nLab Mackey theory

Idea

In representation theory, Mackey theory refers to a set of tools for constructing and classifying irreducible representations of a group $G$ from that of a normal subgroup $N \subset G$, by using induction from the stabilizer groups of the adjoint action of $G$ on the set of irreps of $N$.

(Also referred to as little group-methods, cf. Wigner classification.)

Details

For finite semidirect products with abelian groups

Consider

• $A$ an abelian group

• $G \,\equiv\, H \rtimes A$ a semidirect product group.

and denote:

• $\widetilde A \,\coloneqq\, Hom(A, \mathbb{C}^\times)$ the dual group of group characters, hence, since $A$ is abelian, of irreducible representations of $A$,

• the adjoint action of $G$ on $A$ and the induced action on $\widetilde A$:

  $$\begin{array}{ccc} G \times \widetilde A &\longrightarrow& \widetilde A \\ \big(g, A \xrightarrow{\chi} \mathbb{C}^\times \big) &\mapsto& \chi\big(g^{-1} (\text{-}) g\big) \end{array}$$

• $(\chi_i)_{i \in \widetilde{A}/H}$ a set of representatives of the resulting orbits of $H$ in $\widetilde A$,

• $H_i \,\coloneqq\, Stab_H(\chi_i)$ the corresponding stabilizer subgroups of $H$,

• $G_i \,\coloneqq\, H_i \rtimes A$ the corresponding subgroups of $G$

Finally, denote

$$\widehat{\chi}_i \,\colon\, G_i \twoheadrightarrow A \xrightarrow{\chi_i} \mathbb{C}^{\times} \,,$$

which is a group character on $G_i$, by the definition of $H_i$.

Similarly, for $\rho \in Irr(H_i)$ an irrep of $H_i$, write

$$\widehat{\rho} \,\colon\, G_i \twoheadrightarrow H_i \xrightarrow{\rho} GL(V)$$

and consider the induced representations of the tensor products:

(1)$$\theta_{i,\rho} \;\coloneqq\; Ind_{G_i}^G \big( \widehat{\chi}_i \otimes \widehat{\rho} \big) \;\in\; Rep_{\mathbb{C}}(G) \,.$$

Proposition

1. The representations $\theta_{i,\rho}$ (1) are irreducible.

2. If $\theta_{i,\rho} \simeq \theta_{i',\rho'}$ then $i = i'$ and $\rho \simeq \rho'$.

3. Every irrep of $G$ is of the form $\theta_{i, \rho}$, up to isomorphism.

(Serre 1977 Prop. 25 p 62)

Examples

Example

Consider the semidirect product group $G \equiv \mathbb{Z}_n \rtimes \mathbb{Z}_2$, with

• $A \equiv \mathbb{Z}_n \,\coloneqq\, \mathbb{Z}/n$ the cyclic group of order $n \in \mathbb{N}$,

• $H \equiv \mathbb{Z}_2$ the cyclic group of order 2 acting on $\mathbb{Z}_n$ via the sign involution.

In order find all the (complex) irreps of $\mathbb{Z}_n \rtimes \mathbb{Z}_2$ we unwind the statement of Prop. :

(We denote congruence classes by square brackets: $[-] \colon \mathbb{Z} \twoheadrightarrow \mathbb{Z}_n$.)

First, the complex irreps of the normal subgroup $\mathbb{Z}_n$ are themselves labeled by $[k] \in \mathbb{Z}_n$ and given by

$$\begin{array}{ccc} \mathbb{Z}_n &\xrightarrow{\;\; \chi_{[k]} \;\;}& \mathbb{C}^\times \\ [r] &\mapsto& e^{2 \pi \mathrm{i} \tfrac{r k}{n}} \mathrlap{\,.} \end{array}$$

On these, the nontrivial element $\sigma \in \mathbb{Z}_2$ acts by $\sigma \colon [k] \mapsto [-k]$:

$$\begin{array}{l} (\sigma \chi_{[k]})\big([r]\big) \\ \;\equiv\; \chi_{[k]}\big(\sigma^{-1}\cdot [r] \cdot \sigma\big) \\ \;\equiv\; \chi_{[k]}\big(\sigma^{-1}\cdot \sigma\cdot [-r]\big) \\ \;=\; \chi_{[k]}\big([-r]\big) \\ \;\equiv\; e^{2 \pi \mathrm{i} \tfrac{(-r)k}{n}} \\ \;=\; e^{2 \pi \mathrm{i} \tfrac{r (-k)}{n}} \\ \;\equiv\; \chi_{[-k]}\big([r]\big) \mathrlap{\,.} \end{array}$$

From this we have two kinds of orbits:

1. the singleton orbits with stabilizer $\mathbb{Z}_2$:

   $\big\{ [0] \big\}$, which always exists

   and $\big\{ [n/2] \big\}$, which exists when $n$ is an even number,

2. the pairs with stabilizer the trivial group $1$:

   $\big\{[k],[-k]\big\}$, for $k \in \{1, \cdots, \lfloor(n-1)/2\rfloor\}$ (with $\lfloor-\rfloor$ the floor).

