1 Introduction
In order to improve and generalise the MEAT-AXE algorithm of Richard Parker [14], Holt and Rees [10] suggested the use of a family of matrices defined as follows: an n×n matrix X over a field F=GF(q) is primary cyclic if, for some irreducible polynomial f over F, the nullspace of f(X) in V(n,q)=Fn is an irreducible FX-submodule (see also Definition 2.3).
Given a group G⩽GL(n,F) acting on V=Fn, the irreducibility test in the MEAT-AXE algorithm,
originally due to Simon Norton, tests whether or not G leaves invariant a proper nontrivial subspace of V. The
version of the test used by Holt and Rees in [10] does so by randomly searching for primary cyclic matrices and analysing their actions on V: for the analysis it is crucial to know how abundant primary cyclic matrices are.
Holt and Rees in [10, pp. 7–8] obtain a positive constant lower bound on the proportion of primary cyclic matrices in the full matrix algebra M(n,F), and in [7] Glasby and the second author show that the proportion of primary cyclic matrices in M(n,F) lies in the interval (1-c1qn,1-c2qn) for positive constants c1,c2. Here we focus on irreducible proper subalgebras of M(n,F): any such subalgebra can be identified with the full matrix algebra M(c,K) over some extension field K=GF(qb), where n=bc (see Section 2). We prove an analogous result to the Holt–Rees estimate for these subalgebras.
We treat the case of fixed degree extensions GF(qb) of a field of fixed size q as the dimension n=bc
grows unboundedly. Let PM(c,qb) be the proportion of matrices in M(c,qb) which are primary cyclic in
M(n,q) relative to some irreducible polynomial f of degree b over F (the minimal
possible degree of such an f): then PM(c,qb) is a lower bound for the proportion of primary cyclic matrices in M(c,qb).
Theorem 1.1.
Let q be a prime power, and b,c positive integers with b>1. Then
limc→∞PM(c,qb) exists and equals
PM(∞,qb):=limc→∞PM(c,qb)=1-(1-bq-b(1-q-b)2ω(1,qb)b)N(q,b)
where ω(1,qb)=∏i=1∞(1-q-bi) and N(q,b) is the number of monic irreducible polynomials of degree b over Fq; and
there exists a constant k(q,b) such that, if
c≥(max{b-1,qb/b}log(3/4))2, then
|PM(c,qb)-PM(∞,qb)|<k(q,b)q-bc.
Remark 1.2.
(i) To prove Theorem 1.1, we use generating functions and in particular, we obtain a new
generalisation in Theorem 3.6 of the Kung–Stong cycle index theorem (see [11, 16]).
(ii) Theorem 1.1 shows that, for fixed q,b, the quantity PM(c,qb) approaches its limiting value exponentially quickly. However the expression for the limit is rather complicated. We study the behaviour of the limiting value as qb grows, and prove (in Proposition 5.5) that the limit as qb approaches infinity of PM(∞,qb) exists and equals
limqb→∞PM(∞,qb)=1-e-1.
This is analogous to the original Holt–Rees estimate in [10] for the case b=1.
(iii) We prove Theorem 1.1 (ii) with the following value for the quantity k(q,b):
k(q,b)=83(1-q-b)(bqbqb-122bq2b2)qb/b
(see Proposition 5.10). We believe that this may be far from the best value.
(iv) A different subfamily of primary cyclic matrices was studied in [3],
namely the set of all f-primary cyclic matrices X∈M(c,qb) for irreducible polynomials f of degree strictly greater than half the rank of X. In [3, Theorem 1.4],
an explicit lower bound is given for the proportion of such primary cyclic matrices in M(c,qb)
as a function of q,b and c. For large b,c the bound is close to loge2.
In Section 2 we present essential results on minimal and characteristic polynomials.
In Section 3 we prove the generalisation of the Kung–Stong cycle index theorem and apply it to estimating the proportion of
primary cyclic matrices in M(c,qb).
Section 4 deals with asymptotics and introduces a generating function crucial for the proof of Theorem 1.1. Then in Section 5 we complete the proof of Theorem 1.1, and discuss how to use it.
A consequence of Theorem 1.1 is that, for sufficiently large c, an explicit lower bound on the proportion of primary cyclic matrices can be calculated. Computationally we determine the proportion exactly for small c, see for example, Table 1: combining these two methods, we may address all values of n, so long as the field size qb is bounded.
2 Preliminaries
We first introduce some notation. Let F be a field of order q and let K be an extension field of F of degree b. The Galois groupG=Gal(K/F)⩽AutK is cyclic of order b, generated by the Frobenius automorphism σ0:x↦xq, and has the subfield F as its fixed point set.
Let V=Fn denote the space of n-dimensional row vectors over F, with standard basis {e1,…,en}, and let M(n,q) denote the full endomorphism ring of V, with elements written as n×n matrices with entries in F relative to the standard basis. For a divisor b of n (say n=bc), we can embed the algebra M(c,qb) as an irreducible subalgebra of M(n,q) as follows. The extension field K is an F-vector space of dimension b, having as a basis {1,ω,ω2,…,ωb-1}, where ω is a primitive element of K. If {v1,…,vn} is a basis for V(c,qb)=Kc, then
{ωivj∣0≤i≤b-1,1≤j≤c} is an F-basis for V(c,qb) as an n-dimensional F-vector space, where n=bc, and the mapping φ:ωivj↦e(j-1)b+i+1 extends linearly to an F-vector space isomorphism from V(c,qb)=Kc to V.
Each X∈M(c,qb) defines an F-endomorphism of V(c,K), and so we have an action of M(c,qb) on V=Fn defined by
(v)Xφ:=vφ-1Xφ,
for v∈V. Thus X↦Xφ defines an F-algebra monomorphism M(c,qb)→M(n,q), and we may identify M(c,K) with its image. This image is an irreducibleF-subalgebra of M(n,q), and each irreducible subalgebra arises in this way (by Schur’s lemma, see for example [4]).
Throughout we will have to consider interchangeably the actions of a matrix in M(c,qb) on two vector spaces, Fn and Kc. For this reason we introduce notation to help keep track of which field we are dealing with.
Notation 2.1.
