1 Introduction
The theory of zeta and
$ L $
-functions associated to graphs, originating in the seminal work of Ihara, has revealed striking analogies between the spectral theory of finite graphs and the arithmetic of global fields. Chief among these analogies is the existence of graph zeta functions, which—much like their number-theoretic counterparts—admit Euler product factorisations akin to the Artin formalism. Moreover, the special values of these zeta functions encode graph-theoretic invariants that serve as combinatorial analogues of class numbers, regulators and other arithmetic quantities. For a comprehensive introduction to the subject, the reader is referred to [Reference Terras16].
In recent years, a new direction has emerged in this field with the development of an Iwasawa-theoretic perspective on graphs. This was initiated independently by Vallières [Reference Vallières17] and Gonet [Reference Gonet7, Reference Gonet8], who introduced the notion of
$\mathbb {Z}_\ell $
-towers of finite multigraphs and established analogues of Iwasawa’s classical results on the asymptotic growth of arithmetic invariants. In this graph-theoretic setting, the complexity of a graph—measured as the cardinality of its Jacobian or sandpile group—plays the role of the class number. Along such towers, one observes analogues of the classical
$\mu $
-,
$\lambda $
- and
$\nu $
-invariants, as well as a natural graph-theoretic analogue of the Iwasawa polynomial. This new Iwasawa theory of graphs has rapidly attracted attention, leading to a series of further developments and refinements, as presented in [Reference Dion, Lei, Ray and Valliéres3, Reference DuBose and Vallières4, Reference Kleine and Müller10–Reference McGown and Vallières13, Reference Ray and Vallières15].
In [Reference Ghosh and Ray6], the Iwasawa theory of Cayley graphs associated to finite abelian groups was studied. The present work extends this framework to nonabelian groups. A key result is that the Iwasawa polynomial associated to a tower of Cayley graphs admits a canonical factorisation (see Theorem 3.2), with each factor corresponding to an irreducible representation of the underlying finite group. This observation leads naturally to the definition of a representation-theoretic Iwasawa polynomial associated to any irreducible representation of a finite group, possibly nonabelian. We study some of the properties of these representation-theoretic Iwasawa polynomials and their associated invariants, exploring their structural properties and their behaviour with respect to congruences. We illustrate our results via an illustrative example.
2 Preliminary notions
2.1 Galois theory of graphs and the Artin–Ihara L-functions
The interplay between spectral graph theory and number theory has become increasingly rich and intricate. At the heart of this interaction lies the theory of Ihara zeta functions of graphs, which are combinatorial analogues of the Dedekind zeta functions of number fields. These zeta functions encode spectral and topological data about the graph, and admit explicit determinant expressions reminiscent of the functional equations satisfied by arithmetic L-functions. These considerations will lead us to interesting new connections with the representation theory of groups. In the next section, we lay the groundwork for a systematic study of towers of graph covers and their associated
$\ell $
-adic L-functions, in the spirit of Iwasawa theory. In due course, we shall specialise the discussion to Cayley graphs associated to groups.
The graphs in this paper are finite, undirected and without loops. A graph
$\mathscr {X}$
is a quadruple
$(V_{\mathscr {X}}, E_{\mathscr {X}}^+, i, \iota )$
, where
$V_{\mathscr {X}} = {v_1, \ldots , v_n}$
is the vertex set,
$E_{\mathscr {X}}^+$
the set of oriented edges,
$i: E_{\mathscr {X}}^+ \to V_{\mathscr {X}} \times V_{\mathscr {X}}$
the incidence map, and
$\iota : E_{\mathscr {X}}^+ \to E_{\mathscr {X}}^+$
the edge inversion satisfying
$i \circ \iota = \tau \circ i$
, with
$\tau (v, v') = (v', v)$
. For convenience, we shall occasionally write
$\bar {e}:=\iota (e)$
. An edge
$e \in E_{\mathscr {X}}^+$
joins v to
$v'$
when
$i(e) = (v, v')$
, and
$\iota (e)$
joins
$v'$
to v. We write
$e \sim e'$
if
$e' = \iota (e)$
, and denote by
$E_{\mathscr {X}}$
the set of equivalence classes under this relation. Thus,
$E_{\mathscr {X}}^+$
represents directed edges and
$E_{\mathscr {X}}$
the corresponding undirected ones. The natural projection
$\pi : E_{\mathscr {X}}^+ \to E_{\mathscr {X}}$
sends e to its class. Define the incidence matrix
$A_{\mathscr {X}}=(a_{i,j})$
of the graph
$\mathscr {X}$
, where
$a_{i,j}$
is the number of edges from
$v_i$
to
$v_j$
. The source and target maps
$o, t: E_{\mathscr {X}}^+\rightarrow V_{\mathscr {X}}$
are the compositions of i with the projections to the first and second factor of
$V_{\mathscr {X}}\times V_{\mathscr {X}}$
, respectively. The degree of v is the number of edges emanating from v, that is,
$\mathrm {deg}(v):=\# E_{\mathscr {X},v}^+$
. The Betti numbers of
$\mathscr {X}$
are defined by
$$ \begin{align*}b_i(\mathscr{X}):=\mathrm{rank}_{\mathbb{Z}} H_i(\mathscr{X},\mathbb{Z}).\end{align*} $$
The Euler characteristic is defined by
$\chi (\mathscr {X}):=b_0(\mathscr {X})-b_1(\mathscr {X})$
. When
$\mathscr {X}$
is connected,
$b_0(\mathscr {X})=1$
and
$b_1(\mathscr {X})=\# E_{\mathscr {X}}-\# V_{\mathscr {X}}+1$
, so that
$\chi (\mathscr {X})=\# V_{\mathscr {X}}-\# E_{\mathscr {X}}$
. It is assumed throughout that all multigraphs are connected, with no vertices having degree equal to
$1$
. Moreover, we assume that
$\chi (\mathscr {X})\neq 0$
, that is, the graph is not a cycle graph.