The first singleton in the first case gives rise to two irreps, corresponding to the two irreps of $\mathbb{Z}_2$ (the trivial representation and the sign representation):

$$\underset{Id}{ \underbrace{ Ind ^{\mathbb{Z}_p \rtimes \mathbb{Z}_2} _{\mathbb{Z}_p \rtimes \mathbb{Z}_2} } } \Big( \big( ([r], [s]) \mapsto 1 \big) \otimes \big( ([r], [s]) \mapsto 1 \big) \Big) \;=\; triv$$

and

$$Ind ^{\mathbb{Z}_p \rtimes \mathbb{Z}_2} _{\mathbb{Z}_p \rtimes \mathbb{Z}_2} \Big( \big( ([r], [s]) \mapsto 1 \big) \otimes \big( ([r], [s]) \mapsto e^{\pi \mathrm{i} s} \big) \Big) \;=\; \big( ([r], [s]) \mapsto e^{\pi \mathrm{i} s} \big) \,,$$

while the second singleton in the first case similarly gives

$$Ind ^{\mathbb{Z}_p \rtimes \mathbb{Z}_2} _{\mathbb{Z}_p \rtimes \mathbb{Z}_2} \Big( \big( ([r], [s]) \mapsto e^{\pi \mathrm{i} r} \big) \otimes \big( ([r], [s]) \mapsto 1 \big) \Big) \;=\; \big( ([r], [s]) \mapsto e^{\pi \mathrm{i} r} \big)$$

and

$$Ind ^{\mathbb{Z}_p \rtimes \mathbb{Z}_2} _{\mathbb{Z}_p \rtimes \mathbb{Z}_2} \Big( \big( ([r], [s]) \mapsto e^{\pi \mathrm{i} r} \big) \otimes \big( ([r], [s]) \mapsto e^{\pi \mathrm{i} s} \big) \Big) \;=\; \big( ([r], [s]) \mapsto e^{\pi \mathrm{i} (r + s)} \big) \,.$$

The second case gives rise to one irrep for each $k \in \{1, \cdots, \lfloor(n-1)/2\rfloor\}$:

$$\begin{array}{l} Ind ^{ \mathbb{Z}_p \rtimes \mathbb{Z}_2 } _{ \mathbb{Z}_p } \Big( \big( [r] \mapsto e^{2 \pi \mathrm{i} \tfrac{r k}{n}} \big) \otimes \big( [r] \mapsto 1 \big) \Big) \\ \;\simeq\; \mathbb{C}\big[ \mathbb{Z}_p \rtimes \mathbb{Z}_2 \big] \otimes_{ \mathbb{C}[\mathbb{Z}_p] } \Big( [r] \mapsto e^{2 \pi \mathrm{i} \tfrac{r k}{n}} \Big) \mathrlap{\,.} \end{array}$$

These latter irreps are 2-dimensional, with an evident matrix representation $\widehat{(-)}$ given by

$$\widehat{ \big( [1], [0] \big) } \;=\; \left[ \begin{matrix} e^{+ 2 \pi \mathrm{i} \tfrac{k}{n}} & 0 \\ 0 & e^{- 2 \pi \mathrm{i} \tfrac{k}{n}} \end{matrix} \right] \,,\;\;\;\;\;\; \widehat{ \big( [0], [1] \big) } \;=\; \left[ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right] \mathrlap{\,.}$$

As a consistency check that we found all irreps, we verify that the sum of squares formula holds:

We have found two 1-dimensional irreps when $n$ is odd and four of them when $n$ is even, together with $\lfloor (n-1)/2\rfloor$ irreps of dimension 2, whence the sum of their squares of dimensions is,

for odd $n$:

$$2 \cdot 1^2 \,+\, \lfloor (n-1)/2\rfloor \cdot 2^2 \;=\; 2 + \frac{n-1}{2} 4 \;=\; 2 + 2 (n-1) \;=\; 2 n$$

and for even $n$:

$$4 \cdot 1^2 \,+\, \lfloor (n-1)/2\rfloor \cdot 2^2 \;=\; 2 \cdot 2 \,+\, \frac{n-2}{2} 4 \;=\; 2 \cdot 2 + 2 \cdot (n-2) \;=\; 2 n \,,$$

correctly coinciding with the order of our group :

$${\vert \mathbb{Z}_n \rtimes \mathbb{Z}_2 \vert} \;=\; {\vert \mathbb{Z}_2 \vert} \cdot {\vert \mathbb{Z}_n \vert} \;=\; 2 n \,.$$

Example

For construction of irreps of wreath products of groups see there.

References

The original article:

• George Mackey: Induced representations of groups and quantum mechanics, W. A. Benjamin, New York (1968) [ark:/13960/t6841m201]

General discussion:

• J. M. G. Fell, R. S. Doran: Representations of $\ast$-Algebras, Locally Compact Groups, and Banach $\ast$-Algebraic Bundles, volume 2: Banach $\ast$-Algebraic Bundles, Induced Representations, and the Generalized Mackey Analysis, Academic Press (1988) [ISBN:9780122527227]

Review for the case of finite groups:

• Charles Curtis, Irving Reiner, around Prop. 11.8 of: Representation theory of finite groups and associative algebras, AMS (1962) [ISBN:978-0-8218-4066-5]

• Jean-Pierre Serre, section 8.2 of: Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer (1977) [doi:10.1007/978-1-4684-9458-7, pdf]

• Brian Conrad: Mackey theory and applications [pdf, pdf]

• Amirtanshu Prasad, M. K. Vemuri: Mackey’s Little Group Method [pdf, pdf]

• S. Martin: Mackey Theory, chapter 12 of Representation Theory, notes by Dexter Chua (2016) [pdf]

Further developments:

• Geetha Venkataraman: On irreducibility of induced modules and an adaptation of the Wigner–Mackey method of little groups, J. Korean Math. Soc. 50 6 (2013) 1213-1222 [arXiv:0908.0026, doi:10.4134/JKMS.2013.50.6.1213]

Review for the case of Lie groups (cf. Wigner classification):

• José Figueroa-O’Farrill: The Theory of Induced Representations in Field Theory [pdf, pdf]