Induced representation

In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the (whole) group G itself. Given a representation of H, the induced representation is, in a sense, the 'most general' representation of G that extends the given one. Since it is often easier to find representations of the smaller group H than of G, the operation of forming induced representations is an important tool to construct new representations. In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the (whole) group G itself. Given a representation of H, the induced representation is, in a sense, the 'most general' representation of G that extends the given one. Since it is often easier to find representations of the smaller group H than of G, the operation of forming induced representations is an important tool to construct new representations. Induced representations were initially defined by Frobenius, for linear representations of finite groups. The idea is by no means limited to the case of finite groups, but the theory in that case is particularly well-behaved. Let G be a finite group and H any subgroup of G. Furthermore let (π, V) be a representation of H. Let n = be the index of H in G and let g1, ..., gn be a full set of representatives in G of the left cosets in G/H. The induced representation IndGH π can be thought of as acting on the following space: Here each gi V is an isomorphic copy of the vector space V whose elements are written as gi v with v∈V. For each g in G and each gi there is an hi in H and j(i) in {1, ..., n} such that g gi = gj(i) hi . (This is just another way of saying that g1, ..., gn is a full set of representatives.) Via the induced representation G acts on W as follows: where v i ∈ V {displaystyle v_{i}in V} for each i. Alternatively, one can construct induced representations using the tensor product: any K-linear representation π {displaystyle pi } of the group H can be viewed as a module V over the group ring K. We can then define This latter formula can also be used to define IndGH π for any group G and subgroup H, without requiring any finiteness. For any group, the induced representation of the trivial representation of the trivial subgroup is the right regular representation. More generally the induced representation of the trivial representation of any subgroup is the permutation representation on the cosets of that subgroup. An induced representation of a one dimensional representation is called a monomial representation, because it can be represented as monomial matrices. Some groups have the property that all of their irreducible representations are monomial, the so-called monomial groups.