07.05.16 · representation-theory / symmetric

Wreath products and the representations of the hyperoctahedral group

shipped3 tiersLean: none

Anchor (Master): Specht 1932 Eine Verallgemeinerung der symmetrischen Gruppe; James-Kerber The Representation Theory of the Symmetric Group (1981) Ch. 4; Macdonald Symmetric Functions and Hall Polynomials (2nd ed.) Ch. I App. B

Intuition Beginner

Take a group $G$ and make $n$ copies of it, one sitting in each of $n$ slots. An element of this big group $G^{n}$ is just a list: one element of $G$ for each slot. Now bring in the permutations of $S_{n}$ , the group that reshuffles the $n$ slots. A wreath product lets you do both at once: first permute the slots, then drop a copy of $G$ into each slot. The notation is $G ≀ S_{n}$ , read " $G$ wreath $S_{n}$ ".

The cleanest example replaces $G$ by the two-element group of signs, plus and minus. Each slot now carries a sign, and on top of that you may shuffle the slots. The result is the group of signed permutations: rearrange $n$ items and flip the sign of any of them. This is the hyperoctahedral group $B_{n}$ , and it is exactly the set of symmetries of an $n$ -dimensional cube.

Visual Beginner

Picture a square in the plane with corners labelled. Its symmetries either rotate the square or reflect it. Each symmetry permutes the two coordinate axes and may flip the direction of either axis. That is a signed permutation of two slots, so the square's symmetry group is $B_{2}$ .

Swapping the two axes is the $S_{2}$ part; flipping the arrow on an axis is the sign part. Combine all swaps with all sign choices and you get every symmetry of the square: there are $2^{2} \times 2 = 8$ of them.

Worked example Beginner

Let us count the hyperoctahedral group $B_{3}$ , the symmetries of an ordinary cube, slot by slot.

Step 1. There are 3 slots, the three coordinate axes. The number of ways to permute 3 slots is $3! = 6$ .

Step 2. Each slot independently carries a sign, plus or minus, so there are $2 \times 2 \times 2 = 8$ sign patterns.

Step 3. A signed permutation is a choice of slot-shuffle together with a choice of sign pattern, so the total count is $6 \times 8 = 48$ .

That matches the known number of symmetries of a cube: 48. The general formula is $∣ B_{n} ∣ = 2^{n} \times n!$ . For $n = 3$ this is $8 \times 6 = 48$ , and for $n = 4$ it is $16 \times 24 = 384$ .

A single element of $B_{3}$ might send the first axis to the third axis with a flipped sign, keep the second axis in place, and send the third axis to the first axis. Writing a minus where a sign flips, this is the signed permutation that takes $(1, 2, 3)$ to $(- 3, 2, 1)$ .

Check your understanding Beginner

Formal definition Intermediate+

Let $G$ be a finite group and let $S_{n}$ be the symmetric group on ${1, \dots, n}$ . The symmetric group acts on the direct power $G^{n}$ by permuting coordinates: for $σ \in S_{n}$ and $(g_{1}, \dots, g_{n}) \in G^{n}$ ,

σ \cdot (g_{1}, \dots, g_{n}) = (g_{σ^{- 1} (1)}, \dots, g_{σ^{- 1} (n)}) .

Definition (wreath product). The wreath product $G ≀ S_{n}$ is the semidirect product $G^{n} ⋊ S_{n}$ with respect to this action. Its underlying set is $G^{n} \times S_{n}$ , and multiplication is

((g_{1}, \dots, g_{n}); σ) ((h_{1}, \dots, h_{n}); τ) = ((g_{1} h_{σ^{- 1} (1)}, \dots, g_{n} h_{σ^{- 1} (n)}); σ τ) .

The subgroup $G^{n} = {((g_{1}, \dots, g_{n}); e)}$ is the base group, normal in $G ≀ S_{n}$ , and the subgroup ${(1; σ)}$ is a complement isomorphic to $S_{n}$ . The order is $∣ G ≀ S_{n} ∣ = ∣ G ∣^{n} n!$ .