A graph can be viewed, to some extent, as a discrete analogue of a Riemann surface. For instance, there are graph-theoretic analogues of a Jacobian and the Riemann–Roch theorem [Reference Baker and Norine1].
The divisor group
$\mathrm {Div}(\mathscr {X})$
is the free abelian group on the vertices
$V_{\mathscr {X}}$
, consisting of formal sums
$D = \sum _v n_v v$
with
$n_v \in \mathbb {Z}$
. The degree map
$\deg : \mathrm {Div}(\mathscr {X}) \to \mathbb {Z}$
, given by
$\deg (D) = \sum _v n_v$
, has kernel
$\mathrm {Div}^0(\mathscr {X})$
. Let
$\mathcal {M}(\mathscr {X})$
be the group of
$\mathbb {Z}$
-valued functions on
$V_{\mathscr {X}}$
, freely generated by the characteristic functions
$\chi _v$
. The map
$$ \begin{align*}\mathrm{div}: \mathcal{M}(\mathscr{X}) \to \mathrm{Div}^0(\mathscr{X})\end{align*} $$
is defined by setting
$\mathrm { div}(\chi _v) = \sum _w \rho _w(v) w$
, where
$$ \begin{align*} \rho_w(v) = \begin{cases} \mathrm{val}_{\mathscr{X}}(v) - 2 \cdot (\text{number of loops at } v) & \text{if } w = v ,\\ - \text{ number of edges from } w \text{ to } v & \text{if } w \ne v. \end{cases} \end{align*} $$
For
$f \in \mathcal {M}(\mathscr {X})$
, extending linearly, we obtain the formula
$\mathrm {div}(f) = -\sum _v m_v(f) \cdot v$
, where
$m_v(f) := \sum _{e \in E_{\mathscr {X},v}^+} (f(t(e)) - f(o(e)))$
. The image
$\mathrm {Pr}(\mathscr {X})$
of
$\mathrm {div}$
is the group of principal divisors, and the quotient
$\mathrm {Pic}^0(\mathscr {X}) := \mathrm {Div}^0(\mathscr {X})/\mathrm {Pr}(\mathscr {X})$
is the Jacobian of
$\mathscr {X}$
. Its cardinality
$\kappa _{\mathscr {X}} := \#\mathrm { Pic}^0(\mathscr {X})$
is called the complexity of
$\mathscr {X}$
, analogous to the class number of a number ring (see [Reference Corry and Perkinson2]).
A morphism of graphs
$f: \mathscr {Y} \to \mathscr {X}$
consists of functions
$f_V: V_{\mathscr {Y}} \to V_{\mathscr {X}}$
and
$f_E: E_{\mathscr {Y}}^+ \to E_{\mathscr {X}}^+$
such that
$$ \begin{align*} f_V(o(e)) = o(f_E(e)), \quad f_V(t(e)) = t(f_E(e)), \quad f_E(\iota(e)) = \iota(f_E(e)). \end{align*} $$
It is a cover if
$f_V$
is surjective and, for each
$w \in V_{\mathscr {Y}}$
, the map
$f: E_{\mathscr {Y},w}^+ \to E_{\mathscr {X},f(w)}^+$
is a bijection. A cover is Galois if
$\mathscr {Y}$
and
$\mathscr {X}$
are connected and the group
$\mathrm {Aut}_f(\mathscr {Y}/\mathscr {X})$
acts transitively on each fibre
$f^{-1}(v)$
. We write
$\mathrm {Gal}(\mathscr {Y}/\mathscr {X}) := \mathrm {Aut}_f(\mathscr {Y}/\mathscr {X})$
.
To define Artin–Ihara L-functions, let
$\mathfrak {c} = a_1 \cdots a_k$
be a walk in
$\mathscr {X}$
, where
${t(a_i) = o(a_{i+1})}$
. Such a walk is a cycle if
$o(a_1) = t(a_k)$
, and is prime if it has no backtracks or tails and is not a nontrivial power of a shorter cycle. For a Galois cover
$\mathscr {Y}/\mathscr {X}$
with abelian Galois group G and character
$\psi \in \widehat {G} := \mathrm {Hom}(G, \mathbb {C}^\times )$
, the Artin–Ihara L-function is defined by
$$ \begin{align*} L_{\mathscr{Y}/\mathscr{X}}(u, \psi) := \prod_{\mathfrak{c}} \bigg(1 - \psi\bigg(\bigg(\frac{\mathscr{Y}/\mathscr{X}}{\mathfrak{c}}\bigg)\bigg) u^{l(\mathfrak{c})} \bigg)^{-1}, \end{align*} $$
where the product runs over all prime cycles
$\mathfrak {c}$
in
$\mathscr {X}$
and
$\big(\frac{\mathscr{Y}/\mathscr{X}}{\mathfrak{c}}\big)$
denotes the Frobenius automorphism associated to
$\mathfrak {c}$
(see [Reference Terras16, Definition 16.1]). The special case
$\psi = 1$
and
$\mathscr {Y} = \mathscr {X}$
recovers the Ihara zeta function
$\zeta _{\mathscr {X}}(u)$
.
