SIMILARITY CLASSES OF 3 × 3 MATRICES OVER A LOCAL PRINCIPAL IDEAL RING
NIR AVNI, URI ONN, AMRITANSHU PRASAD, AND LEONID VASERSTEIN

arXiv:0708.1608v2 [math.GR] 5 May 2008

Abstract. In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.

1. Introduction 1.1. Overview. Let A be a local principal ideal commutative ring. Let m denote its maximal ideal and let k = A/m be the residue field. Let ℓ ∈ N ∪ {∞} denote the length of A, that is, the smallest positive integer for which mℓ = 0. Denote by Mn (A) and GLn (A), the ring of matrices over A and its group of units, respectively. Definition 1. Two matrices α and α′ in Mn (A) are called similar if there exists a matrix X ∈ GLn (A) such that Xα = α′ X. The similarity classes of invertible matrices are the conjugacy classes in GLn (A). The classification problem of similarity classes in n × n matrices over rings has been considered by several authors and is considered to be a highly nontrivial quest, unless the ring in hand happens to be a field. For example, in [Nag78, §4] it is proved that already for A = Z/p2 Z, the classification of similarity classes in M4n (A) contains the matrix pair problem in Mn (Z/pZ). The aim of this paper is to classify similarity classes in M3 (A) and GL3 (A). In order to put things into perspective we shall now take a short excursion in some known results on similarity classes. The similarity classes of matrices with entries in a field have been well understood in terms of their rational canonical forms for a long time and are described, for example, by Dickson in [Dic59, Chapter V]. Over rings, only partial results are available; In [Dav68], Davis has shown using Hensel’s method, that two matrices in Mn (Z/pℓ Z) which are zeroes of a common polynomial whose reduction modulo p has no repeated roots are similar if and only if their reductions modulo p are similar. In a similar vein, using an extension of the Sylow theorems (attributed to P. Hall), Pomfret [Pom73] has shown that for a finite local ring, invertible matrices whose orders are coprime to the characteristic of the residue field are similar if and only if their images in the residue field are similar.
2000 Mathematics Subject Classification. 15A21, 15A30, 15A54. Key words and phrases. matrices, similarity, local ring. The second author was supported by the Edmund Landau Minerva Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany) and by ISF and BSF (USIsrael).
1

In [Gru80], Grunewald has given an algorithm for determining whether two matrices in GLn (Q) are conjugate by an element of GLn (Z). For the special case where n = 3, Appelgate and Onishi [AO82] have given a simpler algorithm to solve the same problem. Given any two matrices α and α′ in SLn (Zp ), Appelgate and Onishi [AO83] have given an explicit method to determine a positive integer ℓ such that α and α′ are conjugate in SLn (Zp ) if and only if they are conjugate in SLn (Z/pℓ Z), thereby reducing the conjugacy problem in the uncountable group SLn (Zp ) to a finite one. In [Nec83], Nechaev has classified the similarity classes in the case n = 3 and ℓ = 2. Close to the present article is [Piz83], where Pizarro has given a set of representatives of the similarity classes in M3 (A) modulo scalar shift, when A is a finite quotient of a complete discrete valuation ring. These representatives, however, do not lend themselves to the explicit enumeration of similarity classes when A is finite, which is one of the main goals of this paper. Such explicit classification and enumeration has two implications in representation theory. The first is that together with the orbit method for p-adic Lie groups [How77], the aforementioned classification is an important ingredient in computing the representation zeta function of SL3 (O), where O is the ring of integers of a p-adic field, see [AO07]. The second is a positive indication that the isomorphism type of the group algebra CGLn (A) whenever A is finite, depends only on k, as conjectured in [Onn07], since we prove that dimC Z (CGLn (A)) = |Sim(GLn (A))| depends only on k for n ≤ 3. 1.2. Some notation. Throughout we fix a uniformizing element π ∈ m. For each a ∈ A there is a unique integer 0 ≤ v(a) ≤ ℓ, called the valuation of a, such that a can be written as the product of π v(a) and a unit. For 1 ≤ ı ≤ ℓ we write Aı for the quotient A/mı and A× for its units. We fix a section k = A1 ֒→ A which maps zero to zero with image ı K1 ⊂ A. We can then define compatible sections Aı ֒→ A for all 1 ≤ ı < ℓ, identifying Aı with Kı = { ı−1 aj π j | aj ∈ K1 } ⊂ A as sets. We also have canonical identifications j=0 Aℓ−j → π j A. The main examples of A that we have in mind are the rings of integers of local fields and their finite length quotients. A shorthand notation is used for some commonly occurring matrices. The identity matrix is denoted by I. The symbol xij is used to denote the matrix I + xE ij , where E ij is the elementary matrix with all entries zero, except for a ‘1’ in the ith row and j th column. Given a polynomial f (x) = xn − an−1 xn−1 − · · · − a1 x − a0 , its companion matrix is the matrix  0 1 0 0 0 1 . . . . . . . . . 0 0 0 a0 a1 a2 ··· 0 ··· 0 . .. . . . ··· 1 · · · an−1 

  Cf = C(a0 , . . . , an−1 ) =   

  .  

