Irreducible representations of a family of groups of maximal nilpotency class I: the non-exceptional case
We use a constructive method to obtain all but finitely many -local representation zeta functions of a family of finitely generated nilpotent groups with maximal nilpotency class. For representation dimensions coprime to all primes , we construct all irreducible representations of by defining a standard form for the matrices of these representations and, after taking into account twisting and isomorphism, count these twist isoclasses to obtain our -local zeta functions.
Representation growth is a fairly recent area in group theory where one studies (usually infinite) groups by studying the sequence of the number of (sometimes equivalence classes of) irreducible complex representations of degree for all More formally, for all let be the number of irreducible representations of degree . If all are finite, we can study this sequence by embedding them as coefficients in a zeta function.
In this paper, we deal with the case of a family of finitely generated torsion-free nilpotent groups. Henceforth, we call finitely generated torsion-free nilpotent groups -groups. If is a -group, it is known that all are infinite. However, we can redefine as the number of irreducible representations “up to twisting”.
More formally, let be a -group. Let be a 1-dimensional complex representation and an -dimensional complex representation of . We define the product to be a twist of . Two representations and are twist-equivalent if for some 1-dimensional representation , we have that This twist-equivalence is an equivalence relation on the set of irreducible representations of . In  Lubotzky and Magid call the equivalence classes twist isoclasses. We say the twist isoclass containing and irreducible representation , is of dimension if and only if is an -dimensional representation. They also show that there are only finitely many irreducible -dimensional complex representations up to twisting and that for each there is a finite quotient of such that each -dimensional irreducible representation of is twist-equivalent to one that factors through . Henceforth we call the -dimensional complex representations of simply representations. We denote the number of twist isoclasses of irreducible representations of dimension by or if no confusion will arise.
We now discuss embedding the sequence as coefficients of a zeta function. Consider the formal expression
If converges for a right half plane of , say , where for some , we call the (global) representation zeta function of . For any -group such a always exists [14, Lemma 2.1]. We call the abscissa of convergence of . Let , where
be the -local representation zeta functions of Considering the domain of as above, we say is the -local abscissa of convergence of
We know, by [11, Theorem 6.6] that in each twist isoclass there exists a representation such that factors through a finite quotient. Since is nilpotent, its finite quotients are nilpotent and therefore decompose as a direct product of their Sylow- subgroups. Since the irreducible representations of direct products of finite groups are the tensor products of irreducible representations of their factors, its representation zeta function decomposes into an Eulerian product of its -local representation zeta functions and therefore . Moreover, it was shown by Hrushovski and Martin  that these -local representation zeta functions are rational functions in
Let . Following the conventions of other papers in this area, all commutators that do not appear in (or follow from) the relations are trivial. In this paper, we discuss the irreducible representations and the representation zeta functions of this family of groups. Additionally, we refer often to certain subgroups of It is clear that for the group is isomorphic to a subgroup of With a slight abuse of notation we let the subgroup
2 Related Results
The idea of using zeta functions to study representation growth was introduced in , in which Witten studies compact Lie groups. Later, representation zeta functions were studied in , where Lubotzky and Martin use representation zeta functions to study arithmetic groups, and in  where Jaikin develops a method for calculating the representation zeta functions of compact -adic analytic groups with property FAb. This method uses Howe’s work  on the Kirillov orbit method and the concept of -adic integration to calculate the zeta functions.
Representation zeta functions of -groups were first studied by Hrushovski and Martin in  using model-theoretic methods. The study of representation growth of -groups was expanded by Voll in . In that paper, Voll develops a method for calculating -local representation zeta functions for a given -group This method, like the method that appears in , involves Howe’s work in . Note that a large part of the method that appears in  is also used in that paper to study other types of algebraic growth, including subgroup and subring growth. Stasinski and Voll, in , generalize Voll’s work in  to -groups coming from unipotent group schemes. The authors also generalize the functional equation that appears in .
Representation zeta functions have been used to study other classes of groups. We briefly mention some work done in these areas. In  Avni et al. study compact -adic analytic groups and arithmetic groups. In , Avni et al. study representations of arithmetic lattices and prove a conjecture by Larsen and Lubotzky. In  Avni shows that arithmetic groups have representation growth with rational abscissa of convergence. Bartholdi and de la Harpe, in , study representation zeta functions of wreath products with finite groups. Craven, in , gives lower bounds for representation growth for profinite and pro- groups.
3 Layout of Paper
We employ a constructive method, first used in , to calculate the irreducible representations and representation zeta functions of this family of groups. The constructive method allows us to calculate the -local representation zeta functions of when is an exceptional prime. Informally, we say a prime is exceptional if appears as a denominator in the equations that determine the possible eigenvalues of the linear operators of our representation. We will formally define these primes later.
