Dynamics of Linear and Affine Maps

c 2008 International Press ASIAN J. MATH. Vol. 12, No. 3, pp. 321–344, September 2008 004 DYNAMICS OF LINEAR AND AFFINE MAPS∗ RAVI S. KULKARNI† Abstract. The well-known theory of the “rational canonical form of an operator” describes the invariant factors, or equivalently, elementary divisors, as a complete set of invariants of a similarity class of an operator on a finite-dimensional vector space V over a given field F. A finer part of the theory is the contribution by Frobenius dealing with the structure of the centralizer of an operator. The viewpoint is that of finitely generated modules over a PID, cf. for example [8], ch. 3. In this paper we approach the issue from a “dynamic” viewpoint. We also extend the theory to affine maps. The formulation is in terms of the action of the general linear group GL(n), resp. the group of invertible affine maps GA(n), on the semigroup of all linear, resp. affine, maps by conjugacy. The theory of rational canonical forms is connected with the orbits, and the Frobenius’ theory with the orbit-classes, of the action of GL(n) on the semigroup of linear maps. We describe a parametrization of orbits and orbit-classes of both GL(n)- and GA(n)-actions, and also provide a parametrization of all linear and affine maps themselves, which is independent of the choices of linear or affine coordinate systems, cf. sections 7, 8, 9. An important ingredient in these parametrizations is a certain flag. For a linear map T on V, let ZL (T ) denote its centralizer associative F-algebra, and ZL (T )∗ the multiplicative group of invertible elements in ZL (T ). In this situation, we associate a canonical, maximal, ZL (T )-invariant flag, and precisely describe the orbits of ZL (T )∗ on V, cf. section 3. The classical theory uses only invariance under T , i.e. V is considered only as a module over F[T ]. The finer invariance under ZL (T ), i.e. considering V as a module over ZL (T ), makes the construction of the flag canonical. We believe that this flag has not appeared before in this classical subject. Using this approach, we strengthen the classical theory in a number of ways. Key words. Rational canonical form, centraliser, dynamical types AMS subject classifications. 12E99, 15A01, 15A02, 15A21, 37C45 1. Introduction. Let F be a field, and V an n-dimensional vector space over F. Let L(V) denote the set of all linear maps from V to V. Underlying V there is the affine space A. Intuitively, A has no distinguished base-point which one can call as the“zero”, or the “origin”. However there is a well-defined notion of “difference of points”. When we distinguish a base-point O, and call it the zero, then there is a well-defined notion of addition, making A into a vector space. An affine map of A is a map (A, v) : V → V of the form (A, v)(x) = Ax + v, where A is in L(V), and x, b are in V. Then (1.1) (A1 , v1 ) ◦ (A2 , v2 ) = (A1 ◦ A2 , A1 v2 + v1 ). This formula shows that A(V) is a semigroup with identity under composition, and L(V) is a sub-semigroup of A(V). It is important to note that the representation (A, v) depends on the choice of the base-point. However the semigroup of affine maps, and the form of an affine map is independent of this choice. Indeed, let O be a base-point making A into a vector space V. Let P be another point of A with the associated vector a. Let x resp y be vector representations of a point Q w.r.t. base-points O and P . Then y = x − a. Let f be an affine map of the form (A, v) in the x-representation, and f (Q) = R. Then the x-representation of R is Ax + v = Ay + Aa + v. So the y-representation of R is Ay + Aa + v − a = Ay + w, where w = (A − I)a + v. Hence the y-representation of f is (A, w). The maps induced by the action of the group (V, +) on V, called the ∗ Received April 17, 2007; accepted for publication April 22, 2008. Institute of Technology (Bombay), Powai, Mumbai 400076, India, & Queens College and Graduate Center, City University of New York (punekulk@yahoo.com; kulkarni@math.iitb.ac.in). † Indian 321 322 R. S. KULKARNI translations, have the form τa = (I, a). They form a subgroup T, which is of course isomorphic to V. The above calculation shows that the expression for τa : x 7→ x + a remains the same no matter where we choose the base-point. In other words, “a” in τa has a dynamic as well as affine meaning. When A 6= I, the same calculation shows that “A” remains the same no matter where we take the base-point, but “v” may change. In other words, even when A 6= I, the “A” has a dynamic meaning, but “v” does not. The formula (1.1) shows that we have a well-defined homomorphism l : A(V) → L(V) given by l((A, v)) = A. We shall call A the linear part of (A, v). We shall also call v the translational part of (A, v), with the understanding that this specification depends on the choice of the base-point. Note that the kernel of l, namely l−1 (I) consists precisely of T. Let us also note an inconsistency in the usual terminology. Probably following the usage in the fields such as Transformation Groups, or Transformation Geometry, the phrase “an affine transformation” usually means a bijective affine map. On the other hand, in Linear Algebra, the phrase “a linear transformation” is used for nonbijective linear maps as well. To avoid confusion, and also for brevity, we use a neutral terminology “linear maps” or “affine maps” for not necessarily bijective maps. We may also like to define (1.2) (A1 , v1 ) + (A2 , v2 ) = (A1 + A2 , v1 + v2 ). As is well-known, L(V) becomes an associative F -algebra with this definition of addition, and taking composition as multiplication. However, we note that with the same definitions, in A(V), we do not get left distributivity of multiplication w.r.t addition. So A(V) becomes only a “near ring”, or better a “near F-algebra”, cf. for example, [10]. Let GL(V), resp. GA(V), denote the subsets of L(V), resp A(V) consisting of invertible elements. They form groups under composition, and GL(V) is a subgroup of GA(V). They act on L(V) resp. A(V) by conjugation. Namely f in GL(V), resp. GA(V), acts on L(V), resp. in A(V) by T 7→ f T f −1 . We denote these actions by φL resp φA . When there will be no confusion, we shall also abbreviate them to φ. Our interest in this paper is to study the “dynamics” of L(V) and A(V). We interpret the words “study of dynamics” to mean i) Parametrization of the φ-orbits of GL(V), resp. GA(V), on L(V), resp. A(V), cf. theorem 7.1. ii) In any action of a group G on a set X we have a notion of orbit-equivalence. Namely, x, y in X are orbit-equivalent iff the stabilizer subgroups Gx , and Gy are conjugate, cf. [9], theorem 2.1 for a precise statement on the structure of an orbitequivalence class, as a certain set-theoretic fibration. In the case of the φ-action a stabilizer subgroup at T in GL(V) resp. GA(V) is precisely the centralizer of T in ∗ GL(V) resp. GA(V). We denote this subgroup by ZL∗ (T ), resp. ZA (T ). For short, we call the orbit-equivalence in either the linear or the affine case, the z-equivalence. In this paper one of our main aims is to parametrize the z-equivalence classes of linear or affine maps, cf. theorem 7.2. iii) Parametrizations of linear, resp. affine, maps which depend only on F and dim V = n, and not on the choice of a linear resp. affine coordinate system, cf. theorem 7.3. Interestingly, in this case GL(V), resp. GA(V), are also subsets of L(V), resp. A(V), so there is also a notion of centraliers of T is L(V), resp A(V). We denote these centralizers by ZL (T ), resp ZA (T ). Then ZL (T ) is an F-subalgebra of L(V), and DYNAMICS OF LINEAR AND AFFINE MAPS 323 ∗ ZA (T ) is a sub-near-F-algebra of A(V). In fact, ZL∗ (T ), resp. ZA (T ), are precisely the groups of invertible elements in ZL (T ), resp ZA (T ). A basic notion of “equivalence of dynamics” in our case is the following. First, let Ti be elements of L(Vi ), i = 1, 2. We say that the Ti ’s are “dynamically equivalent” if there is a