The characteristic polynomial of the companion matrix Cf is f (x). Also, recall that a companion matrix represents a cyclic endomorphism, i.e. there exist v ∈ An such that {C i v | 0 ≤ i < n} is a basis for An . A block diagonal matrix is denoted by D(d1 , . . . , dn )
2

where d1 , . . . , dn are the diagonal entries, which may themselves be square matrices or scalars. Finally, another special matrix that will come up often in this paper is   0 πm 0 E(m, a, b, c, d) = 0 0 1 + dI, m ∈ N, a, b, c, d ∈ A. a b c

The special case E(ℓ, 0, 0, c, d) will play an important role and will be denoted J(c, d) for simplicity.

1.3. Acknowledgements. The first and second authors thank Alex Lubotzky for supporting this research. The authors thank the referee for suggesting some improvements to the first draft of this paper. 2. A baby version: 2 × 2 matrices In lack of an adequate reference with complete results (partial results can be found in [Nob77]), and since it serves as a solid basis for the reasoning in n = 3, we now describe the similarity classes in the case n = 2. 2.1. Representatives. Similarity classes in Mn (A) and GLn (A) for n = 2 are considerably easier to tackle than for n > 2. The underlying reason is the following dichotomy: Any element in M2 (k) is either scalar or cyclic. Lemma 2.1. Any element α ∈ M2 (A) can be written in the form α = dI + π  β with  ∈ {0, . . . , ℓ} maximal such that α is congruent to a scalar matrix modulo m , with unique d ∈ K and unique β ∈ M2 (Aℓ− ) cyclic. Proof. The only fact that is not straightforward is that β is cyclic. But the maximality of ¯  implies that β is not a scalar modulo m, and hence its image β ∈ M2 (k) is cyclic. Using Nakayama’s Lemma β must be a cyclic as well. Clearly, α = dI + π  β is similar to α′ = d′ I + π  β ′ if and only if d = d′ and β is similar to β ′ . Every cyclic matrix is conjugate to a companion matrix. Moreover, two companion matrices are conjugate if and only if the polynomials defining them are equal. We thus have Theorem 2.2. For any α ∈ M2 (A), let , d and β be as above. Then α is similar to the matrix dI + π  C(− det(β), tr(β)). Thus  ∈ {0, . . . , ℓ}, d ∈ K and tr(β), det(β) ∈ Aℓ− completely determine the similarity class of α in M2 (A). The similarity classes in GL2 (A) are represented by the subset of these elements such that d ∈ A× for  ≥ 1, or if  = 0 then det(β) ∈ A× . Remark 1. In [Nec83], Nechaev has introduced the notion of a canonically determined matrix. This is a matrix α whose similarity class is completely determined by its Fitting invariants. These are the ideals in A[x] generated by the m × m minors of xI − α (thought of as a matrix in Mn (A[x])), for m = 1, . . . , n. Nechaev conjectured, and proved it in certain cases, that a matrix is canonically determined if and only if all these ideals are principal ideals. For n = 2, the ideal generated by the entries of xI − α is A(x − d) + Aπ  where d
3

and  are as in Lemma 2.1. When ℓ = ∞, the characteristic polynomial of α determines the characteristic polynomial of β. By Theorem 2.2, it follows that the Fitting invariants determine the similarity class of α, which refutes an extension of his conjecture to the case ℓ = ∞. Representing A as a factor ring of a discrete valuation domain, Kurakin [Kur06] has refined the Fitting invariants. 2.2. Enumeration. Assume now that the residue field k is finite of cardinality q. We wish to count the number of similarity classes in M2 (Aı ) and GL2 (Aı ). Although one can count them directly using Theorem 2.2, we shall use a recursive approach which will be very useful later on. Let η : Mn (Aı+1 ) → Mn (Aı ) denote the reduction map. Then, for any similarity class Ω ⊂ Mn (Aı ), the inverse image η −1 (Ω) is a disjoint union of similarity classes in Mn (Aı+1 ). For n = 2 the branching rules are the following. Let aı denote the number of similarity classes which are scalar matrices and let bı denote the number of the other similarity classes. We wish to establish a recursive relation between (aı+1 , bı+1 ) and (aı , bı ). Scalar matrices in M2 (Aı+1 ) necessarily lie over scalar matrices M2 (Aı ), hence aı+1 = qaı . The non-scalar similarity classes in M2 (Aı+1 ) can come from two sources; they can either lie over a scalar matrix in M2 (Aı ), in which case they are q 2 aı in number, or, they can lie over a non-scalar similarity class in M2 (Aı ), in which case they are q 2 bı in number (both assertions follow from Theorem 2.2). We therefore have aı+1 q 0 = 2 2 bı+1 q q The initial values are vM2 = Setting T =
q 0 q2 q2

