Representations with a Unique Monomial Structure

Abstract

In this chapter, we recall the notion of monomial structures (and their isomorphism) on a representation, show a natural monomial structure on induced representations, and introduce solitary characters (characters whose induced representation has a unique monomial structure up to isomorphism); these characters may be used to detect conjugacy of subgroups. We also recall a specific type of wreath product construction and state and prove Bart de Smit’s theorem on the existence of solitary characters for these (and a follow-up result of Pintonello for characters of degree two)—these were previously formulated and used in the context of number theory, but we present them abstractly. We give an application to covering equivalence in a very specific setup of manifolds, and also count the number of required characters, based on a formula for the commutator of a wreath product.

In this chapter, we recall the notion of monomial structures (and their isomorphism) on a representation, show a natural monomial structure on induced representations, and introduce solitary characters (characters whose induced representation has a unique monomial structure up to isomorphism); these characters may be used to detect conjugacy of subgroups. We also recall a specific type of wreath product construction and state and prove Bart de Smit’s theorem on the existence of solitary characters for these (and a follow-up result of Pintonello for characters of degree two)—these were previously formulated and used in the context of number theory, but we present them abstractly. We give an application to covering equivalence in a very specific setup of manifolds, and also count the number of required characters, based on a formula for the commutator of a wreath product.

1 Monomial Structures

Definition 5.1.1

Suppose $\rho \colon G \rightarrow \operatorname {Aut}(V)$ is a representation, and

$$\displaystyle \begin{aligned} V=\bigoplus\limits_{x \in \Omega} \mathscr L_x \end{aligned}$$

is a decomposition of V  into one-dimensional spaces (“lines”) $\mathscr L_x$ for x ∈ Ω, with Ω some index set. If the action of G on V  permutes the lines $\mathscr L_x$, we say that the G-set

$$\displaystyle \begin{aligned} L=\{ \mathscr L_x \colon x \in \Omega \} \end{aligned}$$

is a monomial structure on ρ.

Equivalently, in a basis having precisely one element from each line $\mathscr L_x$, the action of any g ∈ G is given by a matrix having exactly one non-zero entry in each row and column. Note that, contrary to the case of permutation matrices, the non-zero entry in the matrix need not be 1.

An isomorphism of monomial structures L and L′ on two representation of the same group G is an isomorphism of L and L′ as G-sets.

Example 5.1.2

An induced representation $\operatorname {Ind}_H^G \chi$ of a linear character admits (by definition) a monomial structure where Ω = {g ₁, …, g _n} is such that g _i H are the different cosets of H in G, and $\mathscr L_x = \mathbf {C} \cdot xH$. The corresponding matrices have as non-zero entries n-th roots of unity if χ is a character of order n. We call this monomial structure the standard monomial structure on $\operatorname {Ind}_H^G \chi$. This standard monomial structure is isomorphic to G∕H as G-set. $\lozenge$

Definition 5.1.3

A linear character Ξ on a subgroup H of a group G is called G-solitary if $\operatorname {Ind}_H^G \Xi$ has a unique monomial structure up to isomorphism.

Lemma 5.1.4

Let G denote a group with two subgroups H ₁ and H ₂ , and suppose is a G-solitary linear character. There exists a linear character for which there is an isomorphism of representations $\operatorname {Ind}_{H_1}^G \Xi \cong \operatorname {Ind}_{H_2}^G \chi$ if and only if H ₁ and H ₂ are conjugate subgroups of G.

Proof

In this situation, $\operatorname {Ind}_{H_2}^G \chi$ carries two monomial structures: the standard one and the one induced from the standard one on $\operatorname {Ind}_{H_1}^G \Xi$ through the isomorphism of representations. Hence these monomial structures have to be isomorphic. But as G-sets, they are G∕H ₁ and G∕H ₂, respectively (see Example 5.1.2). By Proposition 3.5.1(ii), this means precisely that H ₁ and H ₂ are conjugate in G. □

2 Wreath Product Construction

Definition 5.2.1

Let G denote a finite group and H a subgroup of index n := [G : H] with cosets

$$\displaystyle \begin{aligned} \{g_1H=H,g_2H,\dots,g_nH\} \end{aligned}$$

of cardinality n. For a prime number ℓ, let C = Z∕ℓ Z denote the cyclic group with ℓ elements, and let

$$\displaystyle \begin{aligned} \widetilde G := C^n \rtimes G \end{aligned}$$

denote the wreath product; this is by definition the semidirect product where G acts on the n copies of C by permuting the coordinates in the same way as G permutes the cosets g _i H. In coordinates, this means that if we let e ₁, …, e _n denote the standard basis vectors of C ⁿ, and, as before, define the permutation i↦g(i) of {1, …, n} by gg _i H = g _g(i) H, then the semidirect product is defined by the action

$$\displaystyle \begin{aligned} G \stackrel{\Phi}{\rightarrow} \operatorname{Aut}(C^n) \colon g \mapsto \Phi(g) = \left[ \sum_{j=1}^n k_j e_j \mapsto \sum_{j=1}^n k_j e_{g(j)} \right] \end{aligned} $$

