Representation growth of some torsion-free finitely generated -nilpotent groups.
We devise a new method for computing representation zeta functions of torsion-free finitely generated -nilpotent groups whose associated Lie lattices have an extra smoothness condition. This method is used first to derive intrinsic formulae for the abscissa of convergence of such representation zeta functions; and secondly, as a practical application, to compute global, local and topological representation zeta functions of groups within an infinite family containing the Heisenberg group over rings of integers in number fields.
Let be a torsion-free finitely generated nilpotent group (-group). Let () be the number of irreducible -dimensional complex characters of up to tensoring by one-dimesional characters. The representation zeta function of is
The abscissa of convergence of this Dirichlet series (when finite) gives the degree of polynomial growth of the partial sums of the ’s (cf. e.g. [stavol2011nilpotent, Section 1.1]).
1.1. A family of -groups
The main results of this paper concern representation zeta functions of torsion-free finitely generated -nilpotent groups (-groups). Of particular interest will be those arising from -nilpotent Lie lattices over rings of integers in number fields. Let henceforth be a number field with ring of integers . Stasinski and Voll [stavol2011nilpotent, Section 2.4] associate a unipotent group scheme with any -nilpotent -Lattice . Theorem 2.1 gives a general method to compute when has some additional smoothness property. As this is rather technical, we first record some zeta functions obtained with this method. We shall consider Lie lattices obtained from the following -Lie lattices.
Let . We define the following -Lie lattice:
In this definition, non-specified Lie brackets which do not follow formally from the given ones are understood to be zero.
Let be as above, we define . We denote by the group scheme .
1.1.1. Representation zeta function of
Fix throughout . Without loss of generality we may assume that . Indeed the -Lie lattices and are isomorphic, so also the -group schemes and are.
Let be the ring of integers of a number field , and let be the Dedekind zeta function of . Then
This theorem compares to a number of known results. First, in loc. cit. Stasinski and Voll also computed representation zeta functions of groups in three infinite families arising from their construction. The Lie lattice coincides with their when ; so Theorem A is a generalization of [stavol2011nilpotent, Theorem B] for the groups . In particular the properties of the Dedekind zeta function imply that the results on functional equation and abscissa of convergence in [stavol2011nilpotent, Corollary 1.3] hold, mutatis mutandis, for the representation zeta function of . Also the statement on meromorphic continuation holds but it is now a consequence of the more general [dunvol2017uniform, Theorem A] by Dung and Voll, which studies the analytic properties (such as meromorphic continuation and rationality of the abscissa of convergence) of the representation zeta function of a -group arising from a unipotent group scheme.
Secondly, Theorem A also compares to [carschvol2017traceless, Section 1.3]. There Carnevale, Schechter and Voll consider Lie lattices obtained from by adding an extra linear relation for its generators. Finally, the groups are a generalization of the Grenham’s groups used by Snocken to prove that every rational number may be attained as abscissa of convergence of the representation zeta function of a -group (cf. [sno2012thesis, Theorem 4.22]). In particular Theorem A may be used to recover Snocken’s result. Indeed, if denotes the -fold central product of , it is known that
This clearly has abscissa of convergence . Notice that by [hrumar2015definable, Theorem 1.5] the abscissa of convergence must be rational (cf. also [dunvol2017uniform, Theorem A]).
1.1.2. Local representation zeta function
For a non-zero prime ideal of we denote by the completion of at . Let be a -nilpotent -Lie lattice. By [stavol2011nilpotent, Proposition 2.2] one has the following Euler factorization:
Henceforth and will denote indeterminates in the field . We define the following objects.
Let , we write
Gauss polynomial. For , we define
Let . We write for the set .
Let . We write for an ordered subset of , i.e. a subset such that .
-multinomial coefficient. Let and let . We write
Pochhammer symbol. Let . We define
Let . We define
Let be a non-zero prime ideal of and be the cardinality of the residue field of . Then
where under the summation symbol denotes an ordered subset.
1.1.3. Topological zeta functions
In [ros2015topological] Rossmann defines topological representation zeta functions for -groups arising from unipotent group schemes. The formula in Theorem A (or rather its proof as we shall see) entails a corresponding formula for the topological representation zeta functions.