aı . bı a1 q−1 = 2 . b1 q −q

a1 q = 2 b1 q

and

vGL2 =

, we get that T =
ı

0 ı −1 , q ı+1 qq−1 q 2ı

qı

hence |Sim (M2 (Aı )) | = ǫT ı−1 vM2 = (q 2ı+1 − q ı )/(q − 1) |Sim (GL2 (Aı )) | = ǫT ı−1 vGL2 = q 2ı − q ı−1 where ǫ is the row vector (1, 1). 3. Representatives for similarity classes of 3 × 3 matrices 3.1. Similarity classes over a field. The similarity classes over a field are given by their rational canonical forms [HK71, Chapter 7, Theorem 5]: Theorem 3.1 (Similarity classes in M3 (k)). Every matrix in M3 (k) is similar to exactly one of the following: (1) A scalar matrix aI, with a ∈ k . (2) A matrix of the form D(a, b, b), with a, b ∈ k distinct.
4

(3) A matrix of the form  a 0 0 0 a 1 , a ∈ k. 0 0 a (4) A companion matrix of the form   0 1 0 0 0 1 , a, b, c ∈ k. a b c 3.2. Some reductions. To begin with, we assign some invariants to similarity classes. Proposition 3.2. For any α ∈ M3 (A) let  ≥ 0 be the largest integer for which α is congruent to a scalar matrix modulo m . Then α can be written, in a unique manner, as α = dI + π  β, where d ∈ K and β ∈ M3 (Aℓ− ) is a matrix that is not congruent to a scalar matrix modulo m. Moreover, two such matrices α1 = d1 I + π  β1 and α2 = d2 I + π  β2 are similar if and only if d1 = d2 and β1 is similar to β2 . It follows that the assignment α → (α) is a similarity invariant. Writing α = dI + π  β, we are therefore reduced to classifying similarity classes of matrices β which are not congruent to a scalar modulo m, that is, of types (2), (3) or (4) in Theorem 3.1. If (4) occurs, then arguing as in Lemma 2.1, it follows that β is cyclic, hence determined by its characteristic polynomial. If (2) occurs we are essentially reduced to the n = 2 case. Proposition 3.3. Suppose that the reduction of β modulo m is D(a, b, b), with a, b ∈ k and a = b. Then β is similar to a unique matrix of the form (∗) D(a, bI + π  C(c, d)), where 1 ≤  ≤ ℓ, c, d ∈ Aℓ− , a ∈ A and b ∈ K with a − a ≡ b − b ≡ 0 (mod m). ¯ Proof. Denote the entries of β by βij . The entries in places (1, 2) and (1, 3) of the conjugation of β by a matrix of the form I + xE 12 + yE 13 are (1) and (2) β13 + (β33 − β11 )y + β23 x − β13 y 2 − β23 xy respectively. Let X be the scheme defined by the polynomials (1) and (2). By our assumptions, β23 , β32 ≡ 0 (mod m) and β22 − β11 , β33 − β11 ≡ 0 (mod m). Therefore, the point (0, 0) ∈ k2 is a non-singular point of X ×Spec(A) Spec(k). By the Hensel lemma, it can be lifted to a point (x0 , y0 ) ∈ X(A). Conjugating β by I + x0 E 12 + y0 E 13 , we can assume that β12 = β13 = 0. Note that this conjugation does not change the reduction of β modulo m. Similarly, there are x1 , y1 ∈ A, such that conjugating β by I + x1 E 21 + y1 E 31 makes β21 and β31 equal to zero. Since this last conjugation does not change the entries in places (1, 2) and (1, 3), the result is a block diagonal matrix. The classification of similarity classes for 2 × 2 matrices (Theorem 2.2) shows that β is similar to a matrix of the kind in (∗). That no two distinct matrices of type (∗) are similar follows from Lemma 3.4 below applied to β − bI.
5