Indeed, the family is a good choice of a family of -groups to study using the constructive method for a number of reasons. The relatively simple eigenspace structure of the irreducible representations, due to the large abelian subgroup inside, allows us to construct all of the irreducible representations of each for all but finitely many prime-power dimensions. Additionally, for small , it is tenable to construct all of the irreducible representations for the exceptional-prime-degree representations as well. This gives us all of the irreducible representation theory (and thus the global representation zeta functions) for and . Also, it gives us an infinite number of examples of exceptional -local zeta functions of groups of ”almost arbitrary” nilpotency class; that is, given a number , there is a prime such that and the -local representation zeta function of is known. Furthermore, together with the results of the calculations of the -local, and -local and -local representation zeta functions of and , respectively, we have two examples of global representation zeta functions of -groups of nilpotency class greater than . To the author’s knowledge, these are the only two examples in the literature. For results involving -local representation zeta functions of exceptional primes, see the sequel to this paper .
This paper describes the construction of the irreducible representations of the maximal class group of nilpotency class , denoted . This is achieved by calculating the irreducible representations of -power dimension. The large abelian subgroup allows us to simultaneously diagonalize all but one element of the images of the generators and reflected by this fact is the relatively simple eigenspace structure of the irreducible representations. We note that the calculation is uniform for most primes, in fact primes not less than the nilpotency class ; denominators that appear in the matrices of the representation are smaller than the prime considered and therefore behave as units mod When the prime considered is smaller than , the calculation loses its uniformity and the structure of the matrices of the representation differs from the non-exceptional cases; again, these will be dealt with in this paper’s sequel.
First, we introduce an important lemma that gives us much information about the eigenspace structure of representations of certain nicely behaved -groups, including . Before this lemma, we give a definition regarding eigenspaces of a set of linear operators.
Let be a set of linear operators of a vector space If a subspace is an eigenspace of each then we say that is a mutual eigenspace of
For some set of let where be a -group and let be an irreducible -dimensional representation of Also, let , and for all and Define and . Then the mutual eigenspaces of are one-dimensional and there are distinct mutual eigenspaces.
Let be the set of mutual eigenspaces of We will show that if is an eigenvector of then for any the vector is an eigenvector of for some We then show that acts transitively on and that all are of the same dimension, in fact a -power. Finally, we show that each is one-dimensional, and
It is clear, by definition, that commutes with all Let and let For a given consider Since is central we have that
For all let be such that . Now, as an induction, we choose and assume that, for each is an eigenvector of each with eigenvalue Then
for some Note that the third equality is by the group relations and the final equality is by the inductive hypothesis. Thus is an eigenvector of with eigenvalue This induction tells us that for any mutual eigenvector of that, for any is also a mutual eigenvector.
Let For some and let
It is clear that Now consider
Since must be an eigenvector of we have that and for some Since is invertible it preserves dimension and for Also, since was arbitrary and is irreducible it must be that must be the entire space and thus must act transitively on It follows, by counting, that and, for we have that for such that
Let For an eigenspace let and let Let be a -stable subspace. Let and let be the restriction of to Since is a mutual eigenspace of the each it is clear that for some scalars and thus the -stable subspaces are the -stable subspaces. Consider the -orbit of , say Since is irreducible then and since is -stable it must be that Thus has no proper stable subspaces and is irreducible. By  we have that factors through a finite quotient up to twisting. By assumption and Schur’s Lemma, since is abelian, each must be a scalar matrix. Thus, since is irreducible, It then follows that all mutual eigenspaces of are 1-dimensional and, since acts transitively on we have that ∎
We now introduce some notation for complex roots of unity.
Let be the all complex th roots of unity for all and be the th roots of unity (and note that are the primitive th roots of unity). Define such that if and only if If we say that has depth
The calculation of the irreducible representations of will involve generalizations of triangle numbers, namely -simplex numbers. Let and for and recursively define for The next lemma lists some properties of these numbers that are needed. We state these without proof.
Let and be defined as above.
Let Then for any and such that we have
As a corollary of (iii) we have the following.
Let be a prime, let , let let let such that and, for , let
Then we have that for all such that and all such that
Consider We have that
By Lemma 4.4(iii), and noting that we have that only the term with no factor of survives mod that is,
and thus ∎
5 Representation Structure and Standard Form
We now calculate the -local representation zeta function of each , which we denote , by explicitly constructing representatives of each twist isoclass. Let be a -dimensional irreducible representation of and let and .