Consider an abelian cover
$ \mathscr {Y} \to \mathscr {X} $
with Galois group
$ G = \mathrm {Aut}(\mathscr {Y}/\mathscr {X}) $
. For each
$ i = 1, \ldots , g_{\mathscr {X}} $
, fix a vertex
$ w_i $
in the fibre above
$ v_i \in V_{\mathscr {X}} $
. For
$ \sigma \in G $
, define the matrix
$ A(\sigma ) = (a_{i,j}(\sigma )) $
by
$$ \begin{align*} a_{i,j}(\sigma) = \begin{cases} 2 \cdot \text{(number of loops at } w_i) & \text{if } i = j \text{ and } \sigma = 1 ,\\ \text{number of edges from } w_i \text{ to } w_j^\sigma & \text{otherwise}. \end{cases} \end{align*} $$
For each character
$ \psi \in \widehat {G} $
, define the twisted adjacency matrix
$$ \begin{align*} A_\psi := \sum_{\sigma \in G} \psi(\sigma) A(\sigma). \end{align*} $$
Let
$ D = \mathrm {diag}(\deg (v_1), \ldots , \deg (v_{g_{\mathscr {X}}})) $
. Then, the Artin–Ihara L-function is given by
$$ \begin{align*} L_{\mathscr{Y}/\mathscr{X}}(u, \psi)^{-1} = (1 - u^2)^{-\chi(\mathscr{X})} \cdot \det(I - A_\psi u + (D - I) u^2) \end{align*} $$
(see [Reference Terras16, Theorem 18.15]). Set
$$ \begin{align*} h_{\mathscr{X}}(u, \psi) := \det(I - A_\psi u + (D - I) u^2), \quad h_{\mathscr{X}}(u) := h_{\mathscr{X}}(u, 1). \end{align*} $$
The next result links the derivative of
$ h_{\mathscr {X}} $
at 1 to the complexity
$ \kappa _{\mathscr {X}} $
of the graph.
Theorem 2.1 (See [Reference Hammer, Mattman, Sands and Vallières9, Reference Northshield14])
Assume that
$\mathscr {X} $
is connected and
$\chi (\mathscr {X})\neq 0$
. Then
$h_{\mathscr {X}}'(1) = -2\chi (\mathscr {X})\kappa _{\mathscr {X}}$
.
This result is strikingly parallel to classical class number formulas in number theory, where the zeta function of an extension factors over characters of the Galois group, and the special values encode arithmetic invariants such as regulators and class numbers. Artin formalism gives a factorisation of zeta functions of covers.
Theorem 2.2 (See [Reference Terras16])
If
$ \mathscr {Y} \to \mathscr {X} $
is an abelian Galois cover with group
$ G $
, then
$$ \begin{align*} \zeta_{\mathscr{Y}}(u) = \zeta_{\mathscr{X}}(u) \cdot \prod_{\substack{\psi \in \widehat{G} \\ \psi \neq 1}} L_{\mathscr{Y}/\mathscr{X}}(u, \psi). \end{align*} $$
Evaluating at
$ u = 1 $
gives a relation between complexities.
Corollary 2.3. Under the same assumptions,
$$ \begin{align*} |G|\kappa_{\mathscr{Y}} = \kappa_{\mathscr{X}} \prod_{\substack{\psi \in \widehat{G} \\ \psi \neq 1}} h_{\mathscr{X}}(1, \psi). \end{align*} $$
This result implies in particular that
$ h_{\mathscr {X}}(1, \psi ) \neq 0 $
for nontrivial
$ \psi \in \widehat {G} $
.
2.2 Iwasawa theory of graphs
In this section, we discuss the Iwasawa theory of
$\mathbb {Z}_\ell $
-towers over a connected graph
$\mathscr {X}$
for which it is assumed throughout that
${\chi (\mathscr {X})\neq 0}$
. We begin by explaining how certain Galois covers of
$\mathscr {X}$
may be constructed from combinatorial data known as voltage assignments. Let
$\mathscr {X}$
be a graph and let
$\pi : E_{\mathscr {X}}^+ \to E_{\mathscr {X}}$
denote the natural projection from the set of oriented edges to the set of unoriented edges, which associates to each oriented edge its underlying unoriented edge. Fix a section
$\gamma : E_{\mathscr {X}} \to E_{\mathscr {X}}^+$
of
$\pi $
, so that each unoriented edge is assigned a distinguished orientation. Setting
$S:=\gamma (E_{\mathscr {X}})$
, a voltage assignment is a function
$\alpha :S \to G$
. We extend the voltage assignment
$\alpha $
to all of
$E_{\mathscr {X}}^+$
by declaring
$\alpha (\bar {e}) = \alpha (e)^{-1}$
for every
$e \in E_{\mathscr {X}}^+$
. Given this data, one constructs a multigraph
$\mathscr {X}(G, S, \alpha )$
as follows. The vertex set is
$V = V_{\mathscr {X}} \times G$
, and the set of directed edges is
$E^+ = E_{\mathscr {X}}^+ \times G$
. Each directed edge
$(e, \sigma ) \in E^+$
connects the vertex
$(o(e), \sigma )$
to the vertex
$(t(e), \sigma \cdot \alpha (e))$
, where
$o(e)$
and
$t(e)$
denote the origin and target of the edge e, respectively. The edge-reversal map is defined by
$\overline {(e, \sigma )} = (\bar {e}, \sigma \cdot \alpha (e))$
.
Now suppose
$G_1$
is another finite abelian group and let
$f: G \to G_1$
be a group homomorphism. Then f induces a morphism of multigraphs
$$ \begin{align*} f_* : \mathscr{X}(G, S, \alpha) \to \mathscr{X}(G_1, S, f \circ \alpha), \end{align*} $$
defined on vertices and edges by
$$ \begin{align*} f_*(v, \sigma) = (v, f(\sigma)) \quad \text{and} \quad f_*(e, \sigma) = (e, f(\sigma)). \end{align*} $$
Definition 2.4. Let
$\ell $
be a prime and let
$ \mathscr {X} $
be a connected graph. A
$ \mathbb {Z}_\ell $
-tower over
$ \mathscr {X} $
is a sequence of connected graph covers
$$ \begin{align*} \mathscr{X} = \mathscr{X}_0 \longleftarrow \mathscr{X}_1 \longleftarrow \mathscr{X}_2 \longleftarrow \cdots \end{align*} $$
such that for each
$ n \geq 1 $
, the composite cover
$ \mathscr {X}_n \to \mathscr {X} $
is Galois with Galois group isomorphic to
$ \mathbb {Z}/\ell ^n\mathbb {Z} $
.