β12 + (β22 − β11 )x + β32 y − β12 x2 − β13 xy

Lemma 3.4. Let B and B ′ be two matrices in M2 (A) which are congruent to 0 modulo m, and let a, a′ ∈ A× . Then the two block matrices β= a 0 0 B and β ′ = a′ 0 0 B′

are similar if and only if a = a′ and B is conjugate to B ′ . Proof. Clearly, the condition for similarity in the statement of the lemma is sufficient. To see that it is necessary, suppose X ∈ GL3 (A) is such that Xβ = β ′ X. Write X as a block y matrix x W . Evaluation of the above equality in terms of block matrices gives z xa yB a′ x a′ y = . za W B B′z B′W Since B ≡ 0 mod m and a′ is a unit, comparing the top right entries shows that y ≡ 0 mod m. Similarly, z ≡ 0 mod m. It follows that x and W are invertible, which implies that a = a′ and B is similar to B ′ . It remains to analyze case (3) of Theorem 3.1, to which we dedicate the next subsection. 3.3. The hard case. Assume that β ∈ M3 (A) form  ¯ d 0 ¯ ¯ = 0 d J(0, d) 0 0 is such that its reduction modulo m is of the  0 ¯ 1 , d ∈ k. ¯ d

¯ Proposition 3.5. Any matrix β ∈ M3 (A) which lies above J(0, d) ∈ M3 (k) is a conjugate of a matrix of the form   0 πm 0 E = E(m, a, b, c, d) = 0 0 1 + dI a b c ¯ with m ≥ 1 and a, b, c, d − d ≡ 0 (mod m). Proof. Reduce β to a matrix of the form E(m, a, b, c, d) by the following sequence of similarity transformations: (1) conjugation of β by a diagonal matrix makes the (2, 3)-entry equal to 1. (2) conjugation by (β13 )12 kills the (1, 3)-entry. (3) addition of a scalar matrix kills the (1, 1)-entry. (4) conjugation by (β21 )31 kills the (2, 1)-entry. (5) conjugation by (β22 )32 kills the (2, 2)-entry. (6) conjugating by a diagonal matrix makes the (1, 2)-entry of the form π m for some positive integer m.

In order to avoid redundancies we should check when two representatives E(m1 , a1 , b1 , c1 , d1 ) and E(m2 , a2 , b2 , c2 , d2 ) are conjugate. This question is the core of the difficulty when passing from n = 2 to n = 3. A similar problem is considered in [Piz83]. Note that the four types of essentially cyclic matrices defined by Pizarro are similar to some E(m, a, b, c, d), in particular they are similar to each other. The parametrization given in [Piz83, Thm 2.17]
6

is ineffective in the sense that it does not give a reasonable way of enumerating the classes for finite rings. This is the main point in which we take an alternative route and focus on the branching of classes of level ı in level ı + 1, which is the subject of the next section. Consequently, we shall be able to enumerate the similarity classes in the case of finite rings in Section 5. 4. Analysis of the hard case Let α1 = E(m1 , a1 , b1 , c1 , d1) and α2 = E(m2 , a2 , b2 , c2 , d2) be in M3 (Aı ). The elements α1 and α2 are similar if and only if there exist X = (xij ) ∈ GL3 (Aı ) which satisfies (3) Such a matrix X should satisfy Y21 : x31 = a2 x23 + (d2 − d1 )x21 (4) Y22 : x32 = π m2 x21 + b2 x23 + (d2 − d1 )x22 Y23 : x33 = x22 + (c2 − d1 + d2 )x23 Y13 : x12 = π m1 x23 − c2 x13 + (d1 − d2 )x13 Using equations (4) and the fact that d1 − d2 , π mi , ai , bi , ci are all congruent to 0 mod m, we get that det(X) is congruent to x11 x2 modulo m. Therefore, we get the extra condition 22 (5) det : x11 , x22 ∈ A× . There are five additional equations Y11 : π m1 x21 − a2 x13 + (d1 − d2 )x11 = 0 Y12 : π m1 x22 − π m2 x11 − b2 x13 + (d1 − d2 )x12 = 0 (6) Y31 : a1 x11 + b1 x21 + c1 x31 − a2 x33 + (d1 − d2 )x31 = 0 Y32 : a1 x12 + b1 x22 + c1 x32 − π m2 x31 − b2 x33 + (d1 − d2 )x32 = 0 Y33 : a1 x13 + b1 x23 + c1 x33 − x32 − (c2 − d1 + d2 )x33 = 0 whose solution in general is very complicated. To this end, we shall narrow down possibilities by looking at the centralizers of representatives and then be able to solve these equations. 4.1. Centralizers. In this section we shall take a closer look on the centralizers in GL3 (Aı ) of the elements E(m, a, b, c, d), which are subgroups that are defined over A. We shall use the Greenberg functor F [Gre61, Gre63] which enables us to view them as algebraic groups over k. Taking α = α1 = α2 = E(m, a, b, c, d) in (3) we get that X = (xij ) commutes with α if and only if ax13 = π m x21 (7) bx13 = π m (x22 − x11 ) bx21 = a(x22 − x11 ), with x12 , x31 , x32 , x33 determined, respectively, by equations Y13 , Y21 , Y22 , Y23 above, together with an additional free variable x23 . The system of equations (7) possess symmetries which can be made more transparent. Write a = u1 π t1 and b = u2 π t1 for some invertible elements u1 , u2 ∈ A× and replace m by t3 . Then, the system (7) can be written as ı
7

Y := α1 X − Xα2 = 0.