Definition (hyperoctahedral group). Taking $G = Z_{2} = {+ 1, - 1}$ gives the hyperoctahedral group

B_{n} = Z_{2} ≀ S_{n}, ∣ B_{n} ∣ = 2^{n} n! .

An element is a signed permutation: a permutation $σ \in S_{n}$ together with a sign $ε_{i} \in {+ 1, - 1}$ attached to each $i$ . Concretely $B_{n}$ is the group of $n \times n$ matrices with exactly one nonzero entry in each row and column, every such entry being $+ 1$ or $- 1$ ; it is the symmetry group of the hypercube $[- 1, 1]^{n}$ .

Conjugacy classes. Let $G$ have conjugacy classes $C_{1}, \dots, C_{r}$ . Each $g \in G^{n}$ together with the cycle structure of $σ$ produces, for each cycle of $σ$ , a cycle product in $G$ (the ordered product of the base entries around the cycle), well defined up to conjugacy. A conjugacy class of $G ≀ S_{n}$ is recorded by a function assigning to each class $C_{j}$ of $G$ a partition $μ^{(j)}$ , where the parts of $μ^{(j)}$ are the lengths of the cycles whose cycle product lies in $C_{j}$ , subject to $\sum_{j} ∣ μ^{(j)} ∣ = n$ . These are the $G$ -coloured partitions of $n$ .

Counterexamples to common slips

Order of the factors. $G ≀ S_{n}$ is not $S_{n} ≀ G$ . The base group is the power $G^{n}$ ; the symmetric group acts on it. Reversing the symbols changes the group.
Cycle products, not entries. Two base vectors $g, g^{'} \in G^{n}$ paired with the same $σ$ are conjugate exactly when their cycle products match up to conjugacy in $G$ , not when the individual entries match. Conjugating by an element of the base group slides entries around a cycle and changes them individually while preserving the ordered product.
Signs versus a single global sign. In $B_{n}$ each slot carries its own sign. The diagonal copy of $Z_{2}$ (all signs equal) is a proper subgroup; the full base group $Z_{2}^{n}$ has $2^{n}$ elements, not 2.

Key theorem with proof Intermediate+

The irreducibles of $G ≀ S_{n}$ are built by the little-group method, the Wigner-Mackey construction for a semidirect product with abelian-or-not base, applied through the induced representation 07.01.07 and controlled by Frobenius reciprocity 07.01.08.

Theorem (classification of irreducibles of $G ≀ S_{n}$ ). Let $G = {V_{1}, \dots, V_{r}}$ be the irreducible representations of $G$ . The irreducible representations of $G ≀ S_{n}$ are indexed by $r$ -tuples of partitions $λ = (λ^{(1)}, \dots, λ^{(r)})$ with $\sum_{k} ∣ λ^{(k)} ∣ = n$ . Writing $n_{k} = ∣ λ^{(k)} ∣$ , the irreducible attached to $λ$ is

W^{λ} = Ind_{\prod_{k} (G ≀ S_{n_{k}})}^{G ≀ S_{n}} k ⨂ (V_{k}^{\otimes n_{k}} \otimes S^{λ^{(k)}}),

where $V_{k}^{\otimes n_{k}}$ is the canonical extension of the outer tensor power $V_{k}^{\otimes n_{k}}$ of the base to $G ≀ S_{n_{k}}$ , and $S^{λ^{(k)}}$ is the Specht module of $S_{n_{k}}$ pulled back to $G ≀ S_{n_{k}}$ .