We now describe a natural way to construct such towers using voltage assignments. Fix a finite set
$ S $
of oriented edges of the base graph
$ \mathscr {X} $
and let
$$ \begin{align*} \alpha = (\alpha_1, \alpha_2, \ldots, \alpha_t) \in \mathbb{Z}_\ell^t, \end{align*} $$
where
$ t = |S| $
, and each
$ \alpha _i = \alpha (s_i) $
for a chosen enumeration
$ S = \{s_1, \ldots , s_t\} $
. The map
$ \alpha $
may be interpreted as a continuous homomorphism from the free abelian group on
$ S $
into
$ \mathbb {Z}_\ell $
, that is, a
$\mathbb {Z}_\ell $
-valued voltage assignment.
For each
$ n \geq 1 $
, let
$ \alpha _{/n} $
denote the reduction of
$ \alpha $
modulo
$ \ell ^n $
, taking values in
$ \mathbb {Z}/\ell ^n\mathbb {Z} $
. Applying the voltage graph construction to
$ \alpha _{/n} $
, we obtain a sequence of finite Galois covers
$$ \begin{align*} \mathscr{X}(\mathbb{Z}/\ell^n\mathbb{Z}, S, \alpha_{/n}) \longrightarrow \mathscr{X}. \end{align*} $$
These fit into a tower:
$$ \begin{align*} \mathscr{X} \longleftarrow \mathscr{X}(\mathbb{Z}/\ell\mathbb{Z}, S, \alpha_{/1}) \longleftarrow \mathscr{X}(\mathbb{Z}/\ell^2\mathbb{Z}, S, \alpha_{/2}) \longleftarrow \cdots, \end{align*} $$
which defines a
$ \mathbb {Z}_\ell $
-tower over
$ \mathscr {X} $
in the sense of Definition 2.4.
In the following discussion, we assume that the multigraphs
$ \mathscr {X}(\mathbb {Z}/\ell ^n\mathbb {Z}, S, \alpha _{/n}) $
are connected for all
$ n \geq 0 $
. An explicit condition ensuring the connectedness of such graphs can be described in terms of the fundamental group. Given a walk
${ w = a_1 a_2 \cdots a_n }$
in
$ \mathscr {X} $
, we define the product
$ \alpha (w) := \alpha (e_1) \cdots \alpha (e_n) \in G $
, where
$ {\alpha : S \to G} $
satisfies
$ \alpha (\iota (e)) = \alpha (e)^{-1} $
. It follows that homotopically equivalent walks
$ c_1 $
and
$ c_2 $
have equal image under
$ \alpha $
. Fixing a base vertex
$ v_0 \in V_{\mathscr {X}} $
, the map
$ \alpha $
induces a group homomorphism
$ \rho _\alpha : \pi _1(\mathscr {X}, v_0) \to G $
defined by
$ \rho _\alpha ([\gamma ]) = \alpha (\gamma ) $
. When
$ \mathscr {X} $
is connected, the derived graph
$ \mathscr {X}(G, S, \alpha ) $
is connected if and only if
$ \rho _\alpha $
is surjective; this equivalence is established in [Reference Ray and Vallières15, Theorem 2.11].
Now suppose
$ \mathscr {X} $
is a connected graph with vertex set
$ \{v_1, \ldots , v_{g_{\mathscr {X}}}\} $
. Define the matrix
$ D_{\mathscr {X}} = (d_{i,j}) $
where
$ d_{i,j} = \deg (v_i) $
if
$ i = j $
and
$ 0 $
otherwise. The Laplacian matrix is
$ Q_{\mathscr {X}} := D_{\mathscr {X}} - A_{\mathscr {X}} $
, with
$ A_{\mathscr {X}} $
the adjacency matrix. Let
$ \alpha : S \to \mathbb {Z}_\ell $
be a voltage assignment satisfying
$ \alpha (\iota (e)) = -\alpha (e) $
. This extends to a matrix
$$ \begin{align*}M(x) = M_{\mathscr{X},\alpha}(x) \in \mathbb{Z}_\ell[x;\mathbb{Z}_\ell]^{g_{\mathscr{X}} \times g_{\mathscr{X}}} ,\end{align*} $$
defined by subtracting from
$ D_{\mathscr {X}} $
the matrix whose
$(i,j)$
th entry is
$$ \begin{align*}\sum_{e \in E_{\mathscr{X}}^+,\,i(e) = (v_i, v_j)} x^{\alpha(e)}.\end{align*} $$
Here,
$ \mathbb {Z}_\ell [x; \mathbb {Z}_\ell ] $
consists of expressions
$ \sum _a c_a x^a $
with
$ a \in \mathbb {Z}_\ell $
and
$ c_a \in \mathbb {Z}_\ell $
. The Iwasawa polynomial associated to the tower defined by
$ \alpha $
is 
. Though not necessarily a polynomial, this formal power series becomes a polynomial after multiplying by a suitable power of
$ (1+T) $
. For the tower of derived graphs
$$ \begin{align*} \mathscr{X} \leftarrow \mathscr{X}(\mathbb{Z}/\ell\mathbb{Z}, S, \alpha_{/1}) \leftarrow \mathscr{X}(\mathbb{Z}/\ell^2\mathbb{Z}, S, \alpha_{/2}) \leftarrow \ldots, \end{align*} $$
we have the evaluation
$ f_{\mathscr {X},\alpha }(1 - \zeta _{\ell ^n}) = h_{\mathscr {X}}(1, \psi _n) $
for any primitive
$ \ell ^n $
th root of unity