linear isomorphism h : V1 → V2 such that h ◦ T1 = T2 ◦ h. In this case we shall also say that the pairs (Vi , Ti ), i = 1,2, are dynamically equivalent. Similarly let Ti be elements of A(Vi ), i = 1, 2. We say that the Ti ’s are “dynamically equivalent” if there is an affine isomorphism h : A1 → A2 such that h ◦ T1 = T2 ◦ h. Next, let T be an element of L(V). We say that V is T -decomposable, or the pair (V, T ) is decomposable, or more loosely also that T is decomposable, if V is a direct sum of proper T -invariant subspaces. Otherwise V is said to be T -indecomposable. Since dim V is finite, clearly V is a direct sum of finitely many T -indecomposable invariant subspaces. Also the pair (V, T ) is indecomposable iff any dynamically equivalent pair is indecomposable. So from a dynamic viewpoint, a basic problem is to describe suitable models of indecomposable (V, T )’s, and secondly, given a pair (V, T ) to understand in general all decompositions of V into T -indecomposable subspaces. The first problem is solved by the theory of “rational canonical form of a square matrix” as a special case of modules over principal ideal domains. In this classical approach, the second problem gets obscured in a clever inductive proof. Following a dynamic viewpoint, we shall offer a new view of both the problems which, in some sense, is “dual” to the the classical approach. This approach strengthens the classical theory in a number of ways. In this paper we have considered the following aspects. i) Making an essential use of the ZL (T )-invariant flag, we determine the conjugacy classes, the centralizers, and z-classes of both linear and affine maps. The consideration of affine maps naturally arises in the study of affine ODEs (where F = R), cf. section 6. The texts of ODEs appeal to a general “method of variation of parameters”. In our opinion, the dynamic approach offers a better insight. At the same time, we are not aware of any literature on the general case, from the viewpoint of GA(V)-action on A(V). ii) We derive a necessary and sufficient condition for the existence of “S + N”decomposition of an operator – or its multiplicative analogue, the “SU”-decomposition of an invertible operator – and its relation to lifts of E-structures, cf. section 5. A basic observation going back to Maurer in special cases, cf. [1], [2], [4], is that a linear algebraic group over a field of characteristic 0, contains the semisimple and unipotent parts of each of its elements. If the base-field is of positive characteristic such decomposition in general does not exist. In this context one introduces the notion of perfectness of the base-field, which is a sufficient, but not necessary, condition for the existence of such decomposition. The dynamic viewpoint provides an overall insight on this ticklishly confusing point in the theory of linear algebraic groups. iii) We derive a dynamic interpretation of Frobenius’ “double commutant” theorem, cf, section 4. iv) We prove the following finiteness result: If F has the property that there are only finitely many field-extensions of F of degrees at most n, then there are only finitely many z-classes in L(V), and A(V). For example, if F is an algebraically closed field, a real closed field, or a local field then F has the stated property. This is a major example which illustrates the viewpoint that motivated [9]. In the forthcoming papers we hope to extend this work to transformations in other classical geometries. v) We obtain the generating functions for z-classes of linear maps, in some cases 324 R. S. KULKARNI when there are only finitely many such z-classes in each dimension. They are related to the generating function for partitions in an interesting way. They appear to be a new type of generating functions which have not appeared in number theory before. We only make some elementary observations regarding these generating functions. In some sense, the dynamical viewpoint has brought together two seemingly separate chapters of Linear Algebra, namely, i) “rational canonical forms”, and ii) “Frobenius’s theory of centralisers”, of operators on a finite-dimensional vector spaces. Before closing this introduction, we would like to remark that from a dynamic viewpoint, the minimal polynomial mT (x) is perhaps a more basic invariant than the more easily computable invariant χT (x), the characteristic polynomial of T . One indication of this fact is that mT (x) is defined even when V is infinite-dimensional. Most results of this paper can be suitably extended to the case when V is infinitedimensional, and mT (x) 6= 0. However, to keep a focus we do not elaborate on this direction here, as we had done in [9]. I wish to acknowledge the benefit of many conversations on the contents of this work with Rony Gouraige. His thesis, cf. [6], (City University of New York, 2006) partially extends this work to operators on finite-dimensional vector spaces over skewfields. It is also a pleasure to acknowledge some conversations with I. B. S. Passi, and Surya Ramana at the Harish-Chandra Research Institute, Allahabad, India, regarding the “S + N”-decomposition. 2. Classical Theory for L(V). Let mT (x) denote the minimal polynomial of T . If F[T ] denotes the F-algebra generated by T , then F [T ] ≈ F [x]/(mT (x)). Let mT (x) = Πri=1 pi (x)di be the decomposition into irreducible factors. Here pi (x) is a monic irreducible polynomial in F[x], and pi (x)’s are pairwise distinct. We shall call pi (x)’s the primes associated to T. The first step in the theory provides a decomposition V = ⊕ri=1 Vi into T -invariant subspaces. Here Vi = ker pi (T )di . We observe that this decomposition is in fact invariant under ZL (T ), the F-subalgebra of L(V) consisting of all operators commuting with T. Let Ti denote the restriction of T to Vi . Then mTi (x) = pi (x)di . Moreover we have a canonical F-algebra decomposition. (2.1) ZL (T ) = Πri=1 ZL (Ti ). So to describe the indecomposable pairs (V, T ) we have reduced to the situation where mT (x) = p(x)d , where p(x) is a monic irreducible polynomial in F[x]. At this point, we note a crucial example. Consider the algebra V = F[x]/(p(x)d ), but consider it only as an F-vector space. For u(x) in F[x] let [u(x)]denote the class of u(x) in F[x]/(p(x)d ). Let T = µx be the operator [u(x)] 7→ [xu(x)]. For i = 0, 1, . . . , d, let Vi = {[f (x)p(x)i ]|f (x) ∈ F[x]}. Clearly we have a flag of subspaces 0 = Vd ⊂ Vd−1 . . . ⊂ V1 ⊂ V0 = V. The claim is that Vi ’s are precisely all the T -invariant subspaces of V. Indeed let W be a T -invariant subspace of V. If [f (x)p(x)i ] is in W, then by T -invariance, for all g(x) in F[x], we have [g(x)f (x)p(x)i ] also in W. Let i be the least non-negative integer such that W contains an element of the form [f (x)p(x)i ] such that p(x) does not divide f (x). Then [f (x)] is a unit in the algebra F[x]/(p(x)d ). So [p(x)i ] is in W. It easily follows that W = Vi . Notice that no Vi has a proper complementary subspace. So (V, T ) is an indecomposable pair. For the future, notice that in this case dimF V = deg p(x)d = d deg p(x). DYNAMICS OF LINEAR AND AFFINE MAPS 325 A second and major step in the theory is that the converse of the observation in the above example is true. Theorem 2.1. Let (V, T ) be an indecomposable pair. Then it is dynamically equivalent to (F[x]/(p(x)d ), µx ), for some monic irreducible polynomial p(x) in F[x]. In view of the reduction in the first step, clearly an equivalent statement is the following. Theorem 2.2. Let (V, T ) be a pair such that mT (x) = p(x)d , where p(x) is a monic irreducible polynomial p(x) in F[x], of degree m. Then (V, T ) is a direct sum of T -invariant indecomposable subspaces, each dynamically equivalent to di (F[xi ]/(p(xi )P ), µxi ). Here di ≤ d, for at least one i, we have di = d, and dim V = m i di . We note that in case d = 1, the proof of either of these statements is easier than in the classical approach dealing with the more general situation of