(5.1)

where k _j ∈Z∕ℓ Z. This is the (left) action of g ∈ G on C ⁿ given by

$$\displaystyle \begin{aligned} C^n \ni (k_1,\dots,k_n) \mapsto (k_{g^{-1}(1)},\dots,k_{g^{-1}(n)}) \in C^n. \end{aligned}$$

Define

$$\displaystyle \begin{aligned} \widetilde H:= C^n \rtimes H \end{aligned}$$

to be the subgroup of $\widetilde G$ corresponding to H. The cosets of $\widetilde H$ in $\widetilde G$ are of the form

$$\displaystyle \begin{aligned} \{ \widetilde g_1 \widetilde H = \widetilde H, \widetilde g_2 \widetilde H, \dots, \widetilde g_n \widetilde H \}, \end{aligned}$$

where for g _i ∈ G, we have a corresponding element $\widetilde g_i:=(0,g_i) \in \widetilde G$.

Remark 5.2.2

Recall that $\operatorname {Ind}_{H}^G \mathbf {1}$ is the Z[G]-module corresponding to the permutation representation of G acting on the G-cosets of H. Thus, if we identify C with the additive group of the finite field F _ℓ, the action of G on $C^n \cong {\mathbf {F}}_\ell ^n$ corresponds to the F _ℓ[G]-module $(\operatorname {Ind}_G^H \mathbf {1}) \otimes _{\mathbf {Z}} {\mathbf {F}}_\ell$.

Proposition 5.2.3 (Bart de Smit [28, §10])

For all ℓ ≥ 3, there exists a $\widetilde G$ -solitary character of order ℓ on $\widetilde H$.

Proof

Define Ξ by

$$\displaystyle \begin{aligned} \Xi \colon \widetilde H \rightarrow {\mathbf{C}}^* \colon (k_1,\dots,k_n,g) \mapsto e^{2 \pi i k_1/\ell}. \end{aligned} $$

(5.2)

Let $L =\{ \mathscr L_x\}$ and $L' = \{ \mathscr L^{\prime }_x\}$ denote two monomial structures on $\rho :=\operatorname {Ind}_{\widetilde H}^{\widetilde G} \Xi$, where L is the standard one (see Example 5.1.2). The action of $G \leq \widetilde G$ on L is that of G on G∕H and (after rearranging) the action of $C^n \leq \widetilde G$ is given by

$$\displaystyle \begin{aligned} (k_1,\dots,k_n)\cdot \mathscr L_j = e^{2 \pi i k_j/\ell} \cdot \mathscr L_j,\end{aligned} $$

(5.3)

where we used the simplified notation $\mathscr L_j := \mathscr L_{g_j \widetilde H}$. The character ψ of ρ can be computed using as basis any set of vectors from the lines in L or L′. From the above,

$$\displaystyle \begin{aligned} |\psi((1,0,\dots,0))|=|e^{2 \pi i/\ell}+\underbrace{1+\dots+1}_{n-1}| > n-2, \end{aligned}$$

where the last inequality is strict since ℓ ≥ 3. On the other hand, computing the same trace using a basis from L′, we get a sum of some number, say, m, of ℓ-th roots of unity, where m is the number of lines in L′ that are mapped to itself by (1, 0, …, 0). If there is a line not mapped to itself (a zero diagonal entry in the corresponding matrix), then there are at least two (since every row/column has precisely two non-zero entries), so m = n or m ≤ n − 2. In the latter case, |ψ((1, 0, …, 0))|≤ n − 2, which is impossible. Since C ⁿ is generated by G-conjugates of (1, 0, …, 0), we find that C ⁿ fixes all lines in L′. Hence L′⊆ L, but since $|L|=|L'|=[\widetilde G:\widetilde H]$, we have L = L′. □

Pintonello [80, Theorem 3.2.2] has shown that for ℓ = 2, there does not always exist a solitary character as in Proposition 5.2.3. However, he also proved the following result, of which we give a self-contained proof.