(i) π t1 y3 = π t3 y1 , (8) with new variables y1 = u−1 x21 , 1 y2 = u−1 (x22 − x11 ), 2 y3 = x13 . In order to solve these equations in M3 (Aı ), we may assume, by relabeling the variables, that t1 ≤ t2 ≤ t3 . It then follows that equation (ii) can be omitted. We are left with two equations y2 = π t2 −t1 y1 y3 = π t3 −t1 y1 This gives {(y1 , y2, y3 ) satisfying (8)} = Aı × At1 × At1 ,
ı+2t which using the Greenberg functor can be identified with the affine k-space Ak 1 . To ensure that X ∈ GL3 (Aı ), we need x11 , x22 ∈ A× by (5). The elements of the centralizer of α in ı GL3 (Aı ), are identified under the Greenberg functor with the following k-varieties, depending on the values of the ti ’s, which we now relabel according to the original variables: m, v(a) and v(b). The centralizers depend on the relative value of min{m, v(a)} and v(b) as follows

(ii) π t2 y3 = π t3 y2 , (iii) π t2 y1 = π t1 y2 ,

mod π ı−t1 mod π ı−t1

v(b) ≤ min{m, v(a)}: v(b) > min{m, v(a)}:

F StabGL3 (Aı ) (α) ≃ A× k

2

× Ak

3ı+2v(b)−2

, ,

F StabGL3 (Aı ) (α) ≃ A× × Ak k

3ı+2 min{m,v(a)}−1

where A× = Ak {0} stands for the punctured affine line. The difference between the two k cases arises from (7), as x11 and x22 can be chosen independently if v(b) ≤ min{m, v(a)} but not if v(b) > min{m, v(a)}. We shall now describe the branching rules of these representatives when passing from M3 (Aı−1 ) and M3 (Aı ). We fix a compatible system of representatives in level ı which lie above their reduction in level ı − 1. Claim 4.1. Let J(¯, d) = E(ı − 1, 0, 0, c, d) ∈ M3 (Aı−1 ) with c ≡ 0 mod m. Then the c ¯ ¯ ¯ ¯ ¯ in M3 (Aı ) represent disjoint following four types of representatives which lie above J(¯, d) c similarity classes. Type I  0 0 0 0 0 1 + dI 0 0 c   Type II 0 π ı−1 ǫ 0  0 0 1 + dI ı−1 ı−1 π a π b c  (ǫ ∈ {0, 1})  0 π ı−1 ǫ 0  0 0 1 + dI ı−1 π a 0 c  (a, ǫ) = (0, 0) Type IIIǫ

a ∈ k, b ∈ k× , ǫ ∈ {0, 1}
8

¯ with c, d ∈ Aı such that c − c ≡ d − d ≡ 0 (mod mı−1 ). ¯

5ı−2 Proof. The k-varieties which correspond to each type are F (ZI ) ≃ A× × Ak , F (ZII ) ≃ k 2 5ı−4 5ı−3 A× × Ak and F (ZIII ) ≃ A× × Ak , hence each type remains invariant under conjugak k tion. In order to separate the subtypes III0 and III1 we use equation Y12 = 0, which reads in this case x11 ǫ1 ≡ x22 ǫ2 (mod m), therefore if two matrices of type III are similar they must have the same ǫ.