finitely generated modules over a PID. We sketch it first. The general case d ≥ 2 will be considered in the next section. Indeed observe that E = F[x]/(p(x)) w.r.t. to its standard additive and multiplicative structures is a field. In fact it is a simple field extension of F. Here “simple” means that E is generated over F by a single element [x]. Indeed [x] is a root of p(x) in E, and in the language of field theory [x] is a primitive element of E over F. Thus the operation of T on V, which amounts to multiplication by [x], or [xi ]’s in the standard models (F[xi ]/(p(xi )), µxi ), equips V with the structure of a vector space over E, which extends its structure as a vector space over F. In this E-structure, the T -invariant subspaces are precisely the E-subspaces of V. Also an F-linear operator S is in ZL (T ) iff S is an E-linear operator. It follows that (V, T ) is indecomposable iff dimE V = 1. Equivalently, (V, T ) is decomposable iff dimE V = r ≥ 2. In this case, a choice of an E-basis leads to a T -invariant decomposition of V into T -indecomposable subspaces. The ambiguity in the choice of a T -invariant decomposition of V is precisely the ambiguity of choosing an E-basis. Here ZL (T ) ≈ LE (V), and ZL (T )∗ ≈ GLE (V). The orbits of ZL (T )∗ on V are {0}, and V − {0}. As a module over the associative F-algebra ZL (T ) or the group ZL (T )∗ , V is irreducible. Moreover the T -action is dynamically semi-simple in the sense that every T -invariant subspace has a T -invariant complement. A word of caution regarding the use of the phrase “dynamically semi-simple”. There is another notion of semi-simplicity: namely, T is algebraically semi-simple if it is diagonalizable on V⊗ F̃, where F̃ denotes the algebraic closure of F, cf. [1]. Contrary to some mis-statements in the literature the notions of algebraic semi-simplicity and dynamic semi-simplicity are not equivalent. They differ when the characteristic of F is not 0, and F is not perfect. See section 5. Now note that as an associative E-algebra, ZL (T ) is simple, and E can be recovered from ZL (T ) as its center. Since E is a simple field extension, we have also verified Frobenius’s well-known “bi-commutant theorem”, in the special case d = 1, namely an operator which commutes with every operator which commutes with T is a polynomial in T. The case d ≥ 2 is much more difficult. In this case the dynamical approach provides a different, in some sense “dual”, insight, over the classical theory. We turn to this case in the next section. 326 R. S. KULKARNI 3. Orbits of ZL (T )∗ , and a Canonical Maximal ZL (T )-invariant Flag. Let T ∈ L(V), mT (x) = p(x)d , where p(x) is a monic irreducible polynomial p(x) in F[x]. Let deg p(x) = m. So E = F[x]/(p(x)) is a simple field extension of F , and dimF E = m. Assume d ≥ 2. Let N = p(T ), and Vi = ker N i , i = 0, 1, 2, . . . d. Thus we have a ZL (T )-invariant flag of subspaces 0 = V0 ⊂ V1 ⊂ V2 ⊂ . . . ⊂ Vd = V. We note an immediate consequence. Let T̄i denote the operator induced by T on Vi /Vi−1 , i = 1, 2, . . . d. Then mT̄i (x) = p(x). So by the case d = 1 treated in the previous section we see that Vi /Vi−1 has a canonical E-structure. So dimF Vi /Vi−1 , and finally dimF V is divisible by m. So let n = dim V = ml. We shall obtain a canonical, maximal ZL (T )-invariant refinement of this flag. It will be convenient to use a double-subscript notation Vi,j for the subspaces occurring in this refined flag, with the understanding that Vi = Vi,0 . If we insert k − 1 new terms between Vi and Vi+1 , we shall also denote Vi+1 by Vi,k . Our basic observation is: Vi,j − Vi,j−1 are precisely the ZL (T )∗ -orbits on V. In particular, Vi,j /Vi,j−1 are irreducible, when they are considered as modules either over the group ZL (T )∗ or over the F-algebra ZL (T ). Let T̄i,j denote the operator induced by T on Vi,j /Vi,j−1 . It will turn out that mT̄i,j (x) = p(x). So by the case d = 1 discussed in the last section, we have a canonical E-structure on Vi,j /Vi,j−1 . Let σ = dimE Vi,j /Vi,j−1 , and let Wσ denote an (abstract) vector space of dimension σ over E. As it will turn out, the algebra of operators induced by ZL (T ) on Vi,j /Vi,j−1 is dynamically equivalent to the standard action of LE (Wσ ) on Wσ . Before running into the proofs of these assertions, for the convenience of the reader, let us reconcile, albeit partially, this description with the classical theory. The classical theory attaches to T as above, its elementary divisors, which are polynomials of the form p(x)si , i = 1, 2, . . . r. We may assume that 1 ≤ s1 < s2 < . . . < sr = d are the distinct exponents of these elementary divisors, and σi is the multiplicity of p(x)si . Then l = Σri=1 si σi , where n = dim V = m l, m = deg p(x). According to the classical theory the pair (V, T ) is dynamically equivalent to the direct sum of the pairs of the form (F[x]/(p(x)s , µx ) where s = si occurs σi times, i = 1, 2, . . . , r. It will turn out that the dimensions σ of the (abstract) E-vector spaces Wσ mentioned in the previous paragraph are precisely the multiplicities σi ’s of the elementary divisors in the classical theory. The refined flag mentioned above will independently pick up the exponents si ’s and multiplicities σi′ s of the elementary divisors, subject to the relations, l = Σri=1 si σi , where n = dim V = m l, m = deg p(x). Let us now start building the refined flag. We shall first describe the refined flag where the dimensions of the subspaces in the flag are non-decreasing, and then offer a second description where these dimensions are strictly increasing. Lemma 3.1. i) For i > 0, N = p(T ) maps Vi into Vi−1 , and ii) For i > 1 the map induced by N on Vi /Vi−1 → Vi−1 /Vi−2 is injective. The proof is straightforward, and is omitted. Let (e1 , e2 , . . . ek ) be elements in Vd whose images (ē1 , ē2 , . . . ēk ) form an E-basis of Vd /Vd−1 . Then T j (ei ), 1 ≤ j ≤ m − 1, 1 ≤ i ≤ k are linearly independent over F, as they are indepenedent over F mod Vd−1 . Let Wd denote the F-span of T j (ei ). Notice that by construction, V = Vd = Vd−1 + Wd is a direct sum of subspaces. Among these, Vd−1 is T -invariant, but Wd is not (since we have assumed d ≥ 2). DYNAMICS OF LINEAR AND AFFINE MAPS 327 However by construction, mod Vd−1 it is T -invariant. We shall call such subspace of Vd an almost T -invariant subspace. Now notice that N maps Wd injectively in Vd−1 as a subspace complementary to Vd−2 . Moreover it is easy to check that Vd−2 + N (Wd ) is independent of the choice of Wd . It is a T -invariant, in fact ZL (T )invariant, subspace of Vd−1 . In case Vd−2 + N (Wd ) is a proper subspace of Vd−1 we insert it as an additional subspace in the flag between Vd−2 and Vd−1 . Notice that (Vd−2 + N (Wd ))/Vd−2 is an E-subspace of Vd−1 /Vd−2 . Assume that Vd−2 + N (Wd ) is a proper subspace of Vd−1 . For convenience, denote k = dimE Vd /Vd−1 by kd , and ei by ed,i . Let kd−1 = dimE Vd−1 /Vd−2 − dimE (Vd−2 + N (Wd ))/Vd−2 . If kd−1 6= 0, choose ed−1,i , 1 ≤ i ≤ kd−1 in Vd−1 so that their classes mod Vd−2 form an E-basis of a subspace of Vd−1 /Vd−2 complementary to (Vd−2 + N (Wd ))/Vd−2 . Then T j (ed−1,i ), 1 ≤ j ≤ m − 1, 1 ≤ i ≤ kd−1 are clearly linearly independent over F. Let Wd−1 denote the F-span of T j (ed−1,i )’s. Then Wd−1 is an almost T -invariant subspace of Vd−1 . Then N maps Wd−1 injectively into Vd−2 onto a subspace complementary to Vd−3 + N 2 (Wd ). If Vd−3 + N 2 (Wd ) + N (Wd−1 ) is a proper subspace of Vd−2 we insert it as an additional subspace in the flag between Vd−3 + N 2 (Wd ) and Vd−2 . We note again that two subspaces Vd−3 + N 2 (Wd ) and Vd−3 + N 2 (Wd ) + N (Wd−1 ) are independent of the choices of Wd and Wd−1 , and they are ZL (T )-invariant subspaces of Vd−2 . In case Vd−2 + N (Wd ) is not a proper subspace of Vd−1 , we simply take Wd−1 to be 0, and continue. Proceeding in this way we obtain the following refined flag, where the dimension of the subspaces are non-decreasing. 