In this section we will choose a basis for the image of such that is in the form of a -cycle permutation matrix and such that each is diagonal with each diagonal entry in a certain form, discussed later in the section. It is not necessary to state a basis to understand the eigenspace structure of However, as a canonical basis is easy to determine in this case, we appeal to a basis as an indexing device on the set of mutual eigenspaces of .
We begin by considering twisting. Since are commutators they are invariant under twisting. We can twist and by any complex number. We remind the reader that we can obtain every -power irreducible representation of by twisting a representative from each twist isoclass.
Since all commute they are all simultaneously diagonalizable. By [11, Theorem 6.6] all irreducible representations factor through a finite quotient (up to twist equivalence) and thus by Schur’s lemma the central element is a scalar matrix.
By the group presentation of it is clear we can apply Lemma 4.2. Let Then we know the mutual eigenspaces of are -dimensional and that there are distinct mutual eigenspaces. Also, we have that must permute the eigenspaces of transitively. Thus, must act as a -cycle on the mutual eigenspaces of We choose our basis, with basis vectors such that the are diagonal and
for some By cofactor expansion of , we have that the characteristic equation is and thus the eigenvalues of are all of the th roots of However, we have the freedom to twist by and therefore we can ensure the eigenvalues of are all of the th roots of unity. We can choose a representative of our twist isoclass such that is a true -cycle permutation matrix under some choice of basis, or equivalently, we can ensure that
We set up some notation. Let be the th entry on the diagonal of . We let and . Note that for all By twisting we can ensure that . It will be shown that the , and thus , are determined by the for all such that
We now determine the structure of the matrices and the allowable values for the The next lemma is the base case for the inductive lemma following it. Although we could start the induction with , this case is trivial. For purposes of elucidation, we start this induction with Note that this lemma is true for all primes.
The matrix has the form
Moreover, for any prime , we have that is a th root of unity; that is .
Since by our group relations we have that for and . Combining these equations we have that and therefore ∎
For we have that and thus the matrix has the structure
Moreover we have that
for some By the group relation we have, for some , that
Combining the above equations for each we have that
We have shown that, up to twisting and isomorphism, that any irreducible representation must be of the form given above. We give this a name.
6 Stable Subspaces
In this section we determine possible stable subspaces of a representation We show that if is not irreducible then it must have a certain proper stable subspace; we name this Thus, to determine if is irreducible, we only need to check if is a stable subspace of In this vein, let be the subspace spanned by . Note two things: first, has dimension ; second, if is a stable subspace of then so is for
We define the -tuple where is considered mod
For any if then
By Lemma 5.2 we have that for all and that for all Consider It is clear to see that since is central. Now, as our inductive step, choose such that and assume that for all we have that Consider By Equation 12
Our inductive hypothesis holds for the first factor of the right hand side of Equation 17 and the initial assumption holds for the second factor. Thus we have that
Since is of dimension Lemma 6.1 and elementary counting tells us that if, for some and where then for all such that This can be seen since is a unit in the additive group and thus generates all of This argument gives us the following corollary of Lemma 6.1.
Let be the minimal power such that for all such that and for any Then is a stable subspace of and is not stable.
We define notation to this effect. Let and let be the the minimal stable subspace as in Corollary 6.2, of We say that
We can, in fact, say more about this minimal subspace:
The number is minimal such that if and only if
Let be a representation. Then, for if then for some such that
We know that if is -stable then so is for Thus, we obtain the following corollary:
Let be a representation of The representation is irreducible if and only if is not -stable.
Throughout this paper we use Corollary 6.5 to check if a representation is irreducible. We use Corollary 6.2 to determine the number of isomorphic representations in standard form in one twist isoclass. Corollaries 6.3 and 6.4 are used in the sequel to this paper, but are included here since they follow from Lemma 6.1.
7 Isomorphic Representations in Standard Form
Since representations in the same twist isoclass are equivalent under both twisting and isomorphism, we determine when two representations in standard form are isomorphic. In this vein, we have the following proposition.
Let be irreducible representations of in standard form. Then and are in the same twist isoclass if and only if there is a 1-dimensional representation and a permutation matrix such that
Since one direction is immediate, we prove the other direction with the following lemma.
For any prime , let be irreducible and let be a matrix such that, for , the matrix is diagonal and Then for some and scalar . Furthermore, up to twisting, and are in standard form.
Let We will show that since all elements of are diagonal with -dimensional mutual eigenspaces, must be a generalized permutation matrix. Then we show that since commutes with that must be a power of up to scalars. We then show that it follows that, up to twisting, is in standard form.