Let V be the vector space Kc of c-dimensional row vectors over K=GF(qb), with n=bc. Then, as an F-vector space, V is isomorphic, via φ as defined above, to the vector space Fn. We denote this F-vector space by VF. If there is any ambiguity we use VK to denote the K-vector space V. An element X of M(c,qb) thus acts as a linear transformation of VF in a natural way (via the maps above): again we use the notation XF to denote the action of X on VF (and similarly XK to denote the action on VK if there may be ambiguity).
We denote by F[t], Irr(q) and Irr(q,d) (where d≥0) the ring of polynomials over F, the set of monic irreducible polynomials over F, and the set of monic irreducibles of degree d over F, respectively. Let N(q,d)=|Irr(q,d)|. Denote the characteristic and minimal polynomials of XF by cX,F(t), mX,F(t), respectively, and similarly define K[t], Irr(qb,d), N(qb,d) and cX,K(t), mX,K(t) for the X-action on VK=Kc.
The Galois group G=Gal(K/F) acts faithfully on K[t] and M(c,qb) by acting on the coefficients of a polynomial and the entries of a matrix, respectively. The fixed points of G in these actions are respectively F[t] and M(c,q).
If U is an X-invariant F-subspace of V, then we denote by X|U the restriction of X to U; if in addition U is a K-subspace, then we may write (X|U)F and (X|U)K if we wish to emphasise the field.
Definition 2.2.
Let X∈M(n,q) and let mX,F=∏i=1rfiαi,
with each fi∈Irr(q) and αi>0. A useful X-invariant decomposition of VF is the X-primary decomposition (see [9, Theorem 11.8])
VF=Vf1⊕…⊕Vfr,
where the subspace Vfi is called the fi-primary component of X (on V), and has the property that fi does not divide the minimal polynomial of the restriction of X to ⊕j≠iVfj, and the minimal polynomial of X|Vfj is fiαi. Let
DivF(X):={f1,…,fk}.
If f∈Irr(q)∖DivF(X), we say that the f-primary component is trivial and define Vf={0}.
We also define DivK(X) and the XK-primary decomposition of VK similarly.
Definition 2.3.
Let X∈M(n,q) and f∈Irr(q). Then X is called f-primary cyclic if X|Vf is nontrivial and cyclic. Also, X is called cyclic if X is f-primary cyclic for all f∈DivF(X), or equivalently, if mX,F=cX,F. We also say that
X is primary cyclic if it is f-primary cyclic for some f∈Irr(q). We note that X is f-primary cyclic if and only if the nullspace Nullf(X) is an irreducible FX-submodule of V.
2.1 Minimal and characteristic polynomials
We aim to count matrices X in the subalgebra M(c,qb) of M(n,q) such that XF is primary cyclic. To do so we derive necessary and sufficient conditions for this property which are intrinsic to their action on Kc: that is to say, conditions on XK. Our analysis follows that of [13, Section 5].
We investigate the relationship between the characteristic and minimal polynomials of a matrix X over the two different fields F and K. We call two polynomials g,g′ in K[t]conjugate if there exists σ∈G=Gal(K/F) such that gσ=g′. Recall Notation 2.1.
Lemma 2.4.
Let f∈Irr(q,d), let b≥2, and let G=〈σ0〉=Gal(K/F). Suppose that g∈Irr(qb) is a divisor of f in K[t]. Then the following hold:
f=lcm{gσ0i-1∣1≤i≤b}=∏i=1gcd(b,d)gσ0i-1;
g=gσ0i if and only if i≡0(modgcd(b,d));
f is the unique element of Irr(q) divisible by g in K[t].
Proof.
Part (i) follows immediately from [12, Theorem 3.46]. For (ii) and (iii), observe that since σ0 fixes the field F, the image gσ0 divides fσ0=f, and similarly, for every i we have gσ0i|f, so
lcm{gσ0i-1∣1≤i≤b} divides f.
Since the set {gσ0i-1∣1≤i≤b} is permuted under the action of σ0, its least common multiple is fixed by σ0, and so lies in F[t]. Then by the irreducibility of f, they are equal.
Since degf=d=gcd(b,d)degg, it follows that
{gσ0i-1∣1≤i≤b} has size gcd(b,d), and the stabiliser of each gσ0i-1 in G is 〈σ0gcd(b,d)〉. This implies part (iii) and the last assertion of (ii). Part (iv) follows from part (ii).
∎
We now give a description of f-primary cyclic matrices in terms of their representations over the field K. The following result is derived from
[13, Lemma 5.1 and Corollary 5.2]. Recall the notation for minimal polynomials from Notation 2.1.
Proposition 2.5.
Let f∈Irr(q,d), let G=Gal(K/F), and let X∈M(c,qb) such that f∈DivF(X).
Then XF is f-primary cyclic if and only if b|d and the following hold for some irreducible divisor
g∈K[t] of f of degree d/b:
g∈DivK(X) and XK is g-primary cyclic; and
for each nontrivial σ∈G, the conjugate gσ≠g and
gσ∉DivK(X).
Proof.
By [13, Lemma 5.1],
(2.1)mX,F=lcm{mX,Kσ∣σ∈G}.
If g∈Irr(qb) divides f, then, by Lemma 2.4 (ii), f is the product of the
distinct conjugates of g by elements of G.
Since f|m, it follows from (2.1) that mX,K is divisible by at least one conjugate of g.
Without loss of generality g|mX,K. Note that, by Lemma 2.4 (ii), f=lcm{gσ∣σ∈G}.
Consider the following XK-invariant decomposition of V=VK:
VK=V0⊕V1,
where V0 is the sum of the gσ-primary components of V, for σ∈G,
and V1 is the sum of the other primary components.
Let Yi=X|Vi for i=0,1. Then
g|mY0,K, and the only irreducible divisors of
mY0,K are gσ for certain σ∈G.
Also mY1,K is coprime to lcm{gσ∣σ∈G}=f.
By [13, Lemma 5.1] applied to Y0 acting on V0, we have
mY0,F=lcm{mY0,Kσ∣σ∈G}, and it follows from
Lemma 2.4 (iv), that f is the only irreducible divisor of mY0,F.
Thus V0 is the f-primary component of VF.
By definition, XF is f-primary cyclic if and only if (Y0)F is cyclic
(hence with minimum polynomial mY0,F=f).
By [13, Corollary 5.2] applied to Y0, this holds if and only if
(i) (Y0)K is cyclic, and (ii) mY0,K is coprime with mY0,Kσ for all nontrivial
σ∈G. Recall that g|mY0,K and gσ∤mY1,K for all σ∈G.