0 = V0 ⊂ N d−1 (Wd ) ⊂ N d−1 (Wd ) + N d−2 (Wd−1 ) ⊂ . . . N d−1 (Wd ) + N d−2 (Wd−1 ) + . . . N (W2 ) + W1 = V1 ⊂ V1 + N d−2 (Wd ) ⊂ V1 + N d−2 (Wd ) + N d−3 (Wd−1 ) ⊂ . . . V1 + N d−2 (Wd ) + N d−3 (Wd−1 ) + . . . N (W3 ) + W2 = V2 ⊂ . . . ......... Vd−3 ⊂ Vd−3 + N 2 (Wd ) ⊂ Vd−3 + N 2 (Wd ) + N (Wd−1 ) ⊂ Vd−3 + N 2 (Wd ) + N (Wd−1 ) + Wd−2 = Vd−2 ⊂ Vd−2 + N (Wd ) ⊂ Vd−2 + N (Wd ) + Wd−1 = Vd−1 ⊂ Vd−1 + Wd = Vd . 328 R. S. KULKARNI j Notice that in this flag the sum ⊕d−1 j=0 N (Wd ) forms a T -invariant (but not ZL (T )invariant) subspace dynamically equivalent to kd copies of F[x]/(p(x)d ). More genj erally the sums ⊕s−1 j=0 N (Ws ), s = 1, 2, . . . , d form a T -invariant (but not ZL (T )invariant) subspace dynamically equivalent to ks copies of F[x]/(p(x)s ), where mks = dim Ws . If Ws = 0, then those terms effectively do not occur. By construction, Ws is the F-span of T j es,1 , . . . T j es,ks , 0 ≤ j ≤ m − 1. So N u T j es,1 , . . . N u T j es,ks , 0 ≤ j ≤ j m − 1, 0 ≤ u ≤ s − 1 is a basis of ⊕s−1 j=0 N (Ws ). To get an irredundant flag where the dimesions are strictly increasing we need to proceed as follows. Let 1 ≤ s1 < s2 < . . . < sr = d be integers such that Wsi 6= 0. Let mσi = dim Wsi , 1 ≤ i ≤ r. Let Vi = Vi,0 , and for 0 ≤ i ≤ sr−j+1 , 1 ≤ j ≤ r, set Vi,j = Vi + N sr −i (Wsr ) + N sr−1 −i (Wsr−1 ) + . . . + N sr−j+1 −i (Wsr−j+1 ). A final important observation deals with the ambiguities involved in the choices of Ws where s is one of the s′i s. Notice that by construction, Ws is the F-span of T j es,1 , . . . T j es,ks , 0 ≤ j ≤ m − 1. So N u T j es,1 , . . . N u T j es,ks , 0 ≤ j ≤ m − 1, 0 ≤ j ′ u ≤ s − 1 is a basis of ⊕s−1 j=0 N (Ws ). Let Ws be another choice of almost T-invariant subspace complementary to the subspace previous to Vs+1 in the refined flag. Suppose W′s is constructed starting with e′s,1 , e′s,2 , . . . , e′s,ks . Let N u T j e′s,1 , . . . N u T j e′s,ks , 0 ≤ j ′ j ≤ m − 1, 0 ≤ u ≤ s − 1 be the corresponding basis of ⊕s−1 j=0 N (Ws ). Then consider the F-linear map which sends N u T j es,v to N u T j e′s,v and which is identity on the t−1 j remaining ⊕j=0 N (Wt ) for t 6= s. Clearly this map is invertible, commutes with T , and carries Ws into W′s . In particular, repeating this argument to any two successive terms Vi,j and Vi,j+1 we see that ZL (T )∗ is transitive on Vi,j+1 − Vi,j . In particular, this implies that Vi,j+1 /Vi,j is irreducible as a module over the group ZL (T )∗ or the associative algebra ZL (T ). In particular, this completes the proof of theorem (2.1), or as noted earlier, equivalently, of theorem (2, 2). In the process however, we have strengthened the result which we record in the following form. Theorem 3.2. Let T be in L(V), mT (x) = p(x)d , where p(x) is a monic irreducible polynomial in F[x]. Then V admits a canonical, maximal ZL (T )-invariant flag. A complement of each term appearing in the flag in its succeeding term is an orbit of ZL (T )∗ . In particular the quotient of each term appearing in the flag by its preceding term is an irreducible module over the group ZL (T )∗ , or the F-algebra ZL (T ). As a comparison with the classical approach via modules over PID, we note that classically we regard V only as a module over F[T ], and obtain the “vertical” decomposition of V whose summands are isomorphic to (F[xi ]/(p(xi )di ), µxi ). There is a wide choice in choosing these isomorphisms. They are not canonical. In the dynamical approach, we consider V as a module over ZL (T ), and obtain V as an increasing union of a canonical, “horizontal”, maximal, ZL (T )−invariant flag. In this sense, we have called the dynamical approach giving a “dual” perspective vis-a-vis the classical approach. As a by-product of this proof, we have some interesting dimension-counts, which refine the dimension counts in the well known Frobenuis’ dimension formula. For simplicity, let f (x) = p(x)d , where p(x) is a monic irreducible polynomial inPF[x], n = r dim V, m = deg p(x), and n = m l. Consider a partition of l, namely, l = i=1 si σi , DYNAMICS OF LINEAR AND AFFINE MAPS 329 where si occurs σi times, and we have assumed 1 ≤ s1 < s2 < . . . < sr = d. This data uniquely determines a pair (V, T ) up to dynamical equivalence, with mT (x) = f (x). To start with, we note again • n = dim V = m(sr σr + sr−1 σr−1 + . . . + s1 σ1 .) • The successive sub-quotients associated to the flag, starting from V0 = 0, which are the Jordan constituents of V considered as a module over the associative algebra ZL (T ), have F-dimensions (mσr , mσr−1 , . . . , mσ1 ) occurring s1 times, (mσr , mσr−1 , . . . , mσ2 ), occurring s2 − s1 times, ... (mσr , mσr−1 ) occurring sr−1 − sr−2 times, (mσr ) occurring sr − sr−1 times. • Let τi = σr + σr−1 + . . . + σi , 1 ≤ i ≤ r. Then from the refined flag we see that dim im N = n − mτ1 , and so dim kerN = mτ1 . As an associative F-algebra R = ZL (T ) has its nil-radical nil R, and R/nil R is a semi-simple F-algebra. The elements of R which map Vi,j+1 into Vi,j clearly form a nilpotent ideal I of R. On the other hand, R/I is clearly isomorphic to a direct product of L(Wsi ), i = 1, 2, . . . r. So I is nil R. This is worth recording as a theorem. Theorem 3.3. Let T be in L(V), with mT (x) = p(x)d where p(x) is a monic irreducible polynomial in F[x]. Let E = F[x]/(p(x)). Then R = ZL (T ) considered as an associative algebra has its maximal semisimple quotient isomorphic to a direct sum of matrix rings Mσi (E), where σi are as defined above. In particular, • dim R/nil R = m(σ12 + σ22 + . . . σr2 ). 4. Strongly Commuting Operators. Let T be in L(V). We say that an operator S in L(V) strongly commutes with T if S commutes with T , and leaves invariant every T -invariant subspace of V. It is interesting to compare the following theorem with Frobenius’ bicommutant theorem. It will be useful later on. Theorem 4.1. Let T be in L(V). An operator S in ZL (T ) strongly commutes with T iff S is in F[T ]. Proof. The “if” part is clear. Conversely, suppose that S in ZL (T ) strongly commutes with T. First consider the case when (V, T ) is dynamically equivalent to (F[x]/(p(x)d ), µx ), where p(x) is a monic irreducible polynomial in F[x]. Let S be in ZL (T ), and S(1) = [f (x)]. It is easy to see that S = f (T ). Thus ZL (T ) = F[T ], and so every element in ZL (T ) strongly commutes with T. Next consider the case where mT (x) = p(x)d , and p(x) is a monic irreducible polynomial in F[x]. Then V is a direct sum of T -invariant subspaces Wi , dynamically equivalent to (F[xi ]/(p(xi )di ), µxi ), 1 ≤ i ≤ k, and d = d1 ≥ d2 ≥ . . . ≥ dk . Let ei , 1 ≤ i ≤ k be a T -module generator in Wi . Let S|W1 = q1 (T ) where q1 (x) is a unique polynomial of degree at most d m, m = deg p(x). For j ≥ 2 let qj (x) be the polynomial of degree at most dj m, such that S|Wj = qj (T )ej . Then S(e1 + e2 ) = q1 (T )e1 + qj (T )ej . On the other hand, since S strongly commutes with T , we also have S(e1 +ej ) = u(T )(e1 +ej ) for some polynomial u(x) of degree at most dm. It follows that (q1 (T ) − u(T ))e1 = −(qj (T ) − u(T ))ej . 330 R. S. KULKARNI Since W1 ∩ Wj = 0 we must have (q1 (T ) − u(T )) ≡ (qj (T ) − u(T )) ≡ 0(mod p(x)dj .) So q1 (T ) ≡ qj (T )(mod p(x)dj .) Finally consider the general case. Write mT (x) = Πri=1 pi (x)di , where pi (x)’s are monic irreducible polynomials in F[x]. Let V = ⊕Vi , where Vi = ker pi (x)di be the corresponding primary decomposition of V. Now S leaves each Vi invariant. We have shown S|Vi = qi (T ) where qi (x) is a uniquely determined polynomial mod pi (x)di . By Chinese Remainder Theorem, there exists a uniquely determined polynomial q(x) mod mT (x) which is congruent to qi (x) mod pi (x)di . This completes the proof. 