$ \zeta _{\ell ^n} $
and character
$ \psi _n : \mathbb {Z}/\ell ^n\mathbb {Z} \to \mathbb {C} $
defined by
$ \psi _n(\bar {1}) = \zeta _{\ell ^n} $
, as shown in [Reference McGown and Vallières13, Corollary 5.6]. Since
$ Q_{\mathscr {X}} $
is singular with
$ u = (1, 1, \ldots , 1)^t $
in its kernel, we deduce that
$ f_{\mathscr {X},\alpha }(0) = \det Q_{\mathscr {X}} = 0 $
, and thus
$ T $
divides
$ f_{\mathscr {X},\alpha }(T) $
. Consequently,
$$ \begin{align*}f_{\mathscr{X},\alpha}(T) = T g_{\mathscr{X},\alpha}(T),\end{align*} $$
where 
is a power series and
$ m \in \mathbb {Z}_{\geq 0} $
is minimal such that
$ g_{\mathscr {X},\alpha }(T) $
becomes a polynomial. By the
$ \ell $
-adic Weierstrass preparation theorem, there exists a factorisation
$ g_{\mathscr {X},\alpha }(T) = \ell ^\mu P(T) u(T) $
, where
$ P(T) \in \mathbb {Z}_\ell [T] $
is a distinguished polynomial and 
is a unit, that is,
$ u(0) \in \mathbb {Z}_\ell ^\times $
. The Iwasawa invariants associated to the tower are defined by
$ \mu _\ell (\mathscr {X},\alpha ) := \mu $
and
$ \lambda _\ell (\mathscr {X},\alpha ) := \deg P(T) $
. Finally, a powerful result of Gonet [Reference Gonet7, Reference Gonet8], Vallières [Reference Vallières17] and McGown and Vallières [Reference McGown and Vallières12, Reference McGown and Vallières13] states that if
$ \alpha $
is a voltage assignment satisfying the assumptions above (including our connectivity assumption), then for
$ n \gg 0 $
, the complexity
$ \kappa _\ell (X_n) $
of the derived graph
$ X_n := \mathscr {X}(\mathbb {Z}/\ell ^n\mathbb {Z}, S, \alpha _{/n}) $
satisfies the formula
$$ \begin{align*}\kappa_\ell(X_n) = \ell^{\ell^n \mu + n \lambda + \nu} \end{align*} $$
for some integer
$ \nu $
, as proven in [Reference McGown and Vallières13, Theorem 6.1].
3 Factorisation of the Iwasawa polynomial
In this section, G is a finite group and S is a subset of G such that:
-
•
$g S g^{-1}=S$
for all
$g\in G$
; -
• S generates G;
-
•
$S=S^{-1}$
; -
•
$1\notin S$
.
Let
$\mathscr {X}$
be the Cayley graph
$\mathrm {Cay}(G, S)$
associated with the pair
$(G, S)$
and assume throughout that
$\chi (\mathscr {X})\neq 0$
, that is,
$\mathscr {X}$
is not a cycle graph. Enumerate
$G=\{g_1, \ldots , g_n\}$
and write
$V_{\mathscr {X}}=\{v_1, \ldots , v_n\}$
where
$v_i=v_{g_i}$
is the vertex associated to
$g_i$
. Set
$r:=\# S$
. There is an edge
$e_{i,j}$
joining
$v_i$
to
$v_j$
if
$g_i g_j^{-1}\in S$
. Note that since
$S=S^{-1}$
,
$\mathscr {X}$
is an undirected graph, and since
$1\notin S$
,
$\mathscr {X}$
has no loops. Since S generates G, it follows that there is a walk from
$1$
to any other vertex in
$V_{\mathscr {X}}$
, thus,
$\mathscr {X}$
is connected. There is a natural action of G on
$\mathscr {X}$
, that is, a natural group homomorphism
$$ \begin{align*}\rho: G\rightarrow \mathrm{Aut}(\mathscr{X})\end{align*} $$
where
$g\in G$
sends
$v_{h}$
to
$v_{gh}$
and the edge e joining
$v_i$
to
$v_j$
to the edge
$g(e)$
, which joins
$g(v_i)$
to
$g(v_j)$
. This action is well defined since S is stable with respect to conjugation.
Definition 3.1. We shall consider voltage assignments that arise from functions on S. Let
$\ell $
be a prime number and
$\beta : S\rightarrow \mathbb {Z}_\ell $
be a function such that:
-
(1)
$\beta (g a g^{-1})=\beta (a)$
; -
(2) the image of
$\beta $
generates
$\mathbb {Z}_{\ell }$
(as a
$\mathbb {Z}_\ell $
-module); -
(3)
$\beta (s^{-1})=-\beta (s)$
and
$\beta (1_G)=0$
; -
(4) the image of
$\beta $
lies in
$\mathbb {Z}$
; -
(5) there exist
$m>0$
and a tuple
$(h_1, \ldots , h_m)\in S^m$
such that
$h_1 h_2 \cdots h_m\in S$
and
$$ \begin{align*} \beta(h_1 h_2 \cdots h_m)\not \equiv \sum_{i=1}^m \beta(h_i)\pmod{\ell}. \end{align*} $$
We define a
$\mathbb {Z}_\ell $
-valued voltage assignment
$\alpha =\alpha _\beta : E_{\mathscr {X}}^+\rightarrow \mathbb {Z}_\ell $
by
$\alpha (e):=\beta (g_1 g_2^{-1})$
where e is the edge joining
$v_{g_1}$
to
$v_{g_2}$
.
