nLab wreath product of groups

For disambiguation see at wreath product.

Definition

Definition

Given (discrete) groups $H, G \in Grp(Set)$ and a G-set $S$, the wreath product group

$$H \wr_S G \;\; \coloneqq \;\; H^S \rtimes G$$

is the semidirect product group of

1. the direct product group $H^S$ of ${\vert S \vert}$ copies of $H$

2. with $G$, acting on $H^S$ by permutation of factors:

   $$g \big((h_s)_{s \in S}\big) \;\coloneqq\; \big(h_{g^{-1}(s)}\big)_{s \in S} \,.$$

(cf. BMMN 2006, Def. 8.1, see also James & Kerber 1984 §4.1)

Remark

(Notation) Other notation for “$H \wr_S G$” is “$H Wr_S G$” or “$H wr_S G$”.

Often the choice of $G$-set $S$ is notationally suppressed, writing instead “$H \wr G$”, etc.

But beware that conventions differ:

Some authors understand by default the regular action of $G$ on its underlying set.

But when $G = Sym(n)$ is a symmetric group, then often its defining action on $\{1, \cdots, n\}$ is understood (cf. Ex. below). In this case James & Kerber 1984 §4.2 speak of complete monomial groups over $H$.

Examples

Example

The dihedral group $Dih(4)$ of order $2\cdot 4=8$ is the wreath product $C_2 \wr C_2$ of $C_2$ with itself (in this sense of regular representations).

More generally:

Example

The symmetry group of the $n$-dimensional hypercube is $C_2 \wr Sym(n)$.

Example

The Sylow-$p$-subgroups $P_n$ of the symmetric groups $Sym(n)$ can be recursively described as the wreath product $C_p \wr P_a$ where $C_p$ is the cyclic group of order $p$ and $n = a p+r$ with $0\leq r \lneq p$.

Example

The Sylow-$\ell$-subgroups of $GL_n(q)$ for $gcd(q,\ell)=1$ can recursively be described as a wreath product of the Sylow-$\ell$-subgroup of a strictly smaller $GL_{m}(q)$ and a Sylow-$\ell$-subgroups of a suitable symmetric group.

Example

For a connected topological space $X$, the homotopy quotient of its $n$-fold product topological space by the symmetric group $Sym(n)$ acting by permutation of factors, hence the Borel construction

$$X^n \sslash Sym(n) \;\; \simeq \;\; \big( X^n \times E Sym(n) \big) / Sym(n) \,,$$

has as fundamental group the wreath product with the fundamental group of $X$:

$$\pi_1\big( X^n \sslash Sym(n) \big) \;\simeq\; \pi_1(X) \wr Sym(n) \,.$$

(cf. MO:a/867292)

Example

A framed braid group is the wreath product with the integers $\mathbb{Z}$ of a plain braid group.

Properties

General

Remark

(as a permutation group)
If both $H \subset Sym(X)$ and $G \subset Sym(Y)$ are presented as permutation groups, then the wreath product is itself a permutation group, acting on the Cartesian product set $X \times Y$ by letting $G$ act trivially on the first and naturally on the second factor and letting $H^Y$ act on $X\times Y$ such that the $y$-th component of $H^Y$ permutes $X\times\{y\}$ naturally and fixes everything else pointwise.

(e.g. Holland 1989)

Irreducible representations

Let $G$ be a finite group. Denote by $Irr(G) \simeq \big\{ [\rho_i] \big\}_{i \in I}$ the set of isomorphism classes of its irreducible representations.

For $n \in \mathbb{N}$ and $(n_i \in \mathbb{N})_{i \in I}$ an $I$-partition, in that $n = \sum_i n_i$, write

$$Sym_{(n)} \;\coloneqq\; \textstyle{\prod_{i \in I}} Sym_{n_i} \;\; \subset \;\; Sym_n$$

There is an evident linear representation of

$$\textstyle{\prod_{i \in I}} \big( G \wr Sym_{n_i} \big) \;\; \simeq \;\; G \wr Sym_{(n)} \;\;\subset\;\; G \wr Sym_n$$

on

(1)$$\rho \;\coloneqq\; \displaystyle{\underset{i \in I}{\boxtimes}} \rho_i^{\boxtimes_{n_i}} \;\;\; \in \;\; Rep(G \wr Sym_{(n)}) \,,$$

where the subgroup $G^n \subset G \wr Sym_{(n)}$ acts according to the external tensor product of representations, and the subgroup $Sym_{(n)}$ acts by permutation of tensor factors.

The irreps $\sigma$ of the direct product group $Sym_{(n)}$ are (see there) of the form

(2)$$\sigma \;\simeq\; \underset{i \in I}{\boxtimes} \sigma_{j(i)} \,,$$

for $\sigma_{j(i)}$ an irrep of the symmetric group $Sym_{n_i}$. We may canonically regard $\sigma$ as a representation of $G \wr Sym_{(n)} \twoheadrightarrow Sym_{(n)}$.