5. Lifting T -invariant E-structures, and “S+N”-decomposition. Let T be in L(V), and E an extension field of F. An E-structure on V is an F-algebra homomorphism σE : E → L(V). Such a homomorphism is necessarily injective, and allows one to consider V as a vector space over E, lifting the structure of V as a vector space over F. An E-structure σE is said to be T -invariant if the image of σE lies in ZL (T ). Suppose that mT (x) = p(x)d , where p(x) is a monic irreducible polynomial in F[x]. Let E = F[x]/(p(x)). An interesting problem is to investigate when V admits a T invariant E-structure. When d = 1, T itself induces a canonical E−structure. Namely, F[T ] ≈ E, and the inclusion mapping of F[T ] in ZL [T ] is a T -invariant E-structure. Assume d ≥ 2. Then Vi /Vi−1 admits a canonical T -invariant E-structure, since the minimal polynomial of the operator induced by T on Vi /Vi−1 is p(x). Our concern is whether these canonical E-structures on Vi /Vi−1 ’s can be lifted to a canonical E-structure on V itself. By a “canonical E-structure on V” we mean: i) Each T -invariant subspace is an E-subspace. ii) For each i = 1, 2, . . . , d, the induced E-structure on Vi /Vi−1 coincides with the one induced by T . It will eventually turn out that if V admits an E-structure which satisfies ii) then it also satisfies i). A first basic result in this direction is the following. For its importance in the theory of algebraic groups see below. In the following, for f (x) in F[x], let f ′ (x) denote its formal derivative. Theorem 5.1. Let T be in L(V), mT (x) = p(x)d , where p(x) is a monic irreducible polynomial in F[x], and E = F[x]/(p(x)). Then V admits a T -invariant E-structure iff either d = 1 or p′ (x) is not identically zero. Such structure is unique if it is canonical in the sense that it satisfies i) and ii) stated above. Proof. First we consider the issue of the existence of a T -invariant E-structure. We may assume d ≥ 2. If deg p(x) = 1, we have E = F, and p(x) = x − α for some α in F. On each Vi+1 /Vi , T acts as µα : v 7→ αv. Then clearly µ̃α defined by the same formula acting on V is a T -invariant E-structure on V. Note that we have also p′ (x) ≡ 1 6= 0, and the structure is canonical. So suppose deg p(x) = m ≥ 2. First suppose that p′ (x) is not identically 0. Then deg p′ (x) ≤ m−1. So p(x), p′ (x) are relatively prime. Since (V, T ) is dynamically equivalent to a direct sum of pairs of the form (F[x]/(p(x)e , µx ) where e ≤ d, and µx ([u(x)]) 7→ ([xu(x)]), it suffices to prove the existence of µx -invariant E-structure in the special case of (F[x]/(p(x)e ), µx ), e ≥ 2. In this case ZL (µx ) ≈ F[x]/(p(x)e ). For any y ∈ F[x] let [y] denote its class in F[x]/(p(x)e ). So the assertion of existence of an µx -invariant E-structure amounts to the existence of a polynomial z = u(x) ∈ F[x] such that the corresponding operator µz has minimal polynomial p(x). DYNAMICS OF LINEAR AND AFFINE MAPS 331 Since p(x), p′ (x) are relatively prime, there exist a(x), b(x) ∈ F[x] such that a(x)p(x) + b(x)p′ (x) = 1. Consider y = x − b(x)p(x). (Notice that for any polynomial u(x) in F[x], µu(x)p(x) is nilpotent, and its minimal polynomial is of the form xr , r ≤ d.) Writing ǫ = −b(x)p(x), the formal Taylor’s theorem (for polynomials with coefficients in commutative rings), gives p(y) = p(x + ǫ) = p(x) + ǫp′ (x) + ǫ2 ′′ p (x) + . . . 2 ≡ p(x)(1 − b(x)p′ (x)) + . . . ≡ p(x)(a(x)p(x)) + . . . ≡ 0 (mod p(x)2 ). So p(y)r = 0, for a suitable r < e. It follows that µy has minimal polynomial of the form p(x)r where r < e. So F[[y]] ≈ F[x]/(p(x)r ), and F[[y]] ⊂ F[[x]]. By induction on e it follows that there exists a polynomial z = u(x) ∈ F[x] such that the corresponding operator µz has minimal polynomial p(x). To prove the converse suppose that we have a pair (V, T ), mT (x) = p(x)d , d ≥ 2 where p(x) is a monic irreducible polynomial in F[x], E = F[x]/(p(x)), and V admits a T -invariant E-structure. This implies the existence of S in ZL (T ) with mS (x) = p(x). In the associated flag V2 is S-invariant. We only need to prove that p′ (x) 6≡ 0. So we readily reduce to the case d = 2. To arrive at a contradiction, suppose that p′ (x) ≡ 0. Now notice that for any polynomial u(x) in F[x] we have by the formal Taylor’s theorem, p(S + u(T )) = p(S) + u(T )p′ (S) + . . . = p(S) = 0. But then p(T ) = p(S + T − S) = p(S + T ) − Sp′ (S + T ) + . . . = p(S + T ) = 0. This is a contradiction since we have assumed mT (x) = p(x)2 . So we must have p′ (x) 6≡ 0. Next we consider the issue of uniqueness of a canonical T -invariant E-structure. Let σ1 : E → ZL (T ), σ2 : E → ZL (T ), be two canonical T -invariant E-structures. By passing to a T -invariant subspace, we may reduce to the case when (V, T ) is dynamically equivalent to F[x]/(p(x)d ), µx ). Then ZL (T ) = F(T ). Let α be a primitive element of E over F, and σi (α) = Si , i = 1, 2. Let Si = fi (T ) where fi (x) ∈ F[x] are well-defined polynomials mod p(x)d . Since Si ’s define canonical T -invariant Estructures we see that we must have fi (x) = x mod p(x). In the existence proof we considered polynomials a(x), b(x) satisfying a(x)p(x)+b(x)p′ (x) = 1. Notice that b(x) is uniquely defined by the condition that deg b(x) < m. This is also the unique choice so that p(x − b(x)p(x)) ≡ 0 mod p(x)2 . The same argument shows that the induction procedure used in the existence proof leads to a polynomial f (x) uniquely determined mod p(x)d , such that mf (T ) (x) = p(x). So f (x) = f1 (x) = f2 (x), and hence σ1 = σ2 . This finishes the proof. Remark 5.2. Notice that the condition p′ (x) 6≡ 0 is automatically satisfied if the characteristic of F is 0. Suppose that the characteristic of F is l > 0. Let p(x) = 332 R. S. KULKARNI Pm ai xi , Then p′ (x) ≡ 0 iff ai = 0 unless i is a multiple of l. So if p′ (x) ≡ 0, then Pm′ we may take p(x) = i=0 bi xil , where m = m′ l. When l > 0, F is said to be perfect if u 7→ ul is an isomorphism. For example a finite field, or an algebraically closed filed is automatically perfect. Notice that the condition p′ (x) 6≡ 0 is automatically satisfied l Pm′ if F is perfect. For otherwise, we can write bi = cli and so p(x) = ( i=0 ci xi ) , which will contradict that p(x) is irreducible over F. i=0 Definition 5.3. Let T be in L(V). A “S+N”-decomposition of T is a pair S, N such that i) T = S + N , ii) S is dynamically semi-simple, iii) N is nilpotent, and iv) SN = N S. Remark 5.4. Usually this notion is defined where dynamic semisimplicity is replaced by a stronger condition of algebraic semisimplicity, cf. the remarks in the introduction. This notion is basic in the theory of algebraic groups, cf. [1], [4], cf. also [2] for historical remarks. Theorem 5.5. Let T be in L(V), and mT (x) = Πri=1 pi (x)di , where pi (x)’s are monic irreducible polynomials in F[x]. Then 1) T admits a “S+N”-decomposition iff for each i, either di = 1 or else p′i (x) 6≡ 0. 2) If it exists, a “S+N”-decomposition is unique. 3) If mT (x) = p(x)d , p(x) is a monic irreducible polynomial, E = F[x]/(p(x)), and a “S+N”-decomposition exists, then S defines the canonical T -invariant E-structure on V. In particular S strongly commutes with T , and so S, and hence N , are polynomials in T . Proof. Notice that by the condition iv) in the definition of “S+N”-decomposition, S, N are in ZL (T ). So they leave the T -primary decomposition of V invariant, and their restriction to a T -primary component of V are the semisimple and nilpotent components of the restriction of T. So to investigate the existence of “S+N”-decomposition we reduce to the case where mT (x) = p(x)d , where p(x) is a monic irreducible polynomial in F[x]. First consider the existence issue. If d = 1, then T is semisimple, and taking S = T, and N = 0, we obtain a “S+N”-decomposition. So consider d ≥ 2. Let E = F[x]/(p(x)). First suppose that p′ (x) 6≡ 0. In the previous theorem we observed that under this condition there exists a polynomial f (x) in F[x] such that S = f (T ) defines a canonical T -invariant E-structure on V. In particular mS (x) = p(x), and so S is dynamically semisimple. Let T̄i , S̄i , be the operators induced by T, S respectively on Vi = ker p(T )i , i = 0, 1, 2, . . . , d. Since S defines a canonical T -invariant E-structure we have T̄i = S̄i . It follows that N = T − S is nilpotent, and hence T = S + N is an “S+N”-decomposition of T. Conversely suppose T = S + N is an “S+N”-decomposition of T. Then the induced operatots T̄i , S̄i , on Vi , i = 1, 2, . . . , d, are commuting dynamically semisimple operators. So their nilpotent difference N̄i must be 0. (See equation (5.1) below). So mT̄i (x) = p(x) = mS̄i (x). Since S is dynamically semi-simple, it follows that mS (x) = p(x) also. Let E = F[x]/(p(x)). Thus F[S] ≈ E, and S defines a T -invariant E-structure on V. So p′i (x) 6≡ 0. Now consider the issue of uniqueness of “S+N”-decomposition. Again we reduce to the case when mT (x) = p(x)d . Let d = 1. Then S = T, N = 0 is one “S+N”decomposition. Suppose T = S + N any “S+N”-decomposition. We need to show DYNAMICS OF LINEAR AND AFFINE MAPS 333 that N = 0. Indeed, (5.1) p(T ) = p(S + N ) = p(S) + N p′ (S) + N 2 ′′ p (S) + . . . . 