU (and therefore U itself) is completely determined by χV.
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