We choose an ordering and write
$G=\{g_1, \ldots , g_n\}$
and set
$v_i:=v_{g_i}$
. For
$g\in G$
, set
$$ \begin{align*}\delta_S(g):=\begin{cases} 1 &\text{ if }g\in S ,\\ 0 &\text{ if }g\notin S. \end{cases}\end{align*} $$
Recall that a voltage assignment
$\alpha : E_{\mathscr {X}}^+\rightarrow \mathbb {Z}_\ell $
gives rise to a
$\mathbb {Z}_\ell $
-tower over
$\mathscr {X}$
. It follows from [Reference Ghosh and Ray6, Proposition 4.3] that this tower consists of connected graphs. Further,
$$ \begin{align*} f_{\mathscr{X}, \alpha}(T)=\mathrm{det}(\mathrm{M}_{\mathscr{X}, \alpha}(1+T))=\mathrm{det}(r- \delta_S(g_i g_j^{-1})(1+T)^{\beta(g_i g_j^{-1})})_{i,j} \end{align*} $$
is the associated Iwasawa polynomial.
Choose an embedding of
$\overline{\mathbb{Q}} \hookrightarrow \overline{\mathbb{Q}}_\ell $
and let
$\mathcal {K}$
be a finite extension of
$\mathbb {Q}_\ell $
which contains all nth roots of unity. Let
$\mathcal {O}$
denote the valuation ring of
$\mathcal {K}$
,
$\varpi $
be its uniformiser and
$\kappa :=\mathcal {O}/(\varpi )$
the residue field. Let
$F:=\mathcal {K}((T))$
and
$A:=\overline {F}[G]$
be the group algebra of G over
$\overline {F}$
. Denote by
$\mathrm {Irr}(G)$
the set of irreducible characters
$\chi : G\rightarrow \overline {\mathbb {Q}}_\ell $
. We note that such characters take values in
$\mathcal {O}$
, since they are expressible as sums of nth roots of
$1$
. For
$\chi \in \mathrm { Irr}(G)$
, set
and
$P_{\chi }(T):={Q_\chi (T)}/{\chi (1)}$
.
Theorem 3.2. Let
$\mathscr {X}$
be a Cayley graph associated to the pair
$(G,S)$
and
$\beta $
satisfy the conditions of Definition 3.1. Then there is a factorisation
$$ \begin{align*}f_{\mathscr{X}, \alpha}(T)=\prod_{\chi\in \mathrm{Irr}(G)} P_\chi(T)^{\chi(1)^2}.\end{align*} $$
Proof. For each
$ a \in A $
, define a right multiplication operator
$ \rho _a: A \to A $
by
$$ \begin{align*} \rho_a(g) := g a \quad \text{for } g \in G. \end{align*} $$
We define the adjacency operator
$ \mathrm {ad}: A \to A $
by
$$ \begin{align*} \mathrm{ad}(g) := \sum_{t \in S} x^{\beta(t)} \, t g, \end{align*} $$
where
$ S \subseteq G $
is a fixed subset and
$ \beta : S \to \mathbb {Z} $
is a weight function. Evaluating this at the identity element
$ 1 \in G $
, we obtain
$$ \begin{align*} z := \mathrm{ad}(1) = \sum_{t \in S} x^{\beta(t)} t. \end{align*} $$
This element
$ z \in A $
lies in the centre of
$ A $
, and since
$ \mathrm {ad} $
is defined via left multiplication by
$ z $
, we deduce that
$\mathrm {ad} = \rho _z$
.
Decompose the semisimple algebra
$ A $
as a direct sum of simple two-sided ideals:
$$ \begin{align*} A = A_1 \oplus \cdots \oplus A_s. \end{align*} $$
Since
$ z $
is central, it acts on each simple ideal
$ A_i $
by multiplication by a scalar
$ \lambda _i $
. Let
$ e_i \in A_i $
denote the identity element of the ideal
$ A_i $
, viewed as a central idempotent of
$ A $
. Then we may write
$$ \begin{align*} z = \sum_{i=1}^s \lambda_i e_i. \end{align*} $$
It follows that the eigenvalues of
$ \mathrm {ad} $
(viewed as a linear operator on
$ A $
) are exactly the
$ \lambda _i $
and each
$ \lambda _i $
occurs with multiplicity
$ \dim A_i $
. In fact,
$A_i$
can be identified with the endomorphisms of an irreducible representation of G. This is well known over
$\mathbb {C}$
(see [Reference Farb and Dennis5, page 166]), however, the argument applies verbatim to any algebraically closed field of characteristic zero.
Let
$ \chi _j $
denote the irreducible character of
$ A $
associated to the ideal
$ A_j $
. Then evaluating
$ \chi _j $
on
$ z $
in two different ways gives
$$ \begin{align*} \chi_j(z) = \sum_{t \in S} x^{\beta(t)} \chi_j(t), \end{align*} $$
and also, since
$ \chi _j(e_i) = 0 $
for
$ i \ne j $
and
$ \chi _j(e_j) = \chi _j(1) $
,
$$ \begin{align*} \chi_j(z) = \sum_{i=1}^s \lambda_i \chi_j(e_i) = \lambda_j \chi_j(1). \end{align*} $$
Combining these expressions yields an explicit formula for
$ \lambda _j $
:
$$ \begin{align*} \lambda_j = \frac{\sum_{t \in S} x^{\beta(t)} \chi_j(t)}{\chi_j(1)}. \end{align*} $$
Therefore,
$$ \begin{align*} \begin{aligned} f_{\mathscr{X}, \alpha}(T) = \mathrm{det}(r\cdot \mathrm{Id}-\mathrm{ad})=\prod_{\chi\in \mathrm{Irr}(G)} \mathrm{ det}(r\cdot \mathrm{Id}-\lambda_j)^{\chi_j(1)^2}=\prod_{\chi\in \mathrm{Irr}(G)} P_\chi(T)^{\chi(1)^2}. \end{aligned}\\[-47pt] \end{align*} $$
The preceding result motivates the definition of an Iwasawa polynomial associated to an irreducible representation of
$ G $
.