2! Notice that p(T ) = p(S) = 0, and p′ (S) is invertible. If N 6= 0 then the rank of N is greater than the rank of N i for i ≥ 2. So the above equation is not possible unless N = 0. Now consider the case d ≥ 2. The proof is by induction on d. Let T = S + N, T = S1 + N1 be two “S+N”-decomposition. By induction we may assume S = S1 on Vd−1 , and S, S1 induce the same operators on Vd /Vd−1 . It follows that we must have S1 = S + M where M maps Vd into Vd−1 , and Vd−1 onto 0. Such M must be nilpotent. (5.2) p(S1 ) = p(S + M ) = p(S) + M p′ (S) + M 2 ′′ p (S) + . . . . 2! By the same argument as above we see that M = 0. It follows that “S+N”decomposition, if it exists, is unique. By uniqueness of “S+N”-decomposition it follows that when mT (x) = p(x)d , and “S+N”-decomposition exists, then S defines the canonical T -invariant E-structure. So S strongly commutes with T , and hence it (and so also N ) is a polynomial in T. Remark 5.6. The theorem 5.4 shows that in the definition of canonical T invariant E-structure the condition i) is a consequence of condition ii). On the other hand, to get uniqueness of a T -invariant E-structure it is clearly necessary to impose a condition such as ii). For example, if mT (x) and mS (x) are both irreducible, such that E = F[x]/(mT (x)) ≈ F[x]/(mS (x)) then T and S would usually define different E-structures. Remark 5.7. As remarked earlier, perfectness of F is a sufficient condition for the existence of “S+N”-decomposition. However the author is not aware of a statement of a necessary and sufficient condition for the existence of “S+N”-decomposition in the literature. One may avoid the issue by defining a different notion of semisimplicity, namely “algebraic semi-simplicity” to mean that the operator is diagonalizable over the algebraic closure of F. However, in author’s opinion, it is desirable to have all elements of the orthogonal group with respect to an anisotropic quadratic form to be “semisimple”. This would not be the case if we take the algebraic notion of semisimplicity. We also note that among the fields of positive characteristic, an important class of fields, namely, function fields of algebraic varieties of positive dimensions over finite fields, are not perfect. Lastly we note that there are misleading remarks in the literature that the dynamic and algebraic notions of semisimplicity are equivalent. Remark 5.8. Finally we would like to remark on a forgotten rational canonical form for matrices due to Wedderburn, [11]. In the standard texts on Algebra, such as [8], the authors present a matrix for an operator by choosing a suitable basis, called its rational canonical form. Let T be in L(V), and mT (x) = p(x)d , where p(x) is a monic irreducible polynomial in F[x]. For some authors the matrix presented to represent T involves a companion matrix of p(x)d . This is obviously a poor choice when d ≥ 2. It is better to use only the companion matrix of p(x). It is worth noting that one may use any matrix conjugate to a companion matrix. (This remark is important even in 334 R. S. KULKARNI the basic case when F = R, the field of real numbers, and important for the solutions of linear first order ODEs with constant coefficients, cf. section 6.) But secondly when deg p(x) = m ≥ 2, and d ≥ 2, the matrix presented is written in the form “S+N” where S (the matrix of diagonal blocks) is semisimple and N (the matrix of off-diagonal blocks) is nilpotnt. However this is not the “S+N”-decomposition of T , for these S, N do not commute. In case “S+N”-decomposition exists, a better matrix representation is obtained by choosing the off-diagonal blocks to be identity matrices of size m × m. Such a basis may be constructed starting from an E-basis which gives the usual “Jordan block” over E, and then constructing the corresponding basis over F. This was effectively mentioned already by Wedderburn, cf. [11], and rediscovered by the author early on in this investigation. The author thanks Rony Gouraige for pointing out the reference [11]. 6. The Affine Case. In this section we extend the theory to the affine case, and determine the centralizer of an affine map. Let F and V be as in the introduction, A the underlying affine case, and T = (A, v) an affine map which maps x to Ax + v. As observed there, A in (A, v) has an intrinsic affine meaning, and v has an intrinsic affine meaning if A = I. Let S = (α, a), α ∈ GL(V) be an element of GA(V). Then (6.1) S −1 = (α−1 , −α−1 a), and (6.2) ST S −1 = (αAα−1 , −αAα−1 a + αv + a). Let CL (V) resp. CA (V) denote the orbit-spaces L(V)/GL(V) resp. A(V)/GA(V). For T in L(V) resp. A(V) let [T ]L resp. [T ]A denote its orbit in CL (V) resp. CA (V). We have seen that the map (A, v) 7→ A is a homomorphism l : A(V) → L(V). The formula (6.2) shows that the map [(A, v)]A 7→ [A]L is a well-defined map [l] : CA (V) → CL (V). The main result about the map [l] is Theorem 6.1. [l] is a finite map, that is [l]−1 ([A]) has only finitely many elements. More precisely, for A ∈ L(V) let mA (x) = (x − 1)r g(x), where g(1) 6= 0 be its minimal polynomial. Here r ≥ 0 is an integer. Then [l]−1 ([A]) has r + 1 elements. Proof. First consider the generic case where r = 0. Consider the equation (∗)Ax+ v = x where x is indeterminate, and A ∈ L(V), v ∈ V are known entities. Since r = 0, we have det(I − A) 6= 0. So (*) has a unique solution in x. Let x0 be that unique solution. Let τ = (I, x0 ). Then τ (A, v)τ −1 = (A, 0) = A. So any element in l−1 (A) is conjugate to A. It follows that [l]−1 ([A]) has a unique element. Now suppose r > 0. Then V = V1 + V2 (direct sum) where V1 = ker(A − I)r , and V2 = ker g(A). Consider T = (A, v). Write v = v1 + v2 where vi ∈ Vi , i = 1, 2. Let x0 be the solution in V2 of the equation (∗)Ax + v2 = x. Such solution exists since det(I − A)|V2 6= 0. Let τ = (I, x0 ). Then τ (A, v)τ −1 = (A, v1 ). We have proved that an element (A, v) ∈ l−1 (A) is in the same GA(V)-orbit as an element (A, v1 ), where (A − I)r (v1 ) = 0. Let s be the least non-negative integer, s ≤ r, such that (A − I)s (v1 ) = 0. Now the theorem follows from the following lemma. DYNAMICS OF LINEAR AND AFFINE MAPS 335 Lemma 6.2. Suppose S = (A, v) resp. T = (A, w) be in A(V) such that mA (x) = (x − 1)r . Let s resp. t be the least non-negative integers ≤ r satisfying (A − I)s (v) = 0 resp. (A − I)t (w) = 0. Then S and T are in the same GA(V)-orbit iff s = t. Proof. From (6.2) we see that (α, a) conjugates S into T iff α is in ZL (A)∗ and w = (I − A)a + αv. Since mA (x) = (x − 1)r we are in the situation of the previous section. In particular, set N = I − A, and consider the ZL (A)-invariant refined flag. By symmetry, we may assume s ≤ t. In the notation introduced in the previous section, let Vt = Vt−1,k , and v lies in Vt − Vt−1,k−1 . From the structure of invertible elements in ZL (A), we see that α is in ZL (A)∗ and w = (I − A)a + αv iff s = t. Now we are in a position to determine the centralizer of an affine map. In effect, we describe a good representative of a GA(V)−orbit of the centralizer of an affine map. Let T = (A, v) in A(V). Let S = (B, w) be in ZA (T ). The