The topological representation zeta function of is
1.2. Abscissae of convergence
Fix throughout a non-zero prime ideal of and let be the cardinality of the residue field of . Theorem B is obtained by applying an analogue of [zor2017poincare, Theorem E] to the -Lie lattice . This gives a general method for computing local factors of representation zeta function of -groups when the attached Lie lattice has some extra smoothness property called geometrical smoothness (see Section 2.1). This result is Theorem 2.1. We record here two of its consequences that give an intrinsic formula for the abscissa of convergence of the aforementioned local factors.
Let be an -Lie lattice and let , . Let be the -Lie algebra obtained from by reducing it modulo . For set
Let be a -group. As noted by Voll in [klonikvol2011lectures, Chapter III,§ 3.2], the representation zeta function of has the following Euler factorization:
where . In order to state the next main result we need to recall some facts about the Lie theory of -groups. Namely, with each -group there is associated a -Lie algebra and an injective map such that spans (see [seg1983polycyclic, Chapter 6] and [gruseg1984reflections, grusegsmi1988subgroups]). This construction is such that if is a finite index subgroup of then . The set is not necessarily even an additive subgroup of , but by [grusegsmi1988subgroups, Theorem 4.1], there is such that is LR, i.e. is a -Lie sublattice of . When , by [hrumar2015definable, Lemma 8.5],
Moreover may be chosen so that is such that where is the nilpotency class of . In this case Howe’s Kirillov orbit method [how1977discrete] applies and the representation zeta function may be computed as the Poincaré series of a matrix of linear forms (see [voll2010functional, Proposition 3.1]).
Let be a Lie algebra over . The commutator matrix of a basis of is defined analogously to the commutator matrix for a basis of an -Lie lattice (see Section 2.1). We say that has smooth rank-loci when the rank-loci of a commutator matrix of are smooth schemes over . An analogue of [zor2017poincare, Lemma 2.1] ensures that this definition is independent of the basis chosen to define the commutator matrix.
We shall show that, if is a -group of nilpotency class , having smooth rank-loci implies that, for almost all , is the Poincaré series of a -Lie lattice with geometrically smooth rank-loci, to which Theorem 2.1 will apply. The output will be the following theorem.
Let be a finitely generated torsion-free -nilpotent group. Assume that has smooth rank-loci. Then, for almost all primes, the abscissa of convergence of is where such that is LR and .
1.2.2. Unipotent group schemes
Let be a -nilpotent -Lie lattice and let be the unipotent group scheme associated with it by [stavol2011nilpotent, Section 2.4]. Assume that has smooth rank-loci. Then, for almost all non-zero prime ideals in , has abscissa of convergence .
The first three main results mostly compare, as said, to [stavol2011nilpotent, dunvol2017uniform, ros2015topological]. Zeta functions analogous to the ones considered here have been introduced and studied for -adic analytic groups and arithmetic subgroups of semisimple algebraic groups over number fields. In that case representations are not counted up to twisting by one-dimesional characters (see for instance the work of Lubotzky and Martin [lubmar2004PRG], Jaikin-Zapirain [zap2006zeta], Larsen and Lubotzky [larlub2008reprgrowth], and Avni, Klopsch, Onn and Voll [akov2013representation, akov2]). Recently Stasinski and Häsä have introduced a representation zeta function counting, up to twisting, characters of the general linear group over a compact discrete valuation ring (cf. [hassta2017compact]). Theorem 2.1 applies to analogous situations whenever a Kirillov orbit method applies.
Theorems D and C compare to the upper and lower bounds obtained by Snocken and Ezzat in [sno2012thesis, Theorem 4.24]. In a broader sense they compare to the bounds in the context of -adic analytic groups obtained in [akov2012some_p, Theorem 1.1] and to the bounds for subgroup zeta functions featuring in [grusegsmi1988subgroups, paa2007degree].
Throughout, denotes the integers, the set of positive integers and the set of natural numbers. The set of rational numbers is denoted by . If is a rational prime, and denote the sets of the -adic numbers and of the -adic integers respectively.
Let be a ring, an -Lie lattice is a free finitely generated -module endowed with a Lie bracket. If is an -Lie lattice, we write for its derived Lie sublattice. If not otherwise specified, when is considered as an -lie lattice it is always endowed with the trivial Lie bracket.
Let . The ring of matrices with entries in is denoted by . If the identity matrix is denoted by , the diagonal matrix with on the diagonal is denoted by .
We have consistently denoted tuples by , their components are denoted by respectively. Often we shall represent -tuples of elements in a ring as matrices. In this case the components of the -tuple are denoted by for and .