Proof. The base group $N = G^{n}$ is normal in $H = G ≀ S_{n}$ , so Clifford theory applies. An irreducible $ρ$ of $H$ restricts to $N$ as a direct sum of an $H$ -orbit of irreducibles of $N$ . The irreducibles of $N = G^{n}$ are the outer tensor products $V_{i_{1}} ⊠ \dots ⊠ V_{i_{n}}$ , indexed by functions ${1, \dots, n} \to G$ . The quotient $H / N ≅ S_{n}$ permutes these functions by permuting their arguments, so an $H$ -orbit is a function recorded up to relabelling of slots: equivalently a multiset assigning to each $V_{k} \in G$ a number $n_{k}$ of slots, with $\sum_{k} n_{k} = n$ .

Fix such an orbit, with representative $θ = \bigboxtimes_{k} V_{k}^{⊠ n_{k}}$ (the first $n_{1}$ slots carry $V_{1}$ , the next $n_{2}$ carry $V_{2}$ , and so on). Its stabiliser in $S_{n}$ is the Young subgroup $S_{n_{1}} \times \dots \times S_{n_{r}}$ , so the inertia subgroup is $T = \prod_{k} (G ≀ S_{n_{k}}) \leq H$ . By Clifford theory the irreducibles of $H$ lying over the orbit of $θ$ are obtained by inducing from $T$ the irreducibles of $T$ that restrict on the base to $θ$ .

It remains to extend $θ$ to $T$ and to list the irreducibles of $T$ over it. For each block $G ≀ S_{n_{k}}$ the outer power $V_{k}^{\otimes n_{k}}$ extends canonically to a representation $V_{k}^{\otimes n_{k}}$ of $G ≀ S_{n_{k}}$ : the base acts diagonally through $V_{k}$ and a slot-permutation $σ$ acts by the operator that permutes the tensor factors of $V_{k}^{\otimes n_{k}}$ . One checks the cocycle identity, so this is an honest representation extending $θ$ on that block. Given one extension, the irreducibles of $G ≀ S_{n_{k}}$ over $V_{k}^{\otimes n_{k}}$ are $V_{k}^{\otimes n_{k}} \otimes (pullback of an irreducible of S_{n_{k}})$ , and the irreducibles of $S_{n_{k}}$ are the Specht modules $S^{λ^{(k)}}$ , $λ^{(k)} ⊢ n_{k}$ (the symmetric-group theory of 07.05.01). Taking the outer tensor product over the blocks and inducing to $H$ yields $W^{λ}$ . Distinct $λ$ give inequivalent irreducibles, and counting $\sum_{λ} 1$ over $r$ -tuples summing to $n$ matches the number of $G$ -coloured partitions, hence the number of conjugacy classes. This exhausts $H$ . $□$

Bridge. This classification builds toward the character theory of the hyperoctahedral group and appears again in the symmetric-function description below, where the induction is repackaged as a product of Frobenius characteristics. The foundational reason the little-group method works is that the base group is normal, so Clifford theory forces every irreducible to live over a single $S_{n}$ -orbit of base-characters; this is exactly the orbit-and-stabiliser bookkeeping that induced representations 07.01.07 and Frobenius reciprocity 07.01.08 were built to handle. The construction generalises the bijection {partitions of $n$ } $\leftrightarrow$ {irreducibles of $S_{n}$ } of 07.05.01: taking $G$ to be the one-element group collapses $r = 1$ and recovers the symmetric-group case, so the pattern that organises $S_{n}$ -representations is dual to the pattern here, now indexed by tuples of diagrams. Putting these together, the central insight is that the representation theory of $G ≀ S_{n}$ factors into the representation theory of $G$ (which colours the slots) and the representation theory of $S_{n}$ (which fills each colour-block with a Young diagram).

Exercises Intermediate+

Exercise 4 (medium, symbolic).

The group $G$ has $r$ conjugacy classes. Using the classification theorem, show that the number of irreducibles of $G ≀ S_{n}$ equals the number of $r$ -tuples of partitions $(λ^{(1)}, \dots, λ^{(r)})$ with $\sum_{k} ∣ λ^{(k)} ∣ = n$ , and write the generating function $\sum_{n \geq 0} (# G ≀ S_{n}) t^{n}$ in terms of the partition generating function $P (t) = \prod_{m \geq 1} (1 - t^{m})^{- 1}$ .