equation ST = T S is equivalent to i) BA = AB, i.e. B ∈ ZL (T ). ii) Bv + w = Aw + v, or (B − I)v = (A − I)w. Case 1) Assume that T has a fixed point. Then by conjugation by an element in GA(V) (or what amounts to the same, by an affine change of co-ordinates) we may take v = 0. With this choice, we take the flag associated to A. From ii) we see that ZA (T ) = {(B, w)|B ∈ ZL (A), and w ∈ V1 } where V1 = ker (A − I). Case 2) Assume that T has no fixed point. Then again by change of affine coordinates by theorem (6.1) we may assume that mA (x) = (x − 1)r g(x), g(1) 6= 0 is the minimal polynomial of A, and s is the least positive integer such that (A − I)s v = 0. The equation ii) implies that (A − I)s (B − I)v = (B − I)(A − I)s v = 0 = (A − I)s+1 w. So w is in Vs+1 , where Vi = ker (A − I)i . Conversely, suppose that w is in Vs+1 . Then we show that there exists a B in ZL (A) such that (B, w) is in ZA (T ), and we can precisely determine B’s having this property. Indeed, in the double-subscript notation of the flag, Vs = Vs−1,k (for a suitable k), v is in Vs − Vs−1,k−1 , and (A − I)w is in Vs . So there exists C in ZL (T ) so that Cv = (A − I)w, and all such C’s can be determined from the refined flag. For each such choice of C, we can then take B = C + I. These are precisely the (B, w)’s in ZA (T ). Notice moreover that (B, w) is in ZA (T )∗ iff B is in ZL (A)∗ . Assume that this is the case, then Bv is in Vs − Vs−1,k−1 . Now equation ii) (B − I)v = (A − I)w shows that Bv = v + (A − I)w. Since (A − I)w is in Vs−1,k−1 we see that Bv ≡ v mod Vs−1,k−1 . It follows that the linear map B̄ induced by B on Vs /Vs−1,k−1 has eigenvalue 1. So B also has eigenvalue 1, and the N -images of the corresponding eigen-vector show that the multiplicity of the eigenvalue 1 is at least s. Summarizing, we have proved the following result. Theorem 6.3. Let T = (A, v) be in A(V). Let Vi = ker (A − I)i . 336 R. S. KULKARNI 1) If T has a fixed point then ZA (T ) is conjugate to {(B, w)|B ∈ ZL (A), and w ∈ V1 } 2) Suppose T has no fixed point, and mA (x) = (x − 1)r g(x), g(1) 6= 0 is the minimal polynomial of A. Let s ≤ r be the least positive integer such that (A − I)s v = 0. Then ZA (T ) is conjugate to {(B, w)|B ∈ ZL (A), w ∈ Vs+1 , (B − I)v = (A − I)w.} An element (B, w) in ZA (T )∗ necessarily has eigenvalue 1 with multiplicity at least s. Remark 6.4. Suppose that T = (A, v) in A(V) has no fixed point. Then the explicit forward orbit-structure of T , or the orbit structure of ZA (T )∗ , is quite complicated, compared to the neat answer we obtained in case T has a fixed point. However, next to orbit-structure, for some intuitive understanding, we can enquire about the invariant sets. On this score we have some satisfactory information. Namely, if (B, w) is in ZA (T ) then B preserves the refined flag determined by A in U = ker(A − I)r , where mA (x) = (x − 1)r g(x), g(1) 6= 0, is the minimal polynomial of A. The affine translates of each of the subspaces in the flag may be called a family of affine flags in U. Clearly (B, w) preserves this family of affine flags as a whole. In case the integer s associated to v in (A, v) is 1, one can say a bit more. Namely consider the ZA (T )-invariant family of affine subspaces parallel to V1 . Among these subspaces, there is actually one ZA (T )-invariant subspace. Namely, up to an affine change of co-ordinates we may assume that v is actually an eigenvector of A. Then the eigen-space V1 = ker (A − I) itself is ZA (T )-invariant. Remark 6.5. Consider the case F = R, the field of real numbers, or F = C the field of complex numbers. On the Lie algebra level, one may ask for “normal forms” of the solutions of affine vector fields on A. This amounts to solutions of the ODEs dx = Ax + v, A ∈ L(V), v ∈ V. dt Equivalently one may ask for normal forms of representatives of conjugacy classes of one-parameter subgroups of GA(V). In the texts on ODEs, cf. for example [5], this ODE is solved by the method of variation of parameters. The ideas in this section provide a short-cut. Namely consider the affine map (A, v). If this map has a fixed point, (which is the case if det(A − I) 6= 0), then by an affine change of coordinates we can make v = 0, and the solutions are orbits of the one parameter group t 7→ etA in the new coordinate system. If the map has no fixed point then A must have eigenvalue 1. Write mA (x) = (x − 1)r g(x), g(1) 6= 0. Let Rn = V = V1 + V2 , where V1 = ker (A − I)r , V2 = ker g(A). Let v = v1 + v2 , where vi is in Vi for i = 1, 2. By an affine change of coordinates we can make v2 = 0. Choose the least positive integer s such that (A − I)s v1 = 0. Then in the new coordinate system, the solutions are orbits of the one-parameter group t 7→ (etA , tv1 + t2 t3 ts Av1 + A2 v1 + . . . + As−1 v1 ). 2! 3! s! The point is that one can always make the translational part of a one-parameter group of the affine group a polynomial, rather than an infinite series, in t, either by conjugacy DYNAMICS OF LINEAR AND AFFINE MAPS 337 in GA(V), or what is the same, by an appropriate affine change of coordinates. This “normal form” of a one-parameter group indicates that its orbits, or the orbits of its centralizer, in V are more complicated than in the linear case, when there is an “unavoidable” translational part, which carries an affine meaning. Remark 6.6. From a computational, or algorithmic, perspective the decomposition Rn = V = V1 + V2 , is readily computable. The main issue is the computation of etA . Now mA (x) is algorithmically computable as the last non-zero diagonal entry in the Smith normal form of the characteristic matrix xI − A. Assume that we have a factorization of mA (x) into its irreducible factors. When F = R the irreducible factors are of degree 1 or 2. The (generalized) eigenspaces corresponding to linear factors and the corresponding refined lattice of ZL (A)-invariant subspaces, the corresponding (Jordan) canonical forms and their exponentials are all algorithmically computatble. When F = R and mA (x) has irreducible factors of degree 2, again the corresponding refined lattice of ZL (A)-invariant subspaces is algorithmically computable. However the suggested rational canonical form in the texts of algebra using the companion matrix of an irreducible factor is not useful for computation of the exponential. If the   0 −b 2 2 . irreducible factor is x − 2ax + b, a − b < 0, then its companion matrix is 1 2a   a −c , a2 + c2 = b which is conjugate to It is decisively better to use the matrix c a thecompanion matrix.  For then its exponential becomes readily computable, namely, cos c − sin c ea . Also one should use the (forgotten) rational form as explained in sin c cos c section 6, where the non-diagonal blocks are 2 × 2 identity matrices. This is indicated in the texts and exercises in [5] and [7], without adequate explanation. 