The formulae for the zeta functions computed here were first computed with ad-hoc more elementary methods during my doctoral studies. I wish to thank my PhD supervisor Christopher Voll and I acknowledge financial support by the School of Mathematics of the University of Southampton the Faculty of Mathematics of the University of Bielefeld and CRC 701. I am grateful to Ben Martin for his comments on this work.
I am currently supported by the Research Project G.0939.13N of the Research Foundation - Flanders (FWO).
2. Poincaré series
The main results of this paper all follow from a variant of [zor2017poincare, Theorem E]. In order to treat Poincaré series arising from -groups of class and unipotent group schemes, we shall consider Lie lattices that do not necessarily have finite abelianization. Let, for the rest of the section, be as in Section 1.2. We shall need the following notational conventions.
We briefly recall the definition of commutator matrix and that of its Poincaré series as they are intended in this context.
Let be an -basis of with the last elements in . We define the commutator matrix of with respect to to be
where for , .
Let for the rest of this section be a basis of as in Section 2.1 and . We denote by the matrix obtained from reducing its entries .
Let be a ring. Let with . We define
The definition of geometrical smoothness for the rank-loci of in [zor2017poincare, Definition 1.11] now naturally extends to Lie lattices with infinite abelianization if is defined as above. Henceforth we consider Lie lattice structures on and given by the following identification: tuples in (resp. ) are viewed as coordinates of elements in (resp. ) with respect to (resp. the reduction modulo of ).
We say that has geometrically smooth rank-loci when for all :
every such that has a lift such that has Smith normal form .
For every such that has Smith normal form , is an isolated -submodule of .
The fact that this definition is independent of the choice of is a consequence of the analogue of [zor2017poincare, Lemma 2.1] in this context.
We define the Poincaré series of as in [zor2017poincare, Definition 1.6] with the obvious modifications needed to deal with a matrix of linear forms in variables. Let and let . Let . We define
where are the first elementary divisors of for a lifting . Indeed, taking the minimum with ensures that the this definition is independent of the choice of .
Let . Let also and and
The Poincaré series of is defined as
If is another basis for as in Section 2.1, it is known that , we shall therefore often call the Poincaré series of .
2.2. Technical result
We recall the following definitions from [zor2017poincare]:
An -kernel class is a subset of such that for any two
For an -kernel class we define
A classification by -kernels is a set of -kernel classes which are disjoint and cover . An element of a classification by -kernels is called a kernel -class.
Let be a classification by -kernels and . A set
of kernel -classes such that is called a sequence of kernel -classes.
Let be a sequence of kernel -classes
If we define .
If . We define as the set of -tuples of elements of such that for and :
Recall that . Let . A sequence of kernel -classes with the property that , for , is called an -sequence of kernel -classes. The set of all -sequences of kernel -classes is denoted by .
The following analogue of [zor2017poincare, Theorem E] holds.
Let be an -Lie lattice with and . Let be a commutator matrix of and be a classification by -kernels. Assume that has geometrically smooth rank-loci. Then the Poincaré series of is
The proof closely follows the argument for [zor2017poincare, Theorem E], the only difference is that here the size of the commutator matrix is no longer equal to the number of its variables and one needs to keep them distinct in the proof. This is straightforward, because loc. cit. is a direct consequence of [zor2017poincare, Proposition 3.3], whose proof is independent of the size of : the quantity there, has to be interpreted as the number of variables in , which in the present context is . The current result therefore follows immediately from the analogue of [zor2017poincare, Proposition 3.3] stated with in the place of . Indeed, the rest of the argument applies unchanged to the present situation. ∎
Next result now follows in the same fashion as [zor2017poincare, Theorem A] followed from [zor2017poincare, Theorem E] (see [zor2017poincare, Section 3.3]).
Assume that the rank-loci of are geometrically smooth. Then the Poincaré series of has abscissa of convergence .
In this section we explain how to apply Theorem 2.2 to representation zeta functions of -groups of nilpotency class . From now onwards we assume to be nilpotent of nilpotency class . This implies that the last rows and columns of the commutator matrix of with respect to consist of zeroes. The matrix obtained by discarding them is a antisymmetric matrix of linear forms in variables. To distinguish it from , we call it the trimmed commutator matrix of with respect to and denote it by .
It is clear from its definition that the Poincaré series is the same for and .