Hint

The number of partitions of all sizes has generating function $P (t)$ . An $r$ -tuple of independent partitions multiplies generating functions.

Answer

Each component $λ^{(k)}$ ranges over all partitions independently, contributing a factor $P (t) = \sum_{m \geq 0} p (m) t^{m}$ tracking $∣ λ^{(k)} ∣$ . With $r$ independent components the joint generating function is $P (t)^{r}$ , and the coefficient of $t^{n}$ counts $r$ -tuples with total size $n$ . By the theorem this equals $# G ≀ S_{n}$ , so $\sum_{n \geq 0} (# G ≀ S_{n}) t^{n} = P (t)^{r} = \prod_{m \geq 1} (1 - t^{m})^{- r}$ .

Exercise 6 (medium, multiple choice).

In the little-group construction for $G ≀ S_{n}$ , what plays the role of the inertia subgroup (stabiliser) of a base-orbit representative with $n_{k}$ slots coloured by $V_{k}$ ?

A. The full group $G ≀ S_{n}$ B. The base group $G^{n}$ alone C. The product of smaller wreath products $\prod_{k} (G ≀ S_{n_{k}})$ D. The symmetric group $S_{n}$ alone

Hint

The stabiliser of the colouring in $S_{n}$ is the Young subgroup $\prod_{k} S_{n_{k}}$ ; combine it with the full base.

Answer

C. The stabiliser in $S_{n}$ of the colouring is the Young subgroup $S_{n_{1}} \times \dots \times S_{n_{r}}$ , and together with the full base $G^{n}$ this gives the inertia subgroup $\prod_{k} (G ≀ S_{n_{k}})$ . Inducing irreducibles of this subgroup produces the irreducibles of $G ≀ S_{n}$ . Option A would give nothing to induce; B and D omit half the structure.

Exercise 7 (hard, symbolic).

Derive the dimension formula $dim W^{λ} = \frac{n !}{\prod _{k} n _{k} !} \prod_{k} ((dim V_{k})^{n_{k}} f^{λ^{(k)}})$ for the irreducible of $G ≀ S_{n}$ attached to $λ = (λ^{(1)}, \dots, λ^{(r)})$ , where $n_{k} = ∣ λ^{(k)} ∣$ and $f^{λ^{(k)}}$ is the number of standard Young tableaux of shape $λ^{(k)}$ .

Hint

The dimension of an induced representation is the index of the subgroup times the dimension of the inducing module. The index of $\prod_{k} (G ≀ S_{n_{k}})$ in $G ≀ S_{n}$ is the multinomial $n! / \prod_{k} n_{k}!$ .

Answer

Induction multiplies dimension by the index of the inertia subgroup $T = \prod_{k} (G ≀ S_{n_{k}})$ in $H = G ≀ S_{n}$ . Since $∣ H ∣ = ∣ G ∣^{n} n!$ and $∣ T ∣ = \prod_{k} ∣ G ∣^{n_{k}} n_{k}! = ∣ G ∣^{n} \prod_{k} n_{k}!$ , the index is $[H : T] = n! / \prod_{k} n_{k}!$ . The inducing module is $⨂_{k} (V_{k}^{\otimes n_{k}} \otimes S^{λ^{(k)}})$ , of dimension $\prod_{k} (dim V_{k})^{n_{k}} f^{λ^{(k)}}$ because $dim V_{k}^{\otimes n_{k}} = (dim V_{k})^{n_{k}}$ and $dim S^{λ^{(k)}} = f^{λ^{(k)}}$ . Multiplying, $dim W^{λ} = \frac{n !}{\prod _{k} n _{k} !} \prod_{k} (dim V_{k})^{n_{k}} f^{λ^{(k)}}$ . For $G = Z_{2}$ all $dim V_{k} = 1$ , recovering $dim W^{(λ, μ)} = (∣ λ ∣ n) f^{λ} f^{μ}$ .

