Induced representation

Induction extends a representation of a subgroup to the whole group; Frobenius reciprocity computes its multiplicities.

Embed the order-eight dihedral group as $H=⟨(1234),(12)(34)⟩$ in the symmetric group $S_4$, and let $W$ be its two-dimensional complex representation obtained from its square action. Let $C$ be the three-dimensional complex representation from the rotational symmetry group of a cube. In the class order $1,(12),(12)(34),(123),(1234)$, the induced character is $(6,0,-2,0,0)$: only the identity and the square's half-turn have nonzero character values. This equals the character of $C\otimes (1\oplus \mathrm{sgn})$, since ${\chi }_C=(3,-1,-1,0,1)$. Characters distinguish finite-dimensional complex semisimple representations, giving the displayed isomorphism. Its trivial character multiplicity is $(6-6)/24=0$.

Let $H$ be an open normal subgroup of a compact topological group $G$, and let $\tau :H\to {\mathrm{GL}}_m({\mathbb{Z}}_p)$ be a continuous faithful representation. Choose representatives $t_j$ for the finite left cosets $G/H$. Write $g{t}_j=t_i{h}_{ij}$ and act on the $j$-th summand of ${⨁}_j{\mathbb{Z}}_p^m$ by $\tau (h_{ij})$ into the $i$-th summand. The cocycle identity proves this is a continuous group representation. An element outside $H$ has a nontrivial coset permutation, while an element of $H$ acting trivially has $\tau (g)=1$ on the identity-coset summand. Hence the representation of $G$ is faithful and integral.

For a smooth linear character $\chi$ of a parabolic subgroup $B$, unnormalized induction consists of smooth functions satisfying $f(bg)=\chi (b)f(g)$, with right-translation action. The quotient $B\backslash G$ is compact for the local general linear groups in question. Normalized induction inserts a square root of the parabolic modulus in the covariance, changing formulas for the inducing linear character.

Evaluation of an intertwiner at the identity gives a $B$-equivariant functional. Conversely a functional $\ell$ gives $A(v)(g)=\ell (\pi (g)v)$, which has the required covariance and is fixed on the right by any compact-open subgroup fixing $v$. These inverse constructions prove the displayed adjunction. If $\xi$ is trivial on the unipotent radical, it is equivalent to a character-valued functional on the Jacquet module.

The local principal series of GL2 induces a smooth linear character of its diagonal torus, inflated to its upper-triangular subgroup. Its unnormalized Jacquet module has two one-dimensional composition factors, $\chi$ and ${\chi }^w{\delta }^{-1}$, where $\delta (\mathrm{diag}(a,d))=|d/a|$. When these coincide, the module is a nonsplit self-extension, so one must not count the linear character twice as two independent quotients.

The local Steinberg representation is the quotient of locally constant functions on the projective line by constant functions, with the action induced from GL2. In the unnormalized convention, $V(1)$ fits into $0\to 1\to V(1)\to \mathrm{St}\to 0$. Its scalar endomorphism algebra obstructs a splitting. This local infinite-dimensional representation differs from the finite-field Steinberg representation.

At $\chi ={\chi }^w{\delta }^{-1}$, a determinant-character twist reduces to $\chi (\mathrm{diag}(a,d))=|a|$. In the open Bruhat coordinate, a section $g_0(x)=1$ for $|x|\le 1$ and $g_0(x)=|x{|}^{-1}$ otherwise has $\pi (\mathrm{diag}(\varpi ,1))g_0-q^{-1}{g}_0=(1-q^{-1})1_{\varpi \mathfrak{o}}$. Its additive Haar integral is nonzero, whereas every unipotent translation difference has zero such integral. The difference therefore survives in the Jacquet module, proving that this two-dimensional self-extension is nonsplit and has just one quotient functional of the common linear character.

Every complex irreducible representation of a finite p-group is induced from a one-dimensional representation of a subgroup. Induct on the group order. Nonfaithful representations reduce to a smaller quotient. For a faithful representation of a nonabelian group, choose an abelian normal subgroup not contained in the centre. Restriction cannot be isotypic, since it would make that subgroup scalar and faithfulness would then force it central. The Clifford correspondence induces the representation from a proper subgroup; induction there and transitivity of induction complete the proof. Abelian groups have only one-dimensional complex irreducible representations.

For a character $\psi$ of $H\le G$, the induced character isThis is the character of a representation of $\mathbb{C}[G]{\otimes }_{\mathbb{C}[H]}{V}_{\psi }$. Substituting this expression into the character inner product and putting $g=xh{x}^{-1}$ proves Frobenius reciprocity.

For a finite group $G$, a subgroup $H$, and a character $\chi$ of $H$, each element $h$ of $g^G\cap H$ has exactly $|C_G(g)|$ preimages under $x\mapsto x^{-1}gx$. This follows because one such preimage $x_0$ has full fiber $C_G(g)x_0$. Grouping the character formula for an induced representation by these fibers proves the formula. Grouping again by conjugacy classes of $H$ replaces each class by its size times its character value.

A monomial character is a character induced from a linear character of a subgroup. When discussing monomial finite groups, this property is required for every irreducible character.