2

Remark 2. When k is finite, Claim 4.1 can be proved without using the Greenberg machinery. We just note that if the size of k is q, then the size of the stabilizer of the matrix is (q − 1)2 q 5ı−2 , (q − 1)2 q 5ı−4 , (q − 1)q 5ı−3 in cases I,II, and III respectively. Since no two of these numbers can be equal, types I,II,III are disjoint. In fact, one can show (see [AO07] for complete details) that the centralizers of the types I,II and III modulo m are isomorphic to Autk[x] (k[x]/(x2 ) ⊕ k), respectively. 4.2. Sieving away redundancies. The list of representatives in Claim 4.1 is exhaustive, since we covered all the possible lifts of J(¯, d). To make sure that there are no repetitions c ¯ we need a more delicate analysis. Theorem 4.2. The following is an exhaustive list of non-similar elements lying above J(¯, d) c ¯ (where c ≡ 0 mod m) ¯ ¯ I E(ı, 0, 0, c, d), with d − d ≡ c − c ≡ 0 (mod mı−1 ). ¯ ı−1 ¯ II E(ı − 1, 0, bπ , c, d), with c − c ≡ d − d ≡ 0 (mod mı−1 ), b ∈ k× . ¯ ı−1 III0 E(ı, π , 0, c, 0), with c − c ≡ 0 (mod mı−1 ). ¯ ı−1 ¯ III1 E(ı − 1, aπ , 0, c, d), with d − d ≡ 0 (mod mı−1 ), a ∈ k. ¯ All in all, there is a bijection between the conjugacy classes of elements lying above J(¯, d) c ¯ 2 2 × 2 and the set k ⊔ k × k ⊔ k ⊔ k (corresponding to the classes I, II, III0 , III1 respectively). Proof. Case I. We claim that the matrices α1 = E(ı, 0, 0, c1, d1 ) and α2 = E(ı, 0, 0, c2, d2 ) in M3 (Aı ), with c1 − c2 ≡ d1 − d2 ≡ 0 (mod mı−1 ) are GL3 (Aı )-conjugate if and only if d1 = d2 and c1 = c2 . Indeed, in this case Y11 = (d1 − d2 )x11 , and since x11 is a unit, d1 − d2 = 0. The equality of the cj ’s now follows from cj = tr(αj ). Case II. We first claim that we may assume that ǫ = 1. This follows from the following Lemma 4.3. If v(b) ≤ min{m, v(a)} then E(m, a, b, c, d) ∼ E(v(b), aπ m−v(b) , b, c, d). Proof. The matrix  1 −ec e X = eaπ −v(b) 1 0 , 0 ec 1 where e = b−1 (π m − π v(b) ), realizes the similarity. Note that although b is not invertible, e and aπ −v(b) are well defined as v(b) ≤ v(a), m. We now claim that α = E(ı − 1, aπ ı−1 , bπ ı−1 , c, d) with b ∈ k× is similar to a matrix of the form α′ = E(ı − 1, 0, bπ ı−1 , c′ , d′), i.e. that a can be eliminated. Indeed, one checks that
9

Gm (k)2 × Ga (k) and Gm (k) × Ga (k)2 ,



conjugating α with  1 0 0 1 0 , Xe =  −e 2 ı−1 ı−1 eπ −2eπ 1 gives
−1 Xe αXe = E(ı − 1, (a − eb)π ı−1 , bπ ı−1 , c − 3eπ ı−1 , eπ ı−1 ),



e ∈ Aı ,

and since b is invertible, there exist a choice of e such that a−be ≡ 0 (mod m). Summarizing the last two steps, we may assume that a matrix of type II can be conjugated to a matrix of the form E(ı − 1, 0, bπ ı−1 , c, d) with b ∈ k× . We check that two such matrices are similar if and only if they have the same b, c and d. If α1 = E(ı − 1, 0, b1 π ı−1 , c1 , d1 ) and α2 = E(ı − 1, 0, b2 π ı−1 , c2 , d2 ) are similar and (c1 − c2 ) ≡ (d1 − d2 ) ≡ (b1 − b2 ) ≡ 0 (mod mı−1 ), then Y32 = 0 =⇒ b1 = b2 Y31 = 0 =⇒ x21 ≡ 0 Y11 = 0 =⇒ d1 = d2 (mod m)

tr(α1 ) = tr(α2 ) =⇒ c1 = c2 . Case III0 . First, observe that for any matrix α = E(ı, aπ ı−1 , 0, c, d) of type III0 we may assume that a = 1, since we know that a is invertible, and by conjugating α with D(a, 1, 1) we get E(ı, π ı−1 , 0, c, d). Second, observe that the matrix   1 d2 π ı−1 + dc −d 1 0 , X = 0 ı−1 0 dπ 1 conjugates E(ı, π ı−1 , 0, c, dπ ı−1) to E(ı, π ı−1 , 0, c + 3dπ ı−1 , 0). It follows that any matrix of type III0 is equivalent to a matrix of the form E(ı, π ı−1 , 0, c, d0), where d0 ∈ Aı is a fixed ¯ element which lies above d, and c varies among the lifts of c. Since tr(α) = c + 3d0 , all these ¯ elements are distinct and the assertion is proved. Case III1 . We claim that α1 = E(ı−1, π ı−1 a1 , 0, c1, d1 ) and α2 = E(ı−1, π ı−1 a2 , 0, c2, d2 ), with c1 − c2 ≡ d1 − d2 (mod mı−1 ) are GL3 (Aı )-conjugate if and only if their traces are equal and a1 = a2 . To get the only if part, we use equation Y12 = 0 to deduce that x11 ≡ x22 (mod m), and substituting the latter equality into Y31 = 0 gives that a1 = a2 . The equality of the traces is of course a necessary condition as well. To prove the if part, assuming equality of traces c2 = c1 + 3π ı−1 (d1 − d2 ) and that a1 = a2 , the matrix   1 0 0 1 0 , X =  −δ ı−1 2 ı−1 π δ −2π δ 1 where δ = d1 − d2 , is an invertible solution to the equation α1 X = Xα2 .
10