Exercise 8 (hard, symbolic).

Show that for $G = Z_{2}$ , summing $dim W^{(λ, μ)} = (∣ λ ∣ n) f^{λ} f^{μ}$ over all pairs $(λ, μ)$ with $∣ λ ∣ + ∣ μ ∣ = n$ recovers $\sum (dim W)^{2} = 2^{n} n! = ∣ B_{n} ∣$ , using the identity $\sum_{ν ⊢ m} (f^{ν})^{2} = m!$ .

Hint

Square the dimension, sum over $λ$ of size $k$ and $μ$ of size $n - k$ separately, then sum over $k$ with the binomial coefficient. Use the vandermonde-style identity $\sum_{k} (k n)^{2} k! (n - k)! = ?$ after applying $\sum_{ν ⊢ m} (f^{ν})^{2} = m!$ .

Answer

Fix $∣ λ ∣ = k$ , $∣ μ ∣ = n - k$ . Then $\sum_{∣ λ ∣ = k} \sum_{∣ μ ∣ = n - k} (k n)^{2} (f^{λ})^{2} (f^{μ})^{2} = (k n)^{2} (\sum_{λ ⊢ k} (f^{λ})^{2}) (\sum_{μ ⊢ n - k} (f^{μ})^{2}) = (k n)^{2} k! (n - k)!$ . Now $(k n)^{2} k! (n - k)! = \frac{( n ! ) ^{2}}{( k ! ) ^{2} (( n - k )! ) ^{2}} k! (n - k)! = (n!)^{2} \frac{1}{k ! ( n - k )!} = n! (k n)$ . Summing over $k$ from $0$ to $n$ gives $n! \sum_{k} (k n) = n! 2^{n} = ∣ B_{n} ∣$ . The dimensions of the irreducibles of $B_{n}$ therefore satisfy the sum-of-squares identity.

Advanced results Master

Theorem 1 (signed cycle type and conjugacy in $B_{n}$ ). A signed permutation $w \in B_{n}$ decomposes into signed cycles. Each cycle has a sign equal to the product of the signs of its entries, splitting cycles into positive and negative; the conjugacy class of $w$ is recorded by a pair of partitions $(α, β)$ , where $α$ lists the lengths of the positive cycles and $β$ the lengths of the negative cycles, with $∣ α ∣ + ∣ β ∣ = n$ . The number of conjugacy classes is therefore $\sum_{k = 0}^{n} p (k) p (n - k)$ , equal to the number of irreducibles indexed by pairs $(λ, μ)$ .

This is the $G = Z_{2}$ specialisation of the $G$ -coloured-partition description: the two conjugacy classes of $Z_{2}$ produce the two partitions $α$ (cycle product $+ 1$ ) and $β$ (cycle product $- 1$ ).

Theorem 2 (wreath Frobenius characteristic). Let $Λ$ denote the ring of symmetric functions and let $Λ^{\otimes r}$ carry one copy per irreducible $V_{k}$ of $G$ . There is an isometric isomorphism $$ \mathrm{ch}: \bigoplus_{n \geq 0} R(G \wr S_n) ;\xrightarrow{\ \sim\ }; \Lambda^{\otimes r}, \qquad W^{\boldsymbol{\lambda}} \longmapsto \prod_k s_{\lambda^{(k)}}\big[ X^{(k)} \big], $$ where $R (\cdot)$ is the representation ring, $s_{ν}$ is a Schur function, and $X^{(k)}$ is the alphabet attached to $V_{k}$ . Induction of representations corresponds to multiplication of symmetric functions, and the inner product matching characters corresponds to the Hall inner product on each tensor factor. This is the wreath-product analogue of the Frobenius characteristic for $S_{n}$ and underlies the plethystic substitution $p_{m} \mapsto p_{m} \circ (character of G)$ that converts power sums of $G ≀ S_{n}$ into power sums on the $r$ alphabets.