Definition 3.3. Let
$ \rho : G \to \mathrm {GL}_d(\overline {\mathbb {Q}}_\ell ) $
be an irreducible representation of
$ G $
, and let
$ \chi = \mathrm{tr}\,\rho $
denote its character. Define the element
$$ \begin{align*} P_{\chi}(T) := \frac{Q_{\chi}(T) }{\chi(1)}=\frac{r\chi(1) - \sum_{t \in S} (1 + T)^{\beta(t)} \chi(t)}{\chi(1)}. \end{align*} $$
We call
$ P_{\chi }(T) $
the Iwasawa function attached to
$ \chi $
. It admits a factorisation of the form
$$ \begin{align*} Q_{\chi}(T) = \varpi^{\mu} f(T) u(T), \end{align*} $$
where
$ \mu \in \mathbb {Z} $
,
$ f(T) \in \mathcal {O}[T] $
is a distinguished polynomial and 
is a unit. We define
$$ \begin{align*} \mu_\chi := \mu \quad \text{and} \quad \lambda_\chi := \deg f(T), \end{align*} $$
and refer to
$ \mu _\chi $
and
$ \lambda _\chi $
as the
$\mu $
-invariant and
$\lambda $
-invariant, respectively, associated to
$ \mathscr {X} $
,
$ \chi $
and the function
$ \beta : S \to \mathbb {Z}_\ell $
.
When
$\chi = 1$
is the character of the trivial representation, we obtain
$$ \begin{align*} P_1(T) = r - \sum_{t \in S} (1 + T)^{\beta(t)}. \end{align*} $$
We decompose the set S as a disjoint union
$$ \begin{align*} S = X \sqcup X^{-1} \sqcup X', \end{align*} $$
where
$X = \{h_1, \ldots , h_k\}$
consists of elements
$h_i$
satisfying
$h_i^2 \neq 1$
, and
$X' = \{h_{k+1}, \ldots , h_m\}$
consists of involutions, that is, elements
$h_i$
with
$h_i^2 = 1$
. We set
$\beta _i:=\beta (h_i)$
. Note that
$\beta (h_i^{-1})=-\beta (h_i)$
and thus if
$h_i^2=1$
, then
$\beta _i=\beta (h_i)=0$
. Assume without loss of generality that
$\beta _i\geq 0$
for
$i\leq k$
.
Lemma 3.4. With respect to the notation as above, T divides
$P_1(T)$
.
Proof. Observe that
$$ \begin{align*} P_1(T)= \sum_{i=1}^k(2-(1+T)^{\beta_i}-(1+T)^{-\beta_i}) = -\sum_{i=1}^k (1+T)^{-\beta_i}((1+T)^{\beta_i}-1)^2. \end{align*} $$
Thus, we see that T divides
$P_1(T)$
.
Proposition 3.5. With respect to the notation as above,
$$ \begin{align*} \mu_\ell(\mathscr{X}, \alpha)=\frac{1}{e}\sum_{\chi\in \mathrm{Irr}(G)} \chi(1)^2\mu_\chi \quad\text{and}\quad \lambda_\ell(\mathscr{X}, \alpha)=\sum_{\chi\in \mathrm{Irr}(G)} \chi(1)^2\lambda_\chi -1, \end{align*} $$
where
$(\ell )=(\varpi ^e)$
as ideals in
$\mathcal {O}$
.
Proof. Recall that
$f_{\mathscr {X},\alpha }(T) = T g_{\mathscr {X},\alpha }(T)$
and therefore, by Theorem 3.2,
$$ \begin{align*}g_{\mathscr{X}, \alpha}(T)=P_1(T)/T\times \prod_{1\neq \chi\in \mathrm{Irr}(G)} P_\chi(T)^{\chi(1)^2},\end{align*} $$
from which the result follows easily.
Next, we study the
$\mu $
- and
$\lambda $
-invariants of a character
$\chi \in \mathrm {Irr}(G)$
. The first observation concerns a formula for the constant coefficient of
$P_\chi (T)$
.
Lemma 3.6. Let
$\chi \in \mathrm {Irr}(G)$
and assume for simplicity that
$\ell \nmid \chi (1)$
. The following assertions hold.
-
(1) If
$\chi \neq 1$
, then and
$$ \begin{align*} Q_\chi(0)= \sum_{t\in S}(\chi(1)-\chi(t)), \end{align*} $$
$\ell \nmid Q_\chi (0)$
if and only if
$\mu _\chi =0$
and
$\lambda _\chi =0$
.
-
(2) If
$\chi =1$
, then and
$$ \begin{align*}Q_1'(0)=-\sum_{i=1}^k \beta_i^2, \end{align*} $$
$\ell \nmid Q_1'(0)$
if and only if
$\mu _1=0$
and
$\lambda _1=1$
.
Proof. First, suppose that
$\chi $
is nontrivial. In this case, the computation of
$Q_\chi (0)$
is straightforward and left to the reader. Note that
$\ell \nmid Q_\chi (0)$
if and only if
$Q_\chi (T)$
is a unit in 
and this condition is equivalent to
$\mu _\chi =0$
and
$\lambda _\chi =0$
. Next, consider the case when
$\chi =1$
. Since T divides
$Q_1(T)$
, it follows that
$\lambda _1\geq 1$
. Note that
$$ \begin{align*}Q_1(T)=P_1(T)= -\sum_{i=1}^k (1+T)^{-\beta_i}((1+T)^{\beta_i}-1)^2\end{align*} $$
and therefore
$Q_1'(0)=-\sum _{i=1}^k \beta _i^2$
. From the Weierstrass preparation theorem, it is easy to see that
$\mu _\chi =0$
and
$\lambda _\chi =1$
if and only if
$\ell \nmid Q_1'(0)$
.
Definition 3.7. Let
$\rho _1, \rho _2: G \to \mathrm {GL}_n(\mathcal {O})$
be irreducible representations, and let
$\bar {\rho }_i: G \to \mathrm {GL}_n(\kappa )$
denote the reductions of
$\rho _i$
modulo
$(\varpi )$
. We say that
$\rho _1$
and
$\rho _2$
are congruent if
$\bar {\rho }_1 \simeq \bar {\rho }_2$
. Similarly, two characters
$\chi _1$
and
$\chi _2$
are said to be congruent if
$\chi _1 \equiv \chi _2 \pmod {(\varpi )}$
.
