Induced Representations and Frobenius Reciprocity

Induction builds a representation of a group $G$ from a representation of a subgroup $H$ — the counterpart to the restriction that goes the other way. Together they are adjoint (Frobenius reciprocity), and they are the mechanism by which the little-group representations of Wigner's classification are assembled from the internal symmetry of a single particle. This page builds on reps-basics.md and character-theory.md.

Restriction and induction

Given $H\le G$:

• Restriction ${\mathrm{Res}}_H^{G}V$ takes a representation of $G$ and simply forgets down to $H$ — the same vector space, acted on only by $H$.
• Induction ${\mathrm{Ind}}_H^{G}W$ takes a representation $W$ of $H$ and produces a representation of $G$ of dimension $[G:H]\cdot \dim W$. Concretely it is the space of $W$-valued functions on $G$ satisfying $f(hg)={\rho }_W(h)f(g)$, with $G$ acting by right translation; equivalently ${\mathrm{Ind}}_H^{G}W=\mathbb{C}[G]{\otimes }_{\mathbb{C}[H]}W$.

Inducing the trivial representation of $H$ gives the permutation representation of $G$ on the cosets $G/H$ — the linearised coset action.

Frobenius reciprocity

The two constructions are adjoint:

Frobenius reciprocity. For a representation $V$ of $G$ and $W$ of $H$, $${\mathrm{Hom}}_G({\mathrm{Ind}}_H^{G}W,V)\cong >{\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^{G}V),$$ equivalently, in terms of characters, $⟨{\chi }_{\mathrm{Ind}W},{\chi }_V{⟩}_G=⟨{\chi }_W,>{\chi }_{\mathrm{Res}V}{⟩}_H$.

So the multiplicity of an irrep $V$ of $G$ inside ${\mathrm{Ind}}_H^{G}W$ equals the multiplicity of $W$ inside $V{|}_H$ — a computation about $G$ is traded for one about the smaller $H$. There is an explicit character formula for the induced character as a sum over coset representatives, which makes induced characters computable by hand and drives many entries of a character table.

Mackey theory

Mackey's irreducibility criterion decides exactly when an induced representation ${\mathrm{Ind}}_H^{G}W$ is irreducible, in terms of how $W$ compares with its conjugates under double cosets H</span>G/H. The Mackey machine extends this to classify the irreducibles of a group with a normal abelian subgroup by inducing characters from the subgroup — the finite/discrete prototype of the Poincaré analysis below.

The bridge to Wigner's classification

Induction is not just a finite-group tool. For the Poincaré group $\mathcal{P}={\mathbb{R}}^{1,3}⋊\mathcal{L}$ (a semidirect product with normal abelian translations), the unitary irreps — the particle species — are built by inducing from the little group $G_{\bar{p}}$ (the stabiliser of a reference momentum) up to the full group. A spin state of the little group $SU(2)$ (massive) or $\mathrm{ISO}(2)$ (massless) induces to a one-particle Hilbert space. This is Wigner's classification, and it is exactly Mackey's machine applied to a non-compact group; see QFT/preliminaries.md. The infinite-dimensional unitary irreps that non-compact groups require (compact-groups.md) are produced precisely by this induction.

$S_n$ and Young diagrams

For symmetric groups, all irreducibles are obtained by inducing from Young subgroups $S_{{\lambda }_1}\times \cdots \times S_{{\lambda }_k}$ (products of smaller symmetric groups indexed by a partition $\lambda$) and extracting the irreducible "Specht module" summands — the representation-theoretic side of the cycle-type / partition combinatorics of permutation-groups.md.

References

• Serre, Linear Representations of Finite Groups, Part II.
• Fulton & Harris, Representation Theory, Ch. 3–4.
• Barut & Rączka, Theory of Group Representations and Applications (induced reps and the Poincaré group).