Theorem 3 (Murnaghan-Nakayama for $B_{n}$ ). The irreducible character $χ^{(λ, μ)}$ of $B_{n}$ , evaluated on an element of signed cycle type $(α, β)$ , satisfies a border-strip recursion that removes a positive cycle of length $α_{1}$ from either $λ$ or $μ$ (with a sign from the border-strip height), and removes a negative cycle of length $β_{1}$ from $λ$ minus the same removal from $μ$ (with the type- $B$ sign convention). This refines the symmetric-group rule of 07.05.10: setting $β = \emptyset$ and $μ = \emptyset$ recovers the $S_{n}$ recursion on $λ$ .

Theorem 4 ( $B_{n}$ as a Weyl group). The hyperoctahedral group $B_{n}$ is the Weyl group of the root systems of type $B_{n}$ and $C_{n}$ . It acts on $R^{n}$ by permuting and sign-changing the standard coordinates, with simple reflections the adjacent transpositions $s_{i} = (i, i + 1)$ for $1 \leq i < n$ together with the sign change $s_{n} : e_{n} \mapsto - e_{n}$ . The Coxeter presentation has braid relations of $A_{n - 1}$ among $s_{1}, \dots, s_{n - 1}$ and the single order-4 relation $(s_{n - 1} s_{n})^{4} = e$ . The longest element is $- id$ , and the order $2^{n} n!$ equals the product of the degrees $2, 4, \dots, 2 n$ of the fundamental invariants.

Theorem 5 (random walk on the cube via $B_{n}$ and the Ehrenfest connection). The simple random walk driven by single sign-flips on the base $Z_{2}^{n}$ of $B_{n}$ is the Ehrenfest urn walk on the hypercube; its eigenvalues are $1 - 2 j / n$ for $j = 0, \dots, n$ , the Krawtchouk spectrum, and the upper bound lemma for the abelian group $Z_{2}^{n}$ gives the $\frac{1}{4} n lo g n$ mixing threshold. Walks that also shuffle slots live on the full $B_{n}$ and require the pair-of-partitions character theory above to diagonalise. Diaconis uses exactly this for signed-rank statistics, where data carry a sign and a rank.

Synthesis. The wreath-product construction is the foundational reason that the representation theory of signed permutations parallels that of ordinary permutations one level up. The central insight is that a normal base group forces Clifford theory, and the little-group method then factors every irreducible into a colouring of slots by irreducibles of $G$ and a Young diagram filling each colour-block, which is exactly the bijection {tuples of partitions} $\leftrightarrow$ {irreducibles of $G ≀ S_{n}$ }. Putting these together with the Frobenius characteristic, the entire character theory becomes symmetric-function algebra on $r$ alphabets, and the Murnaghan-Nakayama recursion of 07.05.10 generalises verbatim to border strips removed from a pair of diagrams. The bridge is that $B_{n}$ is simultaneously the symmetry group of the hypercube, the Weyl group of type $B / C$ , and the natural home for signed statistical data, so the same pair-of-partitions index governs cube geometry, root-system combinatorics, and the hyperoctahedral random walks; this construction generalises the $S_{n}$ theory of 07.05.01 and is dual, through the characteristic map, to the plethystic structure of symmetric functions.

Full proof set Master

Proposition 1 (order of the wreath product). For a finite group $G$ , $∣ G ≀ S_{n} ∣ = ∣ G ∣^{n} n!$ , and in particular $∣ B_{n} ∣ = 2^{n} n!$ .