3.1. Proof of Theorem C
Let be such that is LR and is such that . Let be a -basis of such that its last elements span . By [voll2010functional, Proposition 3.1] and Section 3, for all the representation zeta function is the Poincaré series of (i.e. the Poincaré series of when is viewed as a basis of ). In order to prove the theorem it is enough to show that Theorem 2.1 applies, i.e. it suffices to show that, if has smooth rank-loci, then for almost all primes, has geometrically smooth rank-loci. This is immediate as smoothness of the rank-loci of implies smoothness of the rank-loci of for all . Which for almost all primes implies the smoothness of the rank-loci of . This implies geometrical smoothness of the rank-loci as shown in [zor2017poincare, Remark 2.4].
3.2. Proof of Theorem D
Let and . Let also be an -basis of such that its last elements are in . By [stavol2011nilpotent, Proposition 2.18], the representation zeta function is the Poincaré series of the trimmed commutator matrix when is viewed as an basis of . With the same argument used to prove Theorem C, one shows that having smooth rank-loci implies that has geometrically smooth rank-loci for almost all prime ideals . This shows that Theorem 2.1 applies to this context and concludes the proof.
4. An infinite family of groups schemes
As explained in the proof of Theorem D the representation zeta function of is given by the Poincaré series of a trimmed commutator matrix of . So and . We choose to be the basis used in the presentation of in Section 1.1. Accordingly we compute
where is the generic matrix in the variables , for and . For convenience, in what follows we represent -tuples as matrices, therefore the labelling of the variables in .
We shall now show how to apply Theorem 2.1 to the situation in hand. We begin by showing that its hypotheses are satisfied for , i.e. we shall show that
The Lie lattice has geometrically smooth rank-loci.
In what follows we shall see that this is equivalent to a problem involving lifting Smith normal forms of matrices with entries in .
First of all we notice that since , for the matrices and can have -rank at most . Similarly, for , if and are the reductions modulo of and respectively, the matrices and can have -rank at most .
Secondly, by creftype 4.1, has Smith normal form
if and only if and have Smith normal form
respectively. Similarly, has rank (, ) if and only if both and have rank .
The discussion above implies that has geometrically smooth rank-loci if and only if the two following properties are satisfied for all , :
for of rank it is possible to find a lift with Smith normal form .
for with Smith normal form , has isolated derived Lie sublattice.
Both the properties above are easily verified: namely the first one by taking a matrix whose entries are the Teichmüller lifts of the entries of , while the second one because of the following result.
If has Smith normal form (, ), then is isolated and isomorphic to
as an -Lie lattice.
Let and such that . Let
and let be the -module automorphism of whose matrix in the basis is . Choose one each from for . Since is invertible, the set
is an -basis of such that
Moreover, because of the shape of , the -module automorphism defined by for and for , gives a Lie lattice automorphism of . Hence and are isomorphic as Lie lattices and the derived sublattice of the former is isolated if and only if the derived sublattice of the latter is; which is indeed the case as
by definition of . This also shows that is isomorphic to
and we conclude. ∎
4.1. Proof of Theorem B
In order to apply Theorem 2.1 we need to have a classification by -kernels. This is easily obtained from the previous argument for geometrical smoothness. Indeed Section 4 shows that for with the derived subalgebra of is always the same. This implies that the set of loci of constant rank in is a classification by -kernels. For convenience we denote the members of this classification simply by elements of . Namely the kernel class corresponding to rank in will be denoted by . Notice that with this notation we exclude the locus of rank (which is just the zero matrix), this is not a limitation as we are only required to plug primitive coordinate vectors into in order to compute its Poincaré series (cf. Section 2.1).
We shall now rephrase Theorem 2.1 with the current notation and the following convention. As in Section 2, is viewed as an -Lie algebra, so its derived subalgebra is abelian and we identify it with endowed with the trivial Lie bracket. For , is a Lie subalgebra of and so may be viewed as a subalgebra of . First we rephrase Section 2.2 in this notation.
Let be an ordered subset of . If
, we define ,
, we define to be the set of -tuples of matrices in such that for and :
is in the derived Lie subalgebra of when this is viewed inside with the convention above.
as -Lie algebras. It follows that
where denotes a kernel class, as explained at the beginning of this section, and are as in Section 2.2.
In conclusion if the formula in Theorem 2.1 is rewritten with the notation above, one finds that the representation zeta function of is equal to
Notice that here we are summing over sequences of kernel classes with