Theorem

For $G$ a finite group and $n \in \mathbb{N}$, the irreducible representations of the wreath product group $G \wr Sym_n$ are, up to isomorphism, exactly the induced representations along $G \wr Sym_{(n)} \hookrightarrow G \wr Sym_n$ of the tensor products of representations $\rho$ (1) for $(n)$ an $I$-partition as above, with irreps $\sigma$ (2) of $Sym_{(n)}$:

(3)$$\mathbb{C}\big[ G \wr Sym_n \big] \otimes_{ \mathbb{C}\big[ G \wr Sym_{(n)} \big] } \bigg( \Big( \displaystyle{\underset{i \in I}{\boxtimes}} \rho_i^{\boxtimes_{n_i}} \Big) \otimes \sigma \bigg) \,.$$

This is a digest of James & Kerber 1984 Thm. 4.3.34, for the special case that their “$H$” is all of $H = Sym_n$. For proper subgroups $H \subset Sym_n$ the analogous statement holds for the irreps of $G \wr H$, with $Sym_{(n)}$ above replaced throughout by $Sym_{(n)} \cap H$.

Example

(Irreps of $C_k \wr Sym_3$)

For $k \in \mathbb{N}_{\gt 0}$ denote by $C_k$ the cyclic group of order $k$. Its complex irreps $\mathbf{1}_{p}$ are all 1-dimensional, labeled by $p \in C_k$: The element $[1] \in C_k$ acts on $\mathbf{1}_p$ as multiplication with $e^{2 \pi \mathrm{i} \tfrac{p}{k}}$.

The irreps of $Sym_3$ are the trivial representation $\mathbf{1}$, the sign representation $\mathbf{1}_{sgn}$ and the standard representation $\mathbf{2}$.

The irreps of $C_p \wr Sym_3$ according to Thm. may be regarded as falling into 3 classes, according to the nature of the corresponding partition:

Case 1 – Partition involves a single irrep $\mathbf{1}_p$. In this case $Sym_{(3)} \simeq Sym_3$ and the formula (3) produces the representations

$$\mathbf{1}_{p}^{\boxtimes_3} \otimes \sigma \;\; \in \;\; Rep(C_p \wr Sym_3)$$

for $p \in C_p$ and $\sigma \in \big\{\mathbf{1}, \mathbf{1}_{sgn}, \mathbf{2}\big\}$.

Case 2 – Partition involves two distinct irreps, $\mathbf{1}_p$ and $\mathbf{1}_{p'}$. In this case $Sym_{(3)} = 1 \times Sym_2 \simeq Sym_2$ and the formula (3) produces

$$\mathbb{C}\big[ C_k \wr Sym_3 \big] \otimes_{ \mathbb{C}\big[ C_k \times (C_k \wr Sym_2) \big] } \Big( \big( \mathbf{1}_{p} \boxtimes \mathbf{1}_{p'}^{\boxtimes_2} \big) \otimes \sigma \Big)$$

for $p \neq p' \in C_k$ and $\sigma \in \big\{\mathbf{1}, \mathbf{1}_{sgn}\big\}$.

Case 3 – Partition involves three distinct irreps, $\mathbf{1}_{p_1}, \mathbf{1}_{p_2}, \mathbf{1}_{p_3}$. In this case $Sym_{(3)} \simeq 1 \times 1 \times 1 \simeq 1$ and the formula (3) produces

$$\mathbb{C}\big[ C_k \wr Sym_3 \big] \otimes_{ \mathbb{C}\big[ C_k^3 \big] } \big( \mathbf{1}_{p_1} \boxtimes \mathbf{1}_{p_2} \boxtimes \mathbf{1}_{p_3} \big)$$

for $p_i \in C_k$ pairwise distinct.

References

Monographs:

• Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller, Peter M. Neumann: Wreath products, Chapter 8 of: Notes on Infinite Permutation Groups, Lecture Notes in Mathematics 1698, Springer (2006) 67-76 [doi:10.1007/BFb0092558]

See also:

• W. Charles Holland: The characterization of generalized wreath products, J. Alg. 13 (1989) 152-172 [doi:10.1016/0021-8693(69)90069-6, pdf]

In the generality of semigroups:

• Charles Wells: Some applications of the wreath product construction, Amer. Math. Monthly 83 5 (1976) [doi:10.1080/00029890.1976.11994114]

Special cases and applications:

• G. Baumslag: Wreath products and extensions, Math Z 81 (1963) 286-299 [doi:10.1007/BF01111576]

• I. G. Macdonald: Polynomial functors and wreath products, J. Pure Appl. Alg. 18 (1980) 173-204 [doi:10.1016/0022-4049(80)90128-0, pdf]

On the representation theory of wreath products with symmetric groups:

• Gordon D. James, Adalbert Kerber: Representations of Wreath Products, Chapter 4 of: The Representation Theory of the Symmetric Group, Cambridge University Press (1984) [doi:10.1017/CBO9781107340732]

• I. A. Pushkarev: On the representation theory of wreath products of finite groups and symmetric groups, Journal of Mathematical Sciences, 96 (1999) 3590–3599 [doi:10.1007/BF02175835]

  originally in: Representation theory, dynamical systems, combinatorial and algorithmic methods. Part II, Zap. Nauchn. Sem. POMI 240, POMI, St. Petersburg (1997) 229–244 [mathnet:znsl475]

On wreath products of a cyclic with a symmetric group as analogs of (anyon) braid groups in dimension $\gt 2$:

• Michael Freedman, Matthew B. Hastings, Chetan Nayak, Xiao-Liang Qi, Kevin Walker, Zhenghan Wang: Projective Ribbon Permutation Statistics: a Remnant of non-Abelian Braiding in Higher Dimensions, Phys. Rev. B 83 115132 (2011) [doi:10.1103/PhysRevB.83.115132, arXiv:1005.0583]

On wreath products with braid groups (cf. also framed braid group):

• Semidirect products with braid groups and type $F_\infty$ [MO:q/179104]

category: algebra