Proof. As a set $G ≀ S_{n} = G^{n} \times S_{n}$ , a Cartesian product, so its cardinality is $∣ G^{n} ∣ \cdot ∣ S_{n} ∣ = ∣ G ∣^{n} n!$ . The semidirect-product multiplication does not change the underlying set, only the group law. For $G = Z_{2}$ this gives $2^{n} n!$ . $□$

Proposition 2 (the base power $V^{\otimes n}$ extends to $G ≀ S_{n}$ ). Let $V$ be a representation of $G$ on a space $U$ . Define an action of $G ≀ S_{n}$ on $U^{\otimes n}$ by $$ \big((g_1, \dots, g_n); \sigma\big)\cdot (u_1 \otimes \cdots \otimes u_n) = g_1 u_{\sigma^{-1}(1)} \otimes \cdots \otimes g_n u_{\sigma^{-1}(n)}. $$ This is a representation of $G ≀ S_{n}$ extending the outer power $V^{⊠ n}$ of the base.

Proof. Restricting to the base ( $σ = e$ ) gives $(g_{1}, \dots, g_{n}) \cdot (u_{1} \otimes \dots \otimes u_{n}) = g_{1} u_{1} \otimes \dots \otimes g_{n} u_{n}$ , which is $V^{⊠ n}$ . To verify the homomorphism property, apply $((g_{i}); σ)$ then $((h_{i}); τ)$ to a pure tensor and compare with the action of their product $((g_{i} h_{σ^{- 1} (i)}); σ τ)$ . Reindexing the tensor slots shows both routes send $u_{1} \otimes \dots \otimes u_{n}$ to $⨂_{i} g_{i} h_{σ^{- 1} (i)} u_{(σ τ)^{- 1} (i)}$ . The two agree because the slot-permutations compose as $σ τ$ and the group elements multiply within each slot in the order dictated by the wreath law. Hence the assignment is a representation. $□$

Proposition 3 (irreducible count equals conjugacy-class count for $B_{n}$ ). The number of irreducibles of $B_{n}$ equals the number of conjugacy classes, both equal to $\sum_{k = 0}^{n} p (k) p (n - k)$ .

Proof. By Theorem 1 a conjugacy class is a pair $(α, β)$ of partitions with $∣ α ∣ + ∣ β ∣ = n$ , and the count of such pairs is $\sum_{k} p (k) p (n - k)$ on splitting by $∣ α ∣ = k$ . By the classification theorem the irreducibles are indexed by pairs $(λ, μ)$ with $∣ λ ∣ + ∣ μ ∣ = n$ , the same count. Both equal the number of conjugacy classes by the general theorem that a finite group over $C$ has as many irreducibles as conjugacy classes, so the two indexings are matched in cardinality. $□$

Connections Master

Induced representation 07.01.07. The little-group construction of the irreducibles of $G ≀ S_{n}$ is induction from the inertia subgroup $\prod_{k} (G ≀ S_{n_{k}})$ up to the full wreath product. Without the induced representation there is no way to assemble an irreducible of the large group from data living on a stabiliser of a base-orbit.
Frobenius reciprocity 07.01.08. The proof that distinct tuples $λ$ give inequivalent irreducibles, and that the induced modules are themselves irreducible, runs through Frobenius reciprocity: the multiplicity of an induced module in another is computed as a restriction multiplicity on the inertia subgroup, where the orbit-stabiliser structure makes it a single Specht-module pairing.
Symmetric group representation 07.05.01. The wreath product theory contains the symmetric-group theory as the special case $G = {e}$ , $r = 1$ : tuples of partitions collapse to a single partition, and $W^{(λ)}$ is the Specht module $S^{λ}$ . The slots-and-fillings picture here is the colour-graded generalisation of the partition indexing of $S_{n}$ -irreducibles.
Murnaghan-Nakayama rule 07.05.10. The border-strip character recursion for $B_{n}$ in Theorem 3 specialises to the symmetric-group Murnaghan-Nakayama rule when all signs are positive, so the unsigned recursion of 07.05.10 is the $β = \emptyset$ , $μ = \emptyset$ corner of the signed one.