Since all elements of are diagonal and its mutual eigenspaces are 1-dimensional, where is the centralizer of in and are the diagonal matrices. Since is the centralizer of and since has 1-dimensional mutual eigenspaces is also the centralizer of Let be the normalizer of in It is well known that where are the generalized permutation matrices; that is, matrices with precisely one non-zero entry in each row and column. And since the mutual eigenspaces of are all 1-dimensional we have that
By definition of centralizer, it must be that must be in the centralizer of Since is a -cycle we have that for some diagonal matrix But since commutes with and, of course, commutes with must as well. It follows that must be a scalar matrix.
Conjugation of each by which is the same as conjugating by , for each and some , maps to We can twist by some 1-dimensional representation such that and thus by, Proposition 8.2 (or directly by Lemma 4.4(v)), is in standard form.
This ends the proof of the proposition.
Remembering that representations in a twist isoclass are equivalent up to both twisting and isomorphism, we make the following definition.
Let be irreducible and let for such that be in standard form as defined earlier in the section. A shout is a matrix such that, up to twisting, and for are in standard form. The representations and (note that may not be in standard form) are said to be equivalent under shouting.
We now need to count how many representations in standard form are in the same twist isoclass as that is, how many representations in standard form are twist-equivalent to We say that two representations that satisfy these conditions are equivalent under twisting and shouting . If there are twist-and-shout equivalent representations, and if we are just counting representations in standard form then we have overcounted by a factor of Thus, we must take this into account when counting twist isoclasses. In this vein, we now have the following lemma.
Let be the twist isoclass represented by and let Then there are representations in standard form in that are twist-and-shout equivalent to
By Lemma 5.2 the entries of the are determined by the So to determine how many representations are twist-and-shout equivalent to we must count the number of choices of such that such that is in standard form for some -dimensional representation
Let for all and let be the first diagonal entry of By Lemma 7.2 we have that and thus for some we have that for each Since is in standard form it must be that we chose such that
By the argument above, our choice of gives us, up to our choice of twist a representation that is twist-and-shout equivalent to It follows that the number of representations twist-and-shout equivalent to is the size of the set By Corollary 6.2 we have that the size of this set is ∎
Note two things: first, that when we reference this lemma, we say we take shouting into account; and second, since all entries of any differ by products of such that this lemma implies that the depth of has no effect on the number of twist-and-shout equivalent representations.
During the calculation of the -local zeta functions that appear in this section, we break computation into various cases that depend on the depths of the that we choose. We note, without additional special mention, that each case is closed under shouting. For completeness, however, we have the following lemma which can be applied to the various cases to show that they are closed.
For some and let Then, for all
Each where is some product of the roots of unity Also, for we have that The result follows immediately from these two facts. ∎
8 Irreducible Representations for Non-Exceptional Primes
We note that the expressions contain a denominator of . By Lemma 5.2 the maximal value for is . We say a prime is exceptional if and non-exceptional otherwise. For all non exceptional primes the -local zeta function will behave uniformly.
In this section we study the conditions for irreducibility of a -dimensional representation such that If this is the case, then the terms that appear in the calculation of the standard forms have denominators that are all units mod We show that such a representation is irreducible precisely when at least one of the is a primitive th root of unity.
For non-exceptional primes we have, for all , that
Let be matrices in standard form and let be the corresponding representation. We will show that, for non-exceptional primes, is irreducible precisely when one of the has all th roots of unity on its diagonal. This implies that, at least one of the , where is in fact a primitive th root of unity. This will be shown in two stages. First, if for some and for all such that then has all th roots of unity on its diagonal. Secondly, we show a stronger result that implies that if none of the are primitive th roots of unity, that is for all then there is a proper stable subspace. We use the full strength of the second lemma in the sequel to this paper.
We state the above as a proposition. The proof is a consequence of the two lemmas following it.
Let and be a -dimensional representation of with corresponding matrices in standard form. Then is irreducible if and only if there exists a such that where
If and for all such that then all where , are distinct th roots of unity.
Assume that and for We can write these non-primitive , for each , as powers of . Let such that and For ease of display let and . Let Then, by Lemma 5.2,
We show that each diagonal entry is distinct by dividing two of them, say and with , and showing that if then .
Consider the equation
Taking the logarithm base and working mod :
for some . Since all this implies that
and thus and we conclude that . We can now say that each diagonal entry of is distinct. ∎
We prove the necessity of having at least one a primitive root of unity in the following lemma. The idea is to show that if no is a primitive th roots of unity then for and for some , we have that for any Therefore is in fact a proper -stable subspace of . For to be irreducible this cannot be the case.
Let For each let where and