Thus condition (ii) is equivalent to (ii)′mY0,K=gk for some positive integer k and,
for all nontrivial σ∈G, gσ≠g and gσ∤mX,K.
The first assertion of (ii)′ holds if and only if V0 is the
g-primary component of XK. By Lemma 2.4 (iii), the second
assertion in (ii)′ holds if and only if b|d, degg=d/b, and gσ∤mX,K for all nontrivial σ∈G.
Also if (Y0)K is cyclic with minimal polynomial
gk, then k must be 1. Thus both of the conditions (i) and (ii) hold if and only if b|d, degg=d/b,
XK is g-primary cyclic, and gσ≠g and gσ∤cX,K for all nontrivial σ∈G
(recalling that mX,K and cX,K have the same set of irreducible divisors.)
∎
The next corollary follows immediately from Lemmas 2.5 and 2.4 (iii).
Corollary 2.6.
Let X∈M(c,qb)⊆M(n,q), where n=bc, let G=Gal(K/F), and let I={f1,…,fk}⊂Irr(q,b). Then XF is fi-primary cyclic, for every i≤k, if and only if there exists a set I′={g1,…,gk}⊆Irr(qb,1) with |I′|=k satisfying the following for each i∈{1,…,k}:
gi|fi, and XK is gi-primary cyclic;
for each nontrivial σ∈G, we have giσ≠gi, and giσ∉DivK(X).
3 A generalised cycle index for matrix algebras
Our main tool in enumerating matrices is the cycle index of the matrix
algebra M(n,q), introduced by Kung [11] and developed further by Stong [16], and based on Polya’s cycle index (see for example [15]) of a permutation group. We continue to use Notation 2.1. To each pair (h,λ), with h∈Irr(q) and λ a partition of a nonnegative integer, denoted |λ|, with |λ|∈[0,n],
assign an indeterminate xh,λ. Then the cycle index of M(n,q) is
the multivariate polynomial
ZM(n,q)(𝐱):=1|GL(n,q)|∑X∈M(n,q)(∏h∈DivFXxh,λ(X,h)),
where 𝐱 is a vector representing the set of indeterminates xh,λ occurring, and λ(X,h) is a partition (of an integer) uniquely determined by the structure of the action of X on the primary component Vh as described in Definition 3.1 below.
In this section we generalise the cycle index of Kung and Stong to include variables associated with a finite number of irreducible polynomials which do not necessarily divide cX,F(t). We will apply this more general version in our study of primary cyclic matrices. We begin by presenting the original cycle index theorem. In this section V=Fc is viewed solely as an F-space, where, recall, F=GF(q).
Definition 3.1.
Let X∈M(n,q),h∈Irr(q), and let αh be the multiplicity of h in cX,F(t), so that X acts on the h-primary component Vh of VF with characteristic polynomial hαh and αhdegh=dim(Vh)F. (In particular, αh=0 if Vh=0.) There is a direct sum decomposition of Vh into FX-submodules
Vh=Vλ1⊕⋯⊕Vλr
with each Vλi cyclic, such that the restriction of X to Vλi has minimal polynomial hλi, and λi≥λi+1 for all i. The λi are uniquely determined by X (see [9, Theorem 11.19]). Define the partition λ(X,h) as the sequence
λ(X,h):=(λ1,λ2,…,λr,0,0,…).
Then λ(X,h) is a partition of αh, and as this partition is non-increasing, we often omit the ‘trailing zeroes’ and write (λ1,…,λr) if Vh≠{0} and ():=(0,0,…) (the empty partition of the integer zero) if Vh={0}.
In particular, λ(X,h)=() if h∉DivF(X), and otherwise λ(X,h) is determined by the sizes of the blocks in the Frobenius normal form of X|Vh.
See [9] for more information on the cyclic and primary decompositions, and on λ(X,h). Lemma 3.2 follows immediately from the definition of λ(X,h):
Lemma 3.2.
Let X∈M(n,q), h∈Irr(q), and λ=λ(X,h). Then the following hold:
h∉DivF(X) if and only if λ(X,h)=();
h∈DivF(X) and X is h-primary cyclic if and only if λ(X,h)=(λ1), with λ1>0, and in this case λ1 is the multiplicity of h in cX,F(t);
h∈DivF(X) and X is not h-primary cyclic if and only if λ(X,h) has at least two nonzero parts.
Definition 3.3.
Let λ be a partition of an integer |λ|, let h∈Irr(q), and let s=|λ|degh. If λ=(), define c(λ,degh,q)=1. If |λ|≥1, then there exists a matrix X:=Xλ,h∈M(s,q) such that cX,F(t)=h|λ|, and the cyclic decomposition of Fs described in Definition 3.1 determines the partition λ. In this case we define
c(λ,degh,q):=|CGL(s,q)(X)|.
Note that c(λ,degh,q) (the number of matrices in GL(s,q) which commute with X) depends only on degh and λ, since all matrices X with these properties are conjugate under elements of GL(s,q) (see again [9, Theorem 11.19]). The number of such matrices X is |GL(s,q)|/c(λ,degh,q), and this holds also for λ=() if we take GL(0,q) as the trivial group.
The Kung–Stong cycle index theorem is stated in terms of these quantities.
Theorem 3.4 (Cycle index theorem).
The generating function for the cycle index of a matrix algebra M(n,q) satisfies
1+∑n=1∞ZM(n,q)(𝐱)un=∏h∈Irr(q)(1+∑λ≠()xh,λ(h)u|λ|deghc(λ,degh,q)).
Theorem 3.4 assigns to each X∈M(n,q) a monomial
∏h∈DivFXxh,λ(X,h),
and sums over M(n,q). We generalise by forcing a certain finite collection of indeterminates to occur in the monomials for all matrices X, whether or not the corresponding irreducibles divide cX,F(t). The reason for this generalisation will become apparent when we apply this to the proof of Lemma 4.4 in Section 4: it permits us to ask questions about whether some (fixed) f∈Irr(q) divides cX,F(t).
Definition 3.5.
For a finite subset I⊂Irr(q), and partitions λ(X,h) as in Definition 3.1 (X∈M(n,q), h∈Irr(q)), the I-cycle index of M(n,q) is defined as
(3.1)ZM(n,q)(I)(𝐱):=1|GL(n,q)|∑X∈M(n,q)(∏h∈DivF(X)∪Ixh,λ(X,h)),
or equivalently
ZM(n,q)(I)(𝐱):=1|GL(n,q)|
(3.2) ×∑X∈M(n,q)((∏h∈DivF(X)xh,λ(X,h))(∏h∈I∖DivF(X)xh,())).