Historical & philosophical context Master

Wilhelm Specht introduced the representation theory of what he called the generalised symmetric groups $Z_{m} ≀ S_{n}$ in his 1932 paper Eine Verallgemeinerung der symmetrischen Gruppe ^{[Specht 1932]}, extending the partition-indexed theory of $S_{n}$ that Frobenius and Young had built. The systematic treatment of arbitrary wreath products $G ≀ S_{n}$ , the $G$ -coloured-partition description of conjugacy classes, and the little-group classification of irreducibles were consolidated by James and Kerber in their 1981 monograph ^{[James-Kerber 1981]}, which remains the standard reference. Macdonald gave the symmetric-function formulation, with the wreath Frobenius characteristic and its plethystic substitution, in Appendix B to Chapter I of Symmetric Functions and Hall Polynomials ^{[Macdonald 1995]}.

The hyperoctahedral group $B_{n}$ predates this representation theory as a geometric object: it is the symmetry group of the $n$ -cube and the cross-polytope, and it reappears as the Weyl group of the type- $B / C$ root systems in the Cartan-Killing classification. Persi Diaconis brought the wreath-product machinery into probability and statistics, using $B_{n}$ for signed-rank statistics and for random walks on the cube, where the pair-of-partitions characters diagonalise the relevant Markov chains ^{[Diaconis 1988]}.

Bibliography Master

@article{Specht1932,
  author = {Specht, Wilhelm},
  title = {Eine Verallgemeinerung der symmetrischen Gruppe},
  journal = {Schriften des Mathematischen Seminars der Universit\"at Berlin},
  volume = {1},
  year = {1932},
  pages = {1--32},
}

@book{JamesKerber1981,
  author = {James, Gordon and Kerber, Adalbert},
  title = {The Representation Theory of the Symmetric Group},
  publisher = {Addison-Wesley},
  year = {1981},
  series = {Encyclopedia of Mathematics and its Applications},
  volume = {16},
}

@book{Macdonald1995,
  author = {Macdonald, Ian G.},
  title = {Symmetric Functions and Hall Polynomials},
  edition = {2nd},
  publisher = {Oxford University Press},
  year = {1995},
  series = {Oxford Mathematical Monographs},
}

@book{Serre1977,
  author = {Serre, Jean-Pierre},
  title = {Linear Representations of Finite Groups},
  publisher = {Springer},
  year = {1977},
  series = {Graduate Texts in Mathematics},
  volume = {42},
}

@book{Diaconis1988wreath,
  author = {Diaconis, Persi},
  title = {Group Representations in Probability and Statistics},
  publisher = {Institute of Mathematical Statistics},
  year = {1988},
  series = {IMS Lecture Notes--Monograph Series},
  volume = {11},
}

Prerequisites

07.01.07
07.01.08
07.05.01
07.05.10

Tier anchors

beginner: James-Kerber The Representation Theory of the Symmetric Group (1981) Ch. 4 informal; the signed-permutation symmetries of a cube as a hands-on example
intermediate: Serre Linear Representations of Finite Groups Sec. 8.2 (semidirect products / little groups); James-Kerber Ch. 4
master: Specht 1932 Eine Verallgemeinerung der symmetrischen Gruppe; James-Kerber The Representation Theory of the Symmetric Group (1981) Ch. 4; Macdonald Symmetric Functions and Hall Polynomials (2nd ed.) Ch. I App. B

References

Specht, W. — Eine Verallgemeinerung der symmetrischen Gruppe · Schriften des Mathematischen Seminars der Universität Berlin 1 (1932), 1-32
James, G. and Kerber, A. — The Representation Theory of the Symmetric Group · Encyclopedia of Mathematics and its Applications 16, Addison-Wesley (1981), Ch. 4
Macdonald, I. G. — Symmetric Functions and Hall Polynomials (2nd ed.) · Oxford University Press (1995), Ch. I, Appendix B
Serre, J.-P. — Linear Representations of Finite Groups · Graduate Texts in Mathematics 42, Springer (1977), Sec. 8.2
Diaconis, P. — Group Representations in Probability and Statistics · IMS Lecture Notes-Monograph Series 11 (1988), Ch. 7

Estimated time

beginner: 18m
intermediate: 45m
master: 85m