4.3. The branching rules. Classes of types I, II and III branch in the following way when increasing the level level (9) level ı+1 I ı
}I ooo DDDD ooo}}} DD ooo }} DD ooo }}} D ooo o

II

III0 III0

III1 III1 .

II

III0

III1

II

This follows from the fact that the relation between v(b) and min{v(a), m} remains unchanged when the level increases for types II and III. Moreover, from Claim 4.1 it follows that types II and III lift in a ‘regular’ fashion. Namely, for each similarity class C in M3 (Aı ), the set of similarity classes in M3 (Aı+1 ) that lie over C is in bijection with k3 , which is the same behavior as of cyclic elements. 5. Enumeration of similarity classes We now specialize to the case where k is a finite field with q elements, and count the number of similarity classes in M3 (Aı ) and in GL3 (Aı ) for all ı ∈ N. Given a matrix α ∈ M3 (Aı ), recall that  is the maximal integer such that α is scalar modulo m . Then modulo m+1 , α is conjugate to a matrix dI + π  β where modulo m, β is either cyclic, D(a, b, b) with a = b or equal to J(0, e) . If β ≡ D(a, b, b) (mod m) then it is conjugate to a matrix in one of the following forms     a 0 0 a 0 0 0 b 0  0 b 0 + π m 0 0 or 0 C 0 0 b 0 0 b where C ∈ M2 (Aı−m ) is cyclic and ı > m > . If β equals J(0, e) modulo m then it is of type I above or it lies over types II, III0 , III1 above. We divide the set of matrices to the following classes: (i) dI. (ii) dI + π  D(a, b, b), (a = b). (iii) dI + π  J(c, d). (iv) The rest of the options. Before giving the precise analysis, observe right away that these possibilities are partially ordered: (i) ≺ (ii), (iii), (iv), (ii) ≺ (iv), and (iii) ≺ (iv), in the sense that each possibility may branch only to possibilities greater or equal to it. Indeed, • A similarity class of type (i) in level ı splits into all the other similarity classes in level ı + 1, these are precisely the similarity classes in M3 (k): there are q similarity classes of type (i), q 2 − q similarity classes of type (ii), q similarity classes of type (iii) (all are of the form dI + π ı J(0, e)) and q 3 similarity classes of type (iv) (all of the form dI + π ı β where β is cyclic). • A similarity class of type (ii) splits into similarity classes of type (ii) and (iv) only: there are q 2 similarity classes of type (ii) over it and q 3 similarity classes of type (iv) over it, all are of the form   a 0 0  0 b 0 + π ı 0 0 0 C 0 0 b
11

with C cyclic in M2 (k). • For type (iii) we already computed: there are q 2 classes of type (iii) and (q 3 − q 2 ) + q + q 2 = q 3 + q classes of type (iv) above it. • Although type (iv) contains various different subtypes, every conjugacy class of type (iv), splits into q 3 similarity classes of type (iv) in level ı + 1. Altogether, the numbers ηıi , ηıii ηıiii and ηıiv of similarity classes of type (i),(ii),(iii) and (iv), respectively, in level ı satisfy the following recursion:  i   i   i  ηı q 0 0 0 ηı ηı+1 ii   ηıii   ηıii  q 2 − q q 2 ηı+1  0 0   iii  = T  iii  =  ηı   q ηı+1  0 q2 0  ηıiii  iv ηıiv q3 q3 + q q3 q3 ηıiv ηı+1 with the initial condition   i  q η1 ii  η1  q 2 − q  vM3 =  iii =  η1   q  iv q3 η1 Thus, if ǫ = (1, 1, 1, 1), we have (10) sM3 (ı) = |Sim(M3 (Aı ))| = ǫT ı−1 vM3 , sGL3 (ı) = |Sim(GL3 (Aı ))| = ǫT ı−1 vGL3 .    i q−1 η1 ii  η1  (q − 1)(q − 2)  =  iii =   η1   q−1 3 2 iv q −q η1 