The Kung–Stong cycle index is precisely the I-cycle index with I=∅. We now prove the I-cycle index theorem.
Theorem 3.6 (I-cycle index theorem).
For a finite subset I⊆Irr(q) and λ(X,h), c(λ,degh,q) as in Definitions 3.1 and 3.3, the generating function for the I-cycle index of M(n,q) satisfies
∏h∈Ixh,()+∑n=1∞ZM(n,q)(I)(𝐱)un=∏h∈Irr(qb)∖I(1+∑λ≠()xh,λu|λ|deghc(λ,degh,q))
(3.3) ×∏h∈I(xh,()+∑λ≠()xh,λu|λ|deghc(λ,degh,q)).
Proof.
Our proof follows that of Stong in [16]. We consider the quantities in (3.3) as power series in the variables xh,λ, and treat u as a constant. Note that since I is finite, and for X∈M(n,q) the set DivFX is finite, each ZM(n,q)(I)(𝐱) on the left-hand side of (3.3), when expressed as in (3.2), is clearly a sum of products of finitely many of the xh,λ. Recall that c((),degh,q)=1 for all h∈Irr(q), and so
xh,()=xh,()u0.deghc((),degh,q).
Let {hi∣1≤i≤t}⊆Irr(q), and let {λi∣1≤i≤t} be a multiset of partitions such that λi may be () if hi∈I, and otherwise λi≠(). For each i, let ni=|λi|deghi, and let
n=∑i=1tni. The coefficient of ∏i=1txhi,λi on the right-hand side of (3.3) is
(3.4)(∏i=1t1c(λi,deghi,q))un.
On the other hand, the coefficient of ∏i=1nxhi,λi on the left-hand side of (3.3) is equal to 1 if n=0, and otherwise is un/|GL(n,q)| times the number of matrices X∈M(n,q) having characteristic polynomial ∏i=1thi|λi|, with λ(X,hi)=λi for each i. Each of these matrices X is uniquely determined by the following data:
its primary decomposition V=Vh1⊕⋯⊕Vhn is such that dimVhi=ni, noting that we may have λ(X,hi)=() if hi∈I; and
for each primary component Vhi, the partition λi=λ(Xhi,hi).
There are exactly
|GL(n,q)|∏i=1n|GL(ni,q)|
direct sum decompositions of V with the appropriate dimensions, and on each of the parts
Vhi, there are exactly
|GL(ni,q)|/c(λi,deghi,q) matrices Xhi such that λ(Xhi,hi)=λi, as noted in Definition 3.3. Thus the coefficient of ∏i=1txhi,λi on the left-hand side of (3.3) is
un|GL(n,q)|⋅|GL(n,q)|∏1≤i≤t|GL(ni,q)|⋅∏1≤i≤t|GL(ni,q)|c(λi,deghi,q)
=∏1≤i≤t1c(λi,deghi,q)un,
which equals (3.4).
∎
4 Counting
By evaluating (3.3) in Theorem 3.6 at different values of 𝐱, we can enumerate subsets of M(c,qb) having certain properties based on their minimal polynomials. In particular, we wish to count matrices in M(c,qb)⊆M(n,q) which are f-primary cyclic for some f∈Irr(q,b) (recall that by Proposition 2.5, b is the smallest degree for which such f-primary matrices exist). We begin this section by introducing some quantities which will simplify our rather complicated calculations.
Note that while the I-cycle index theorem was presented for the full matrix algebra M(n,q), it may be applied directly to the irreducible subalgebra M(c,qb), provided that we treat M(c,qb) in its own right, rather than as a subalgebra of M(bc,q).
Definition 4.1.
Define the following quantities:
ωn(u,q):=∏i=1n(1-uq-i) for {u∈ℂ:|u|<q};
ω(u,q):=∏i=1∞(1-uq-i) for {u∈ℂ:|u|<q};
G(u,q,n):=1+∑λ≠()u|λ|c(λ,n,q) for {u∈ℂ:|u|<1};
P(u,q):=1+∑n=1∞unωn(1,q) for {u∈ℂ:|u|<1};
S(u,q):=∑n=1∞unqn(1-q-1) for {u∈ℂ:|u|<q};
where c(λ,n,q) is as in Definition 3.3.
Note that
ωn(1,q)=|GL(n,q)||M(n,q)|,
and that ω(1,q)=limn→∞|GL(n,q)|/|M(n,q)| exists.
These definitions simplify our rather complicated calculations later. The following results will be used to manipulate the generating functions:
Lemma 4.2.
The following relations hold between the quantities in Definition 4.1, for |u|<1, and in case (4.3) for |u|<q:
(4.1)G(u,q,1)=P(uq-1,q);
(4.2)∏h∈Irr(q)G(udegh,q,degh)=P(u,q);
(4.3)P(u,q)=11-uP(uq-1,q)=∏i=0∞(1-uq-i)-1;
(4.4)S(u,qb)=1(qb-1)u(1-uq-b);
Proof.
(4.1): In equation (3.3) set I=∅, and for all λ, set xh,λ=0 if h≠t-1 and xt-1,λ=1. Using (3.1), we see that the right-hand side of (3.3) is equal to G(u,q,1), while the left-hand side is
1+∑n=1∞un⋅(# unipotent elements in M(n,q)|GL(n,q)|)
which
by Steinberg’s theorem [2, Theorem 6.6.1] is equal to
1+∑n=1∞unqn(n-1)|GL(n,q)|=P(uq-1,q).
(4.2): The left-hand side of equation (4.2) is equal to the right-hand side of
(3.3) if we set I=∅ and all the xh,λ=1. Thus by (3.3), using also (3.1) and Definition 4.1,
this is equal to
1+∑n=1∞|M(n,q)||GL(n,q)|un=P(u,q).
(4.3): In [1, p. 19] we find the equality, for |u|<q,
∏r=1∞(1-uq-r)-1=1+∑n=1∞unqn(n-1)/2∏i=1n(qi-1),
the right-hand side of which is equal to P(uq-1,q). This proves the second equality of (4.2), and the first equality follows on substituting u for uq-1 into the second equality.
(4.4): This is a routine geometric series calculation.