Extend a class function $\psi$ on $H$ by zero outside $H$. For coset representatives $x_1,\dots ,x_m$ of $G/H$, ${\mathrm{Ind}}_H^G\psi (g)={\sum }_j{\psi }_0(x_j^{-1}g{x}_j)$. Each summand has squared normalized $G$-norm $\Vert \psi {\Vert }_H^2/m$. Pointwise Cauchy-Schwarz inequality and summation give $\Vert \mathrm{Ind}\psi {\Vert }_G^2\le m\Vert \psi {\Vert }_H^2$. If $\psi$ is an irreducible character, its norm is one, and the sum of squared induction multiplicities is at most the subgroup index.

An index-two subgroup is normal. Conjugation by an outside element gives an involution on its irreducible characters. If ${\psi }^g\ne \psi$, the squared norm of induction is one, so induction is irreducible of twice the degree and restriction is $\psi +{\psi }^g$. If ${\psi }^g=\psi$, the squared norm is two, so induction has exactly two distinct constituents, each once. Each restricts to a representation containing $\psi$, and their total degree is twice that of $\psi$, forcing both degrees to equal that of $\psi$. This proves the fixed-orbit versus two-element-orbit classification without assuming extension of $\psi$ in advance.

In the compact picture of a principal-series group representation of the Lorentz spinor double cover, functions on SU(2) have a fixed right-torus character $e^{-in\vartheta }$. Expanding into spin matrix coefficients leaves one right weight in each allowed spin, giving the displayed multiplicity-one rotation decomposition. The boost action couples these spaces. The central element acts by $(-1{)}^n$; even $n$ yields integer rotation spins and a group representation of the connected Lorentz group.

For an $H$-representation $V$ and $g\in G$, the conjugate representation ${}^{g}V$ of $gH{g}^{-1}$ is defined by $(gh{g}^{-1})v=hv$. This transports the acting subgroup; it is distinct from conjugating complex matrix entries. It occurs in the Mackey restriction formula.

For $H\le G$ and an $H$-character $\chi$,

Let $N◃G$ be abelian and let a linear character $\theta$ of $N$ have stabilizer exactly $N$ under conjugation by $G$. Then ${\mathrm{Ind}}_N^G\theta$ is irreducible of degree $[G:N]$. Its value is zero outside $N$, while on $n\in N$ it is the sum of the characters in the $G$-orbit of $\theta$.

For $G=C_{11}⋊C_5$ with faithful action, the abelianization $G/C_{11}\cong C_5$ supplies five linear characters. The two orbits of size five on the nontrivial characters of $C_{11}$ induce two irreducible characters of degree five. Their squared degrees satisfyso these seven characters are all the irreducible characters of $G$.

For a $G$-representation $W$ and an $H$-representation $V$,On group-algebra tensors the isomorphism sends $x\otimes (w\otimes v)$ to $xw\otimes (x\otimes v)$.

For $H\le G$, an $H$-representation $V$, and a $G$-representation $W$,Equivalently, induction and restriction are adjoint for character inner products.

Mackey theory studies how induced representations behave under restriction and conjugation of subgroups.

For $H,K\le G$ and a $K$-representation $V$,where ${}^{x}V(xk{x}^{-1})=V(k)$.

To prove the formula, write induction as $\mathbb{C}[G]{\otimes }_{\mathbb{C}[K]}V$ and decompose the $(H,K)$-bimoduleFor $L_x=H\cap xK{x}^{-1}$, the mapis a well-defined $H$-equivariant isomorphism. Taking the direct sum over the double cosets proves the result.

For an irreducible $H$-representation $V$, the induced representation ${\mathrm{Ind}}_H^{G}V$ is irreducible exactly when, for every nonidentity double coset representative $x\in H\backslash G/H$, the restrictions of $V$ and its $x$-conjugate to $H\cap xH{x}^{-1}$ have inner product zero.

This is the finite special-linear-group instance of the general Bruhat decomposition.

For the upper-triangular subgroup $B$ of $S{L}_2({\mathbb{F}}_p)$ andthe two double cosets are $B$ and $BwB$, and $B\cap wB{w}^{-1}=T$, the diagonal subgroup.

For one-dimensional representations $\theta ,\phi$ of the upper-triangular subgroup $B\le S{L}_2({\mathbb{F}}_p)$,where ${\theta }^w(t)=\theta (w^{-1}tw)$.

Every linear character of the upper-triangular subgroup $B\le S{L}_2({\mathbb{F}}_q)$ with $q\ge 4$ is obtained from a multiplicative character $\theta$ of ${\mathbb{F}}_q^{\times }$. Its induction to $S{L}_2({\mathbb{F}}_q)$ is irreducible exactly when ${\theta }^2\ne 1$.

The diagonal subgroup of $S{L}_2({\mathbb{F}}_p)$ conjugates $u(x)$ to $u(a^{2}x)$. It therefore has two orbits, each of size $(p-1)/2$, on the nontrivial characters $\chi (v\cdot )$ of the unipotent subgroup: one indexed by squares and one by nonsquares.

If $B=U⋊T$ is the upper-triangular subgroup of $S{L}_2({\mathbb{F}}_p)$ and $\theta$ is one-dimensional on $T$, thenis the direct sum of three pairwise nonisomorphic irreducibles. Their restrictions to $U$ have character supports given by the trivial character, the square orbit, and the nonsquare orbit, so their dimensions are $1,(p-1)/2,(p-1)/2$.