Proposition 3.8. With respect to the notation as above, suppose that characters
$\chi _1$
and
$\chi _2$
are congruent and that
$n:=\chi _i(1)$
is prime to
$\ell $
. Then
$$ \begin{align*}\mu_{\chi_1}=0\Leftrightarrow \mu_{\chi_2}=0\end{align*} $$
and if the above conditions hold then
$\lambda _{\chi _1}=\lambda _{\chi _2}$
.
Proof. If
$\chi _1\equiv \chi _2\pmod {(\varpi )}$
then
$Q_{\chi _1}\equiv Q_{\chi _2} \pmod {(\varpi )}$
, and the result clearly follows.
Example 3.9. Let us conclude with a concrete example. Let
$\mathbb {F}_q$
be a finite field, and let
$G := \mathrm {GL}_2(\mathbb {F}_q)$
be the group of invertible
$2 \times 2$
matrices over
$\mathbb {F}_q$
. Fix a prime number
$\ell $
and let
$\mathcal {K}$
be a sufficiently large finite extension of
$\mathbb {Q}_\ell $
such that every irreducible representation
$\rho : G \to \mathrm {GL}_n(\overline {\mathbb {Q}}_\ell )$
is defined over
$\mathcal {K}$
.
Let us denote by
$S := G \setminus \{\mathrm {Id}\}$
the set of all nonidentity elements of G. We define a function
$\beta : S \to \mathbb {Z}_\ell $
as follows. First, we partition the multiplicative group
$\mathbb {F}_q^\times $
as
$$ \begin{align*} \mathbb{F}_q^\times = \{a_1, \ldots, a_k\} \cup \{a_1^{-1}, \ldots, a_k^{-1}\} \cup \{a_{k+1}, \ldots, a_\ell\}, \end{align*} $$
where the elements
$a_i$
are chosen such that
$a_i^2 \ne 1$
for
$i \leq k$
and
$a_i^2 = 1$
for
$i> k$
.
Now define
$\beta : S \to \mathbb {Z}_\ell $
by the following rule:
-
• if
$g \in S$
is not a scalar matrix, then
$\beta (g) := 0$
; -
• if
$g = a_i \cdot \mathrm {Id}$
with
$i \leq k$
, then set
$\beta (g) := 1$
; -
• if
$g = a_i^{-1} \cdot \mathrm {Id}$
with
$i \leq k$
, then set
$\beta (g) := -1$
; -
• if
$g = a_i \cdot \mathrm {Id}$
with
$i> k$
, then set
$\beta (g) := 0$
.
It is straightforward to verify that the function
$\beta $
satisfies the conditions of Definition 3.1. First, consider the natural permutation representation of G on the projective line
$\mathbb {P}^1(\mathbb {F}_q)$
. This yields a representation of dimension
$q+1$
, which contains the trivial representation as a subrepresentation. Let V be the unique complementary q-dimensional subrepresentation and let
$\chi _V$
be the character of V. Then for any scalar matrix
$a \cdot \mathrm {Id} \in G$
, we have
$\chi _V(a \cdot \mathrm {Id}) = q$
. Therefore, the corresponding polynomial becomes
$$ \begin{align*} P_{\chi_V}(T) = (q - 2) - \sum_{1 \ne a \in \mathbb{F}_q^\times} (1+T)^{\beta(a \cdot \mathrm{Id})} \chi_V(a \cdot \mathrm{Id}) = (q - 2) - q \sum_{1 \ne a \in \mathbb{F}_q^\times} (1+T)^{\beta(a \cdot \mathrm{Id})}. \end{align*} $$
We find that
$$ \begin{align*} \sum_{1 \ne a \in \mathbb{F}_q^\times} (1+T)^{\beta(a \cdot \mathrm{Id})} = k(1+T) + k(1+T)^{-1} + (q - 2 - 2k), \end{align*} $$
and so
$$ \begin{align*} P_{\chi_V}(T) = (q - 2) - [k(1+T) + k(1+T)^{-1} + (q - 2 - 2k)] = -T(1+T)(2+T)\cdot k. \end{align*} $$
Consider the family of irreducible representations obtained by inducing characters from the Borel subgroup. Let
$\alpha , \beta : \mathbb {F}_q^\times \to \mathcal {K}^\times $
be two distinct characters. Let
$B \subset G$
be the Borel subgroup consisting of upper triangular matrices
$$ \begin{align*} B := \left\{\!\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} : a, c \in \mathbb{F}_q^\times, b \in \mathbb{F}_q \right\}. \end{align*} $$
Define a character
$\alpha \otimes \beta : B \to \mathcal {K}^\times $
by
$$ \begin{align*} (\alpha \otimes \beta)\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} := \alpha(a)\beta(c). \end{align*} $$
Let
$W_{\alpha , \beta } := \mathrm {Ind}_B^G(\alpha \otimes \beta )$
be the induced representation. When
$\alpha \ne \beta $
, this representation is irreducible. Let
$\chi _{\alpha , \beta }$
denote its character. Then for any scalar matrix
$a \cdot \mathrm {Id} \in G$
,
$$ \begin{align*} \chi_{\alpha, \beta}(a \cdot \mathrm{Id}) = (q + 1)\alpha(a)\beta(a). \end{align*} $$
Therefore, the associated polynomial becomes
$$ \begin{align*} P_{\alpha, \beta}(T) = (q - 2) - \sum_{1 \ne a \in \mathbb{F}_q^\times} (1+T)^{\beta(a \cdot \mathrm{Id})} \cdot \alpha(a)\beta(a). \end{align*} $$