and

vGL3

A straightforward induction on ı gives the following formula for T ı . Claim 5.1.  qı 0 0 0 q 2ı − q ı q 2ı 0 0  T ı =  q ı qı −1 2ı  q−1 0 q 0 ı −1 ı −1 θı q 2ı+1 qq−1 q 2ı−1 (q 2 + 1) qq−1 q 3ı  qı − 1 q−1 q4 + 1 q−1 qı + 1 q+1 − q3 + 1 q−1

where θı is given by

θı = q ı−1

Combining Claim 5.1 and (10) gives Theorem 5.2. The number of similarity classes of matrices in M3 (Aı ) is sM3 (ı) = q 3ı+3 + q 3ı−1 − q 2ı+2 − q 2ı+1 − q 2ı − q 2ı−1 + 2q ı . (q − 1)(q 2 − 1)

The number of similarity classes of elements in GL3 (Aı ) is sGL3 (ı) = q 3ı+2 − q 3ı + 2q 3ı−2 − q 2ı+1 − q 2ı−1 − 2q 2ı−2 + 2q ı−1 . q2 − 1
12

These can be packed in terms of generating functions
∞

ZM3 (z) =
ı=0 ∞

sM3 (ı)z ı = sGL3 (ı)z ı =
ı=0

1 (q − 1)(q 2 − 1) 1 2 −1 q

q 3 + q −1 q 2 + q + 1 + q −1 2 − + 1 − q3z 1 − q2z 1 − qz .

ZGL3 (z) =

q 2 − 1 + 2q −2 q + q −1 + 2q −2 2q −1 − + 1 − q3z 1 − q2z 1 − qz References

[AO82] Harry Appelgate and Hironori Onishi, The similarity problem for 3 × 3 integer matrices, Linear Algebra Appl. 42 (1982), 159–174. [AO83] H. Appelgate and H. Onishi, Similarity problem over SL(n, Zp ), Proc. Amer. Math. Soc. 87 (1983), no. 2, 233–238. [AO07] Nir Avni and Uri Onn, Representation Zeta functions for SL3 , in preparation, 2007. [Dav68] Ronald W. Davis, Certain matrix equations over rings of integers, Duke Math. J. 35 (1968), 49–59. [Dic59] Leonard E. Dickson, Algebraic theories, Dover Publications Inc., New York, 1959. [Gre61] Marvin J. Greenberg, Schemata over local rings, Ann. of Math. (2) 73 (1961), 624–648. , Schemata over local rings. II, Ann. of Math. (2) 78 (1963), 256–266. [Gre63] [Gru80] Fritz J. Grunewald, Solution of the conjugacy problem in certain arithmetic groups, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Stud. Logic Foundations Math., vol. 95, North-Holland, Amsterdam, 1980, pp. 101–139. [HK71] Kenneth Hoffman and Ray Kunze, Linear algebra, Second edition, Prentice-Hall Inc., Englewood Cliffs, N.J., 1971. [How77] Roger E. Howe, Kirillov theory for compact p-adic groups, Pacific J. Math. 73 (1977), no. 2, 365– 381. [Kur06] V. L. Kurakin, Similarity invariants for matrices over a commutative Artinian chain ring, Mat. Zametki 80 (2006), no. 3, 403–412. [Nag78] S. V. Nagorny˘ Complex representations of the general linear group of degree three modulo a power ı, of a prime, Zap. Nauˇn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 75 (1978), 143–150, c 197–198, Rings and linear groups. [Nec83] A. A. Nechaev, Similarity of matrices over a commutative local Artinian ring, Trudy Sem. Petrovsk. (1983), no. 9, 81–101. [Nob77] Alexandre Nobs, Die irreduziblen Darstellungen von GL2 (Zp ), insbesondere GL2 (Z2 ), Math. Ann. 229 (1977), no. 2, 113–133. [Onn07] Uri Onn, Representations of automorphism groups of rank two finite O-modules, math.RT/0611383 (2007). [Piz83] Antonio Pizarro, Similarity classes of 3 × 3 matrices over a discrete valuation ring, Linear Algebra Appl. 54 (1983), 29–51. [Pom73] J. Pomfret, Similarity of matrices over finite rings, Proc. Amer. Math. Soc. 37 (1973), 421–422. Nir Avni Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond Safra Campus, Givat Ram, Jerusalem 91904, Israel avni.nir@gmail.com Uri Onn Ben-Gurion university of the Negev, Beer-Sheva 84105, Israel urionn@math.bgu.ac.il
13

Amritanshu Prasad The Institute of Mathematical Sciences, CIT campus, Chennai 600 113, India amri@imsc.res.in Leonid Vaserstein Department of Mathematics, Penn State University, University Park PA 16802-6401, USA vstein@math.psu.edu

14