Proposition 5.2.4 (Pintonello [80, Theorem 2.3.1])

Given a group G with two subgroups H ₁ and H ₂ , consider the corresponding wreath products $\widetilde G, \widetilde H_1$ and $\widetilde H_2$ with C = Z∕2Z . Set $\Xi \colon \widetilde H_1 \rightarrow {\mathbf {C}}^* \colon (k_1,\dots ,k_n,g) \mapsto (-1)^{k_1},$ and assume that both

$$\displaystyle \begin{aligned} \$ \operatorname{Ind}^{\widetilde G}_{\widetilde H_1} \mathbf{1} \cong \operatorname{Ind}^{\widetilde G}_{\widetilde H_2} \mathbf{1} \mathit{\mbox{ and }}{} \end{aligned} $$

(5.4)

$$\displaystyle \begin{aligned} \$ \operatorname{Ind}_{\widetilde H_1}^{\widetilde G} \Xi \cong \operatorname{Ind}_{\widetilde H_2}^{\widetilde G} \chi, {} \end{aligned} $$

(5.5)

for some linear character χ on $\widetilde H_2$ . Then $\widetilde H_1$ and $\widetilde H_2$ are conjugate in G.

Proof

Equality (5.5) induces two monomial structures L ₁ and L ₂ on $\rho :=\operatorname {Ind}_{\widetilde H_1}^{\widetilde G} \Xi$, where L _i is isomorphic to $\widetilde G / \widetilde H_i$. As in (5.3), $\varepsilon :=(1,0,\dots ,0) \in C^n \leq \widetilde G$ fixes all lines in L ₁. Note that the number of lines in L _i fixed by ε is the value of the character of $\operatorname {Ind}^{\widetilde G}_{\widetilde H_i} \mathbf {1}$ at ε, given in (3.3), and by (5.4), these are equal for i = 1 and i = 2. Hence all lines in L ₂ are fixed by ε, and as in the previous proof, we conclude that C ⁿ fixes all lines in L ₂. Hence L ₂ ⊆ L ₁, but since $|L_1|=|L_2|=[\widetilde G:\widetilde H_i]$, we have L ₁ = L ₂. □

3 Application to Manifolds

We deduce the following intermediate result.

Corollary 5.3.1

Suppose we have a diagram ( 1.2 ). Let C := Z∕ℓ Z denote a cyclic group of prime order ℓ ≥ 3. Let $\widetilde G$ and $\widetilde H_1$ denote the wreath products as in Definition 5.2.1 (with H = H ₁ ) and $\widetilde H_2 := C^n \rtimes H_2$ (with the same action defined via the H ₁ -cosets), and assume that there exists a diagram of Riemannian coverings

(5.6)

Then M ₁ and M ₂ are equivalent Riemannian covers of M ₀ if and only if for a $\widetilde G$ -solitary character Ξ on $\widetilde H_1$ and for some linear character χ on $\widetilde H_2$ , the multiplicity of zero is equal in the two spectra

$$\displaystyle \begin{aligned} \sigma_{M_1}(\overline \Xi \otimes \operatorname{Res}_{\widetilde H_1}^{\widetilde G} \operatorname{Ind}_{\widetilde H_1}^{\widetilde G} \Xi) \mathit{\mbox{ and }} \sigma_{M_2}(\overline \chi \otimes \operatorname{Res}_{\widetilde H_2}^{\widetilde G} \operatorname{Ind}_{\widetilde H_1}^{\widetilde G} \Xi) \end{aligned}$$

and in the two spectra

$$\displaystyle \begin{aligned} \sigma_{M_1}(\overline \Xi \otimes \operatorname{Res}_{\widetilde H_1}^{\widetilde G} \operatorname{Ind}_{\widetilde H_2}^{\widetilde G} \chi) \mathit{\mbox{ and }} \sigma_{M_2}(\overline \chi \otimes \operatorname{Res}_{\widetilde H_2}^{\widetilde G} \operatorname{Ind}_{\widetilde H_2}^{\widetilde G} \chi). \end{aligned}$$

Proof

First of all, since ℓ ≥ 3, a $\widetilde G$-solitary character Ξ on H ₁ exists, by Proposition 5.2.3. By Proposition 4.1.1, the equalities of multiplicities of zero is equivalent to $\operatorname {Ind}_{\widetilde H_1}^{\widetilde G} \Xi \cong \operatorname {Ind}_{\widetilde H_2}^{\widetilde G} \chi$. Since Ξ is $\widetilde G$-solitary, we conclude by Lemma 5.1.4 that $\widetilde H_1$ and $\widetilde H_2$ are conjugate in $\widetilde G$. As C ⁿ is normal in $\widetilde H_2$ with quotient H ₂, we find that $\widetilde H_2 \backslash M' = H_2 \backslash M = M_2$ and hence this conjugacy defines an isometry from M ₁ to M ₂ that is the identity on M ₀. □

Since χ runs over linear characters of $\widetilde H_2$, the “less abelian” the extension is, the less spectra need to be compared. A more precise statement is the following, where we use the abelianisation $H^{\mathrm {ab}}_2$ of H ₂, defined as the quotient of H ₂ by the subgroup generated by commutators (equivalently, the largest abelian quotient of H ₂; equivalently, $H^{\mathrm {ab}}_2 \cong \operatorname {Hom}(H_2,{\mathbf {C}}^*)$). The two extremes are then: if H ₂ is abelian, $H^{\mathrm {ab}}_2$ is as large as H ₂; but if H ₂ is non-abelian simple, then $|H^{\mathrm {ab}}_2|=1$.