∎
Definition 4.3.
Noting that Irr(q,b) is a finite set,
for a nonempty subset I⊆Irr(q,b), define
pcbI(I,c,qb):={X∈M(c,qb)∣XF is f-primary cyclic for all f∈I};
define
pcb(c,qb):=⋃I⊆Irr(q,b)I≠∅pcbI(I,c,qb);
so PM(c,qb)=|pcb(c,qb)|/|M(c,qb)|;
for nonempty I⊆Irr(q,b), define generating functions for pcbI and pcb:
PCBI(I,u,qb):=1+∑c=1∞|pcbI(I,c,qb)||GL(c,qb)|uc,
PCB(u,qb):=1+∑c=1∞|pcb(c,qb)||GL(c,qb)|uc.
Note that pcb(c,qb) is the set of matrices X∈M(c,qb) such that XF is f-primary cyclic for some f∈Irr(q,b): hence the name ‘primary cyclic, degree b’. Our end goal is to find and investigate PCB(u,qb): to do so we compute a formula for PCBI(I,u,qb), depending only on the size of I and the parameters q,b, and a relationship between the functions PCB,PCBI.
Lemma 4.4.
Let I={f1,…,fk}⊆Irr(q,b), with |I|=k.
Then for the generating function PCBI(I,u,qb) as in Definition 4.3 and |u|<1, we have
PCBI(I,u,qb)=P(u,qb)H(u,qb)k,
where H(u,qb):=bP(u,qb)-b(1-u)-bS(u,qb), with P(u,qb),S(u,qb) as in Definition 4.1.
Proof.
Let G=Gal(K/F). By Corollary 2.6, a matrix XF is fi-primary cyclic for all i∈I if and only if there exists a subset I′={g1,…,gk}⊆Irr(qb,1) with |I′|=k such that, for each i≤k, gi divides fi,
the gi-primary component of XK is cyclic, and for 1≠σ∈G, giσ does not divide mX,K. For such a subset I′ and, for h∈Irr(qb), set
xh,λ={0if h∈I′, and either λ=(), or λ≠(|λ|,0,…) with |λ|>0;0if for some nontrivial σ∈G, hσ∈I′;1if h∈I′, λ=(|λ|,0,…) with |λ|>0, and hσ∉I′ for 1≠σ∈G;1if h∉⋃σ∈G(I′)σ.
Let X∈M(c,qb): then X contributes 1 to the I′-cycle index (3.1), evaluated at 𝐱, if and only if, for every gi∈I′, λ(X,gi)=(|λ|,0,…), with |λ|>0, and λ(X,giσ)=() for all nontrivial σ∈G; and X contributes zero otherwise. This is precisely the set of matrices which, for every gi∈I′ and nontrivial σ, are gi-primary cyclic and giσ∤mX,K(t).
Arguing as in the proof of Theorem 3.6 (and in particular noting (3.4)), the number of matrices X which contribute 1 to the I′-cycle index of M(c,qb) is the same for each choice of the k-element set I′. There are bk subsets I′ corresponding to a given k-subset I⊆Irr(q,b), and by Corollary 2.6, each member of pcbI(I,c,qb) contributes 1 for exactly one of these subsets I′.
Hence the number of X∈M(c,qb) for which (3.1) evaluates to 1 with the above assignment of the xh,λ is therefore |pcbI(I,c,qb)|/bk. Set I*=⋃σ∈G(I′)σ. Then since by Corollary 2.6 we have gσ≠g for each g∈I′ and each nontrivial σ∈G, we have |I*|=bk. Hence, by Theorem 3.6,
PCBI(u,qb)=bk∏h∈(Irr(qb)∖I*)(1+∑λ≠()u|λ|deghc(λ,degh,qb))
×∏h∈I′(∑λ=(|λ|,0,…)≠()u|λ|deghc(λ,degh,qb)).
Now since every polynomial in I′ is linear, and since by [7, Table 1] we have that c((|λ|,0,…),1,qb)=q|λ|b(1-q-b), it follows that
∏h∈I′(∑λ=(|λ|,0,…)≠()u|λ|deghc(λ,degh,qb))=∏h∈I′(∑α=1∞uαqαb(1-q-b))
=S(u,qb)k.
Then by Definition 4.1 and Lemma 4.2, and since |I*|=bk,
PCBI(u,qb)
=bkS(u,qb)k(∏h∈(Irr(qb)∖I*)G(udegh,qb,degh))
=bkS(u,qb)k(∏h∈Irr(qb)G(udegh,qb,degh))(∏h∈I*G(u,qb,1))-1
=bkS(u,qb)kP(u,qb)P(uq-b,qb)-bk
=bkS(u,qb)kP(u,qb)((1-u)P(u,qb))-bk
=P(u,qb)(bS(u,qb)(1-u)-bP(u,qb)-b)k
and the result follows.
∎
5 Combining results
The function PCBI(I,u,qb) counts the number of elements of M(c,qb) which are f-primary cyclic (when viewed as elements of the larger algebra M(bc,q)) for all the irreducibles f in the k-subset I⊆Irr(q,b). We seek the proportion of matrices which are f-primary cyclic for somef∈Irr(q,b). The inclusion-exclusion principle yields the following:
Theorem 5.1.
For any q,b, let H(u,qb)=bP(u,qb)-b(1-u)-bS(u,qb), where S(u,qb),P(u,qb) are as in Definition 4.1, and let N=|Irr(q,b)|. Then
PCB(u,qb)=P(u,qb)(1-(1-H(u,qb)N).
Proof.
Any X∈M(c,qb) which is primary cyclic as an element of M(n,q), relative to some element of Irr(q,b), lies in pcbI(I,c,qb) for at least one nonempty subset I of Irr(q,b). Thus for every positive integer c,
pcb(c,qb)=⋃I⊆Irr(q,b)I≠∅pcbI(I,c,qb),
and by the inclusion-exclusion principle, setting N=|Irr(q,b)|,
|pcb(c,qb)|=∑i=1N(-1)i+1(∑I⊆Irr(q,b),|I|=i|pcbI(I,c,qb)|).
By Lemma 4.4, the value of |pcbI(I,c,qb)| depends only on |I|. Thus choosing an i-element subset Ii of Irr(q,b), we have
∑I⊆Irr(q,b),|I|=i|pcbI(I,c,qb)|=(Ni)|pcbI(Ii,c,qb)|.