Proposition 5.3.2

In the setup of Corollary 5.3.1 , the dimension of the representations of which the spectra are being compared is the index [G : H ₂]. Furthermore, the number of spectral equalities to be checked in Corollary 5.3.1 by using all possible linear characters on $\widetilde H_2$ is bounded above by $2 \ell \cdot |H^{\mathrm {ab}}_2|$.

In Corollary 5.3.1 and Proposition 5.3.2, one may interchange the roles of H ₁ and H ₂, which could lead to tighter results.

Proof

The dimension of the representations we are considering, as induced representations, is the index $[\widetilde G: \widetilde H_2]=[G:H_2]$.

The spectral criterion in the proposition requires testing of 2 equalities of spectra for each linear character on $\widetilde H_2$, so there are at most $2 | \widetilde {H}^{\mathrm {ab}}_2 |$ equalities to be checked.

The commutator subgroup of a wreath product $\widetilde H_2 = C^n \rtimes H_2$ is computed in [69, Cor. 4.9], and we find that in our case, with Ω = {g ₁, …, g _n} a set of representatives for the cosets,

$$\displaystyle \begin{aligned} |[\widetilde H_2, \widetilde H_2]| = |[H_2,H_2]| \cdot |\{ f \colon \Omega \rightarrow C \colon \sum_{y \in \Omega} f(y) = 0 \}|; \end{aligned}$$

where, with |C| = ℓ, the second factor is ℓ ^{| Ω|−1}. Hence we find $|\widetilde H_2^{\mathrm {ab}}| = |H_2^{\mathrm {ab}}| \cdot \ell ,$ and the result follows. □

Remark 5.3.3

Using Proposition 4.1.1 to reformulate spectrally the extra assumption in Proposition 5.2.4 (where ℓ = 2), we find that in this case, the number of equalities to check is at most $2+4|H_2^{\mathrm {ab}}|$.

Remark 5.3.4

By Lemma 3.9.1, the multiplicity of zero in the spectrum σ _M(ρ) can be computed purely representation theoretically as the multiplicity of the trivial representation in ρ, which is in principle possible by Mackey theory (cf. Remark 4.2.6), but this would be going in reverse (from spectra to group theory instead of the other way around). Knowing the group G and its subgroups H ₁ and H ₂, Riemannian equivalence of M ₁ and M ₂ over M ₀ can be checked by a finite computation, verifying that H ₁ and H ₂ are conjugate in G. Corollary 5.3.1 translates this into a spectral statement (in the special setup where the group $\widetilde G$ is realised as indicated there).

Remark 5.3.5

One may strip all geometric analysis from the results so far, and formulate the following purely group theoretical result. Given a finite group G and two subgroups H ₁ and H ₂, then

$$\displaystyle \begin{aligned} H_1 \mbox{ and }H_2 \mbox{ are conjugate in }G \mbox{ if and only if }\operatorname{Ind}_{\widetilde H_1}^{\widetilde G} \Xi = \operatorname{Ind}_{\widetilde H_2}^{\widetilde G} \chi \end{aligned}$$

for some linear character χ on $\widetilde H_2$. Here, $\widetilde G$ denotes the wreath product corresponding to the action of G on the G-cosets of H ₁, and Ξ denotes a solitary character of order 3 on $\widetilde H_1$ (which exists by Proposition 5.2.3). The proof is immediate from Lemma 5.1.4 and the final sentence in the proof of Proposition 5.3.1. Observe that the construction of the wreath products and of Ξ is completely explicit, and the linear characters on $\widetilde H_2$ can be described in terms of those on H ₂ via the results used in the proof of Proposition 5.3.2.

In the next chapters, we study under which circumstances we have a cover as in Corollary 5.3.1, i.e., we deal with the realisation problem for the wreath product as isometry group of a cover, given an isometric free action of G on a closed manifold M. This is analogous to the inverse problem of Galois theory, realising the wreath product as Galois group of a number field. In manifolds, some condition is necessary on M for such an extension to be possible at all.