Hence
|pcb(c,qb)|=∑i=1N(-1)i+1(Ni)|pcbI(Ii,c,qb)|,
and a similar relationship holds for the generating functions:
PCB(u,qb)=∑i=1N(-1)i+1(Ni)|PCBI(Ii,u,qb)|.
Now by Lemma 4.4, writing P=P(u,qb) and H=H(u,qb), we have
PCB(u,qb)=P(∑i=1N(-1)i+1(Ni)PHi)
=P(1-∑i=0N(-1)i(Ni)Hi)
=P(1-(1-H)N).∎
Theorem 5.1 shows us how to compute easily (using, e.g., Mathematica [17]) the Taylor coefficients of PCB(u,qb), and hence values of |pcb(c,qb)|/|M(c,qb)| for small c.
We summarise some small cases in Table 1. The data suggests that the proportion has a nonzero constant term. If this were true in general, then for every triple (c,q,b) the proportion would be nontrivial. We use methods from complex analysis to examine the asymptotic behaviour as c→∞. The following fact can be found, for example, in [6, Lemma 1.3.3].
Lemma 5.2.
Suppose that g(u)=∑anun and g(u)=f(u)/(1-u) for |u|<1. If f(u) is analytic with a radius of convergence R>1, then an→f(1) as n→∞, and |an-f(1)|=O(d-n) for any d<R.
We apply this lemma to PCB(u,qb) to obtain one of our main results:
Table 1
The proportion PM(c,qb) of f-primary cyclic matrices in a subalgebra M(c,qb) of M(bc,q), relative to some f∈Irr(q,b). As qb grows, PM(c,qb) rapidly approaches a positive constant.
| c | 𝑷𝑴(𝒄,𝒒𝒃) |
| 1 | 1-qq-b |
| 2 | 12+(32-b2)q-b+(-b2-q+bq2-q22)q-2b |
| +(-1+bq2-q22)q-3b+qq-4b |
| 3 | 23+(13-q2)q-b+(43-b2-b26+q-bq2)q-2b |
| +(-13-b23-bq2+b2q6-q2+bq22-q36)q-3b |
| +(-1-b23+q2-bq+b2q3-q2+bq2-q33)q-4b |
| +(-1+b2-b26-bq2+b2q3+bq2-q33)q-5b |
| +(-bq2+b2q6+q2+bq22-q36)q-6b+(1+q2)q-7b-qq-8b |
Proof of Theorem 1.1 (i).
By Lemma 5.1, writing N=|Irr(q,b)|=N(q,b),
PCB(u,qb)=P(u,qb)(1-(1-H(u,qb))N).
Set L(u,qb)=(1-u)PCB(u,qb). By (4.3) and Definition 4.1 we have
L(u,qb)=ω(1,qb)-1(1-(1-H(u,qb))N).
Now by Lemma 4.2, writing S=S(u,qb) and P=P(u,qb) for brevity,
(5.1)H(u,qb)=bP-b(1-u)-bS=bqb-1u1-uq-b∏i=1∞(1-uq-bi)b
which converges for all |u|<qb. In particular, H(1,qb) exists and satisfies
(5.2)H(1,qb)=bq-b(1-q-b)2ω(1,qb)b.
It follows that
L(1,qb)=ω(1,qb)-1(1-(1-H(1,qb))N).
By Lemma 5.2, we have limc→∞|pcb(c,qb)|/|GL(c,qb)|=L(1,qb), and so
PM(∞,qb)=limc→∞|pcb(c,qb)||M(c,qb)|
=ω(1,qb)limc→∞|pcb(c,qb)||GL(c,qb)|
=1-(1-H(1,qb))N.
Theorem 1.1 (i) is proved.
∎
The following lemma is used to study the asymptotics of PM(∞,qb) as qb grows:
Lemma 5.3.
If x∈[0,13), then
∏i=1∞(1-xi)>1-x-x2>59.
If x∈[0,12] and b is a positive integer, then
1-2bx≤(1-x-x2)b.
If x>1, then xlogx>x1/2.
If x∈(0,12), then
11-x<1+x+2x2.
Proof.
(i) This is proved in [13, Lemma 3.5].
(ii) We prove this inductively on b. For b=1 the inequality holds since 0≤x<1. Suppose that b≥1 and
1-2bx≤(1-x-x2)b. Then
(1-x-x2)b+1≥(1-x-x2)(1-2bx)=1-(2b+1)x+(2b-1)x2-2bx3,
and this is at least 1-2(b+1)x-x2 since 2bx2(1-x)≥0. Then since 0≤x≤12, we have -x2≥-x, which yields the required inequality, and hence the result is proved by induction.
(iii) Since x>1, the required inequality is equivalent to x1/2>logx. Examining the derivative of f(x):=x1/2-logx, we see that, for x>1, f(x) has a unique minimum at x=4. Then since f(4)>0, it follows that f(x)>0 for all x>1.
(iv) Let f(x)=(1+x+2x2)(1-x). The required inequality holds if and only if
f(x)>1 (since x∈(0,12)).
On multiplying we find f(x)=1+x2-2x3 and this is greater than 1 since x2-2x3=x2(1-2x)>0.
∎
Lemma 5.4.
Let t≥1, ϵ∈(0,1), and suppose that
c>max{1,(tlog(1-ϵ))2}. Then
ct≤(1-ϵ)-c.
Proof.
The inequality is equivalent to
tlogc≤-clog(1-ϵ).
Since logc>0, and since 0<1-ϵ<1 implies log(1-ϵ)<0, this holds if and only if
(5.3)-tlog(1-ϵ)≤clogc.
By Lemma 5.3 (iii), c/logc>c1/2, and by assumption c1/2≥-t/log(1-ϵ),
yielding inequality (5.3).
∎
Proposition 5.5.
Let PM(∞,qb)=limc→∞|pcb(c,qb)|/|M(c,qb)|,
where b≥2 and qb>4. Then
-4beqb/2<PM(∞,qb)-(1-e-1)<1+beqb+2(1+b)2eq2b,
so that
|PM(∞,qb)-(1-e-1)|<4e-1bq-b/2.
Proof.
By Theorem 1.1 (i), PM(∞,qb)=1-(1-H(1,qb))N, with H(1,qb) as in (5.2) above. We consider the behaviour of (1-H(1,qb))N as q and b grow. Since ω(1,qb)=∏i=1∞(1-q-bi), and since q-b≤14, it follows from Lemma 5.3 (i) that
1-q-b-q-2b<ω(1,qb)<1-q-b.
Applying Lemma 5.3 (ii) with x=q-b gives
(5.4)1-2bq-b<ω(1,qb)b<1-q-b.
Now as N:=N(q,b)=1b∑d|bμ(d)qd/b, we have
1b(qb-2qb/2)≤N(q,b)≤qbb.
Thus
(1-H(1,qb))(1/b)qb≤(1-H(1,qb))N≤(1-H(1,qb))(1/b)(qb-2qb/2),
and so (with H denoting H(1,qb) for simplicity):
qbblog(1-H)≤Nlog(1-H)≤1b(qb-2qb/2)log(1-H).
Using the inequality 1-1x≤logx≤x-1, which holds for all x>0, we have
qbbHH-1≤Nlog(1-H)≤-1b(qb-2qb/2)H.
Substituting for H using (5.2) and rearranging gives
-ω(1,qb)b(1-q-b)2-bq-bω(1,qb)b≤Nlog(1-H)
≤-1b(qb-2qb/2)bq-b(1-q-b)2ω(1,qb)b.
Using the right inequality of (5.4) and observing a geometric series gives
-ω(1,qb)b(1-q-b)2-bq-bω(1,qb)b>-(1-q-b)(1-q-b)2-bq-b(1-q-b)
=-11-q-b-bq-b
=-11-(1+b)q-b.
If qb≥9, then applying Lemma 5.3 (iv) with x=(1+b)q-b gives
-11-(1+b)q-b≥-1-(1+b)q-b-2(1+b)2q-2b,
and this is true also (with equality) if qb=8. Thus for all qb>4, we have
Nlog(1-H)>-1-(1+b)q-b-2(1+b)2q-2b.
On the other hand, we have, using the left inequality in (5.4), and since qb>4 implies
1(1-q-b)2<1(3/4)2=169<2,
that
-1b(qb-2qb/2)bq-b(1-q-b)2ω(1,qb)b
=-(1-2q-b/2)ω(1,qb)b(1-q-b)2
<-(1-2q-b/2)(1-2bq-b)(1-q-b)2
=-1+2q-b/2+2(b-1)q-b-4bq-3b/2+q-2b(1-q-b)2
<-1+2(2q-b/2+2(b-1)q-b-4bq-3b/2+q-2b).
Since -4bq-3b/2 is negative, and 2q-b>q-2b, this is less than
-1+4q-b/2+4bq-b. Thus we have proved that
-1-(1+b)q-b-2(1+b)2q-2b<Nlog(1-H)
<-1+4q-b/2+4bq-b,
and so exponentiating,
exp(-1-(1+b)q-b-2(1+b)2q-2b)<(1-H)N
<exp(-1+4q-b/2+4bq-b).
Now for 0≤x≤1 we have ex≤1+x+34x2 and e-x>1-x (see for example [8, Lemma 2.3]). The first inequality implies that
(1-H)N<e-1(1+4q-b/2+4bq-b+34(4q-b/2+4bq-b)2)
=e-1+4e-1q-b/2+4e-1(b+3)q-b+24e-1bq-3b/2
+12e-1b2q-2b
<e-1+4be-1q-b/2,
and the second inequality gives
(1-H)N>e-1(1-(1+b)q-b-2(1+b)2q-2b)
=e-1-e-1(1+b)q-b-2e-1(1+b)2q-2b.
Recalling that PM(∞,qb)=1-(1-H)N, the first inequality in the statement is proved by subtracting these two values from 1. The second inequality follows immediately from the first.
∎
5.1 Proof of Theorem 1.1 (ii)
Finally, we apply the method of Wall (see [6]) to M(c,qb) to prove the second part of our main result, which gives a useful lower bound on |pcb(c,qb)|/|M(c,qb)| for sufficiently large c. The inequality we require is proved in Proposition 5.10, thus completing the proof of Theorem 1.1. We introduce the following notation, following Fulman in [5]: for a function X(u) of a complex variable, we denote by [uc]X the coefficient of uc in the Maclaurin series of X.
Lemma 5.6.
Let X(u) be an analytic function of a complex variable, and let t be a positive integer.
For all c≥1, we have
[uc](X(u)1-u)=∑i=0c[ui]X(u).
Suppose there exist constants a1,a2 such that |[uc]X(u)|≤a1a2-c, for all c≥0. Then for all c≥0, we have
|[uc](X(u)t)|≤a1t(c+1)t-1a2-c.
Proof.
(i) Let xi:=[ui]X(u). Then
X(u)1-u=(x0+x1u+⋯)(1+u+u2+⋯)
=x0+(x0+x1)u+(x0+x1+x2)u2+⋯
and (i) follows.
(ii) We proceed by induction on t. The result holds for t=1 by assumption. Let xij:=[uj]X(u)i, and suppose that t≥2 and that part (ii) holds for X(u)t-1. Then
X(u)t=X(u)t-1X(u)
=(xt-1,0+xt-1,1u+⋯)(x10+x11u+⋯)
=∑c=0∞∑i=0c(xt-1,i)(x1,c-i)uc,
and so by induction
|[uc]X(u)t|=|∑i=0cxt-1,ix1,c-i|
≤∑i=0c(a1t-1(i+1)t-2a2-i).(a1a2-(c-i))
=a1t∑i=0c((i+1)t-2a2-c)
≤a1t(c+1)t-1a2-c,
since
∑j=1c+1jt-2≤(c+1)t-1. The result now follows by induction.
∎
Lemma 5.7.
Let J(u,qb)=(1-uqb)PCB(uqb,qb). Then for c≥2, we have
[uc]J(u,qb)=(|pcb(c,qb)||M(c,qb)|-|pcb(c-1,qb)||M(c-1,qb)|)qbc.
Proof.
By definition of J(u,qb) we have
J(u,qb)=(1-uqb)∑c=1∞|pcb(c,qb)||M(c,qb)|(uqb)c
=|pcb(1,qb)||M(1,qb)|uqb+∑c=2∞(|pcb(c,qb)||M(c,qb)|-pcb(c-1,qb)|M(c-1,qb)|)qbcuc,
which completes the proof.
∎
The remainder of this section is devoted to finding an upper bound on the coefficient [uc]J(u,qb), and using this to prove Theorem 1.1 (ii).
Lemma 5.8.
Define L(u,qb):=∏i=1∞(1-uq-bi)=(P(u,qb)(1-u))-1, and suppose b>1. Then
L(u,qb)=11-u(1+∑c=1∞(-1)cqbcuc∏i=1c(qbi-1))
and for all c≥1, we have
|[uc]L(u,qb)|≤aLq-bc,
where aL=2qb.
Proof.
The first assertion follows from [1, Corollary 2.2]. For the second, observe that
[uc]L=1+∑k=1c(-1)kqbk∏i=1k(qbi-1)
=1+∑k=1c((-1)k(qbk-1)∏i=1k(qbi-1)+(-1)k∏i=1k(qbi-1))
=1+∑k=1c((-1)k∏i=1k-1(qbi-1)+(-1)k∏i=1k(qbi-1))
=1-1+(-1)c∏i=1c(qbi-1)
=(-1)cq-bc(c-1)/2∏i=1c(1-q-bi),
as all but the first and last terms of the alternating sum cancel. Now for all c, we have both q-bc(c-1)≤qb.q-bc, and
∏i=1c(1-q-bi)>∏i=1∞(1-q-bi)>12
by Lemma 5.3 (i), and so |[uc]L|≤2qb.q-bc.
∎
Lemma 5.9.
Let J(u,qb) be as defined in Lemma 5.7, and suppose that b>1. Let
Mqb=(max{b-1,qb/b}log(3/4))2.
Then for c≥Mqb, and
aJ=83(bqbqb-12b(2qb)bqb2)qb/b
we have
|[uc]J(u,qb)|<aJ,
and hence
|pcb(c+1,qb)|M(c+1,qb)|-pcb(c,qb)|M(c,qb)||<aJq-bc.
Proof.
Using Theorem 5.1, the fact that P(uqb,qb)=P(u,qb)(1-uqb)-1, the definition of H(uqb,qb) from the right-hand side of (5.1), and (4.3), we have (with N=|Irr(q,b)|)
J(u,qb)=(1-uqb)P(uqb,qb)(1-(1-H(uqb,qb))N)
=P(u,qb)[1-(1-bqbqb-1u1-u∏i=1∞(1-uqb-bi)b)N]
=P(u,qb)[1-(1-bqbqb-1u1-u∏i=0∞(1-uq-bi)b)N]
=P(u,qb)[1-(1-bqbqb-1u1-uP(u,qb)-b)N]
(5.5)=P(u,qb)[1-(1-bqbqb-1u(1-u)b-1L(u,qb)b)N],
since L(u,qb)=((1-u)P(u,qb))-1 by definition.
By Lemma 5.8, we have |[uc]L|≤aLq-bc, where aL=2qb, and hence by Lemma 5.6 (ii), |[uc]Lb| is bounded above by aLb(c+1)b-1q-bc. Then
|[uc]((1-u)b-1Lb)|≤∑k=0b(bk)aLb(c-k+1)b-1q-b(c-k)
<∑k=0b(bk)aLb(c+1)b-1q-b(c-b)
=aLb(c+1)b-1q-b(c-b)(∑k=0b(bk))
=2baLbqb2(c+1)b-1q-bc.
Multiplication by u ‘shifts’ the coefficients, so that c is replaced with c-1: that is,
|[uc](u(1-u)b-1L(u,qb)b)|<2baLbqb2+bcb-1q-bc.
It follows that
|[uc](bqbqb-1u(1-u)b-1L(u,qb)b)|<bq2bqb-12baLbqb2cb-1q-bc,
and since subtracting the function from 1 has no effect on the absolute value of any coefficients when c≥1, we have (for c>1) that
|[uc](1-bqbqb-1u(1-u)b-1L(u,qb)b)|<bqbqb-12baLbqb2cb-1q-bc,
and so by Lemma 5.4 with t=b-1, ϵ=14, we have, for
c≥(b-1log(3/4))2 (and hence c>1),
|[uc](1-bqbqb-1u(1-u)b-1L(u,qb)b)|<bqbqb-12baLbqb2(3qb4)-c.
Again applying Lemma 5.6 (ii), with t=N, and since by [12], N≤qb/b, we have
|[uc](1-bqbqb-1u(1-u)b-1L(u,qb)b)N|
<(bqbqb-12baLbqb2)qb/b(c+1)qb/b(3qb4)-c.
Then setting
aJ=83(bqbqb-12baLbqb2)qb/b
and again applying Lemma 5.4 (with c+1 in place of c and t=qb/b), we have, for c>(qbblog(3/4))2, that
(c+1)qb/b<(1-14)-c-1=43(34)-c,
and so
|[uc](1-bqbqb-1u(1-u)b-1L(u,qb)b)N|<3aJ8.43(9qb16)-c
=aJ2(9qb16)-c.
Now by (5.5), we may attain an expression for J(u,qb) by multiplying the above equation by P(u,qb): doing so, and recalling that by definition
[uc]P(u,qb)=ω(c,qb)-1=∏j=1c(1-q-bj),
gives
|[uc]J(u,qb)|<∑i=0c∏j=ic(1-q-bj)aJ2(9qb16)-i
<aJ2(∑i=0c(9qb16)-i)<aJ,
since ∑i=0c(9qb/16)-i<2 when qb≥4.
The second assertion follows directly from Lemma 5.7.
∎
Proposition 5.10.
Suppose b≥2, and let aJ,Mqb be as defined in Lemma 5.9. Then for c>Mqb,
|PM(c,qb)-PM(∞,qb)|=||pcb(c,qb)||M(c,qb)|-limn→∞|pcb(c′,qb)||M(c′,qb)||
≤aJ1-q-bq-bc.
In particular, Theorem 1.1
(ii) holds.
Proof.
By Lemma 5.9, we have
|pcb(c+1,qb)|M(c+1,qb)|-pcb(c,qb)|M(c,qb)||<aJq-bc,
and so for every c′>c>Mqb we have
||pcb(c′,qb)||M(c′,qb)|-|pcb(c,qb)||M(c,qb)||≤∑m=cc′-1||pcb(m+1,qb)||M(m+1,qb)|-|pcb(m,qb)||M(m,qb)||
<∑m=cc′-1aJq-bm
=q-bcaJ(∑m=0c′-c-1q-bm)
<q-bcaJ(∑m=0∞q-bm)
=q-bcaJ(11-q-b).∎
and
Cheryl E. Praeger