7. Parametrization Theorems. As stated in the introduction, we have interpreted the phrase “understanding the dynamics” in our set-up to mean the parametrizations of similarity classes, z-classes, and finally the elements in L(V) and A(V) themselves in terms of objects having significance independent of the choices of linear or affine co-ordinate systems. Here the word “parametrization” is used in the following sense. The sets L(V) and A(V) are the “unknown” sets which we wish to understand in terms of the “known” sets F and V, and the “universally known” sets such as natural numbers, integers, rational numbers, and if one wishes, also real and complex numbers, and any other similar sets, and the sets derived from such sets by applying the allowable constructions in the model of “naive” set theory. Loosely speaking, the parameters having values in abelian groups are called “numerical parameters”, and the others, such as decompositions into subspaces or flags, are called “spatial” parameters. In more abstract terms they are made precise in theorem 2.1 of [9]. The parametrizations that are obtained here are in terms of the “arithmetic” of F as reflected in the monic irreducible polynomials, and subspaces of V. The datum of irreducible polynomial in F[x] of degree m is equivalent to the datum of a simple field extension E of F such that [E : F] = m, and a primitive element α of E over F. Starting with a monic irreducible polynomial p(x) ∈ F[x] we have E = F[x]/(p(x)), and α = [x], the class of x in F[x]/(p(x)). Conversely, given (E, α), we get p(x) as the minimal polynomial of µα where µα : E → E, µα (u) = αu. Here µα is regarded as a F-linear map of the vector space E over F. To be completely precise, to obtain a 338 R. S. KULKARNI one-to-one correspondence between p(x) and pairs (E, α) we need to consider the Fisomorphism classes of (E, α)’s. Namely, the pairs (E1 , α1 ), (E2 , α2 ), are F-isomorphic if there exists an F-isomorphism carrying α1 to α2 . In particular if we fix E in its isomorphism class of field extensions of F, then α is defined only up to the action of G(E/F), the group of F-automorphisms of E. Pr Let n = dim V and π : n = i=1 ni be a partition of n. A decomposition Dπ patterned on the partition π of V is a direct sum decomposition V = ⊕ri=1 Vi into subspaces, where dim Vi = ni . Let n = dim V. Let m be a divisor of n, and n = ml. Let r be a natural number, and r pairs of natural numbers {(s1 , σ1 ), (s2 , σ2 ), . . . (sr , σr )} such that s1 < s2 < . . . < sr , and l = Σri=1 si σi . A flag of type (n, m; {(s1 , σ1 ), (s2 , σ2 ), . . . (sr , σr )}) is an increasing family of subspaces Vi,j , 0 ≤ i ≤ r, 0 ≤ j such that the successive quotients have dimensions: (mσr , mσr−1 , . . . , mσ1 ) occurring s1 times, (mσr , mσr−1 , . . . , mσ2 ), occurring s2 −s1 times, . . . (mσr , mσr−1 ) occurring sr−1 −sr−2 times, (mσr ) occurring sr − sr−1 times. Such a flag is denoted by F((n, m; {(s1 , σ1 ), (s2 , σ2 ), . . . (sr , σr )}) Now suppose that there exists a simple field extension E of F such that [E : F] = m. Then since the dimension of each successive sub-quotient of F(n, m; {(s1 , σ1 ), (s2 , σ2 ), . . . (sr , σr )}) is divisible by m, it has a structure of a vector space over E. We denote such a choice of an E-structure, a bit loosely, by JE . When we wish to emphasize the sub-quotient W we shall specify JE,W . The choices of JE,W ’s are by no means unique. In fact GL(W) clearly acts on the set of E-structures on W. An important point, which is easy to see, is that the action of GL(W) on the set of E-structures is transitive. Now we define an important notion of compatibility of JE,W ’s. In the flag, we have special components Vi , 1 ≤ i ≤ sr = d. The compatibility of JE,W ’s for the successive sub-quotients in the flag means that there are E-structures on Vi+1 /Vi , 0 ≤ i < d such that for all the components Vi,j in the chain from Vi to Vi+1 the sub-quotients Vi,j /Vi are E-subspaces of Vi+1 /Vi , and the E-structure on W = Vi,j /Vi,j−1 coincides with JE,W . One may enquire whether the E-structures on Vi+1 /Vi , are similarly compatible with a single E-structure on V. As discussed in section 5, this turns out to be a subtle point related to the existence of “S + N ”-decomposition. With this preparation, we are in a position to describe our parametrizations. Theorem 7.1. A) A GL(V)-orbit in its action on L(V) is parametrized by the following data. Pr i) A primary partition π : n = P i=1 ni , ni = mi li . ri si,j σi,j , where si,1 < si,2 < . . . si,ri . ii) The secondary partitions li = j=1 iii) An F-isomorphism class of pairs (Ei , αi ), where Ei is a simple field extension of F of degree mi with αi as its primitive element, for i = 1, 2, ...., r. B) A GA(V)-orbit in its action on A(V) is parametrized by the data i), ii), iii) as in A) and with m(x) = (x − 1)u g(x), g(1) 6= 0, iv) A non-negative integer s ≤ u. Theorem 7.2. : A) A z-class in the GL(V)-action on L(V) is parametrized by the following data. Pr i) A primary partition π : n = P i=1 ni , ni = mi li , ri si,j σi,j , where si,1 < si,2 < . . . si,ri . ii) The secondary partitions li = j=1 DYNAMICS OF LINEAR AND AFFINE MAPS 339 iii) Simple field extensions Ei , 1 ≤ i ≤ r of F, [Ei : Fi ] = mi . B) A z-class of (A, v) in the GA(V)-action on A(V) is parametrized by the data i), ii), iii) as in A) in case mA (x) does not have 1 as an eigenvalue. In case mA (x) = (x − 1)u g(x), g(1) 6= 0, and u > 0, then the z-class of (A, v) is parametrized by the data i), ii), iii) as in A) and iv) A non-negative integer s ≤ u. Theorem 7.3. A) An element of L(V) is uniquely determined by the following data. The data i), ii) , iii) of part A) in theorem 7.1, in particular the field extensions Ei = F[x]/(pi (x)), and the primitive elements αi . iv) A decomposition Dπ : V = ⊕ri=1 Vi of V patterned on the primary partition π. v) Flags F((ni , mi ; {(si,1 , σi,1 ), (si,2 , σi,2 ), . . . (si,ri , σi,ri )}) of subspaces in Vi , patterned on the secondary partitions. vi) Compatible Ei -structures on the sub-quotients in the flag in each Vi . B) An element T of A(V) is uniquely determined by the following data. Case 1. (T has a fixed point): Choose a fixed point as the origin. So T may be identified with an element in L(V). The data i), ... , vi) in part A) is independent of the choice of the fixed point. These data and the affine subspace of fixed points determine T . Case 2. (T has no fixed point): Express T as (B, v) so that there exists s a least positive integer such that (I − B)s v = 0. Then the invariants i), ... , vi) in part A) associated to B and v uniquely determine T . The proofs of theorems 7.1-7.3 are given in the next two sections. A major consequence of theorem 7.2, cf. also section 10, is the following theorem. Theorem 7.4. Let V be an n-dimensional vector space over a field F. Suppose F has the property that there are only finitely many extensions of F of degree at most n. Then there are finitely many z-classes of GL(V)-, resp. GA(V)-, actions on L(V), resp. A(V). 8. Proof of Parametrization Theorems 7.1 and 7.3. We begin with the proof of Theorem 7.1. Notice that the data in ii), and iii) in part A) is just the numerical data regarding the exponents and multiplicities in the elementary divisors in the classical theory, which can be independently read from the refined flag. Given an element T in L(V), we associate to it i) the minimal polynomial m(x) =P mT (x) = Πri=1 pi (x)di , r ii) the primary partition dim V = i=1 dim Vi where Vi = ker pi (T )di , and iii) the secondary partitions with si,j ’s being the exponents in the elementary divisors pi (x)si,j s, and σi,j s being the multiplicities of pi (x)si,j s. Conversely suppose we have the data i), ii), iii). We first show that there actually exists T in L(V) which realizes this data, and secondly that any two elements in L(V) having the same data are in the same GL(V)-orbit. Take an arbitrary decomposition V = ⊕ri=1 Vi patterned over the primary partition. Next construct an appropriate flag in each Vi with type given by the pairs (si,j , σi,j )’s. Let Ei = F[x]/(pi (x)), and α = [x]. Equip the sub-quotients in the flag in Vi with a compatible family of Ei -structures. Take an arbitrary Ei -basis (e1 , e2 , . . . , ek ) in the component V0,1 of the flag. (We have actually k = σsr .) Then 340 R. S. KULKARNI (e1 , αe1 , α2 e1 , . . . , αmi −1 e1 , e2 , αe2 , . . . . . . , αmi −1 ek ) is an F-basis of V0,1 . Moreover we can define the operator T on V0,1 which is multiplication by α. We can continue this process to all the components in the chain ending in V1 , and define the operator T on V1 having the minimal polynomial p(x). Next we consider the component V1,1 in the flag. Notice that by construction dimF V1,1 /V1 is mi k, and V1,1 /V1 has an Ei -structure. Choose (e′1 , e′2 , . . . , e′k ) in V1,1 whose classes [e′i ] modulo V1 form an Ei -basis. Define T j e′u , 1 ≤ j ≤ m−1, 1 ≤ u ≤ k in V1,1 so that their classes [T j e′u ] modulo V1 are [αj eu ]. Now a crucial point is to define p(T )e′i = ei in V0,1 , and more generally p(T )T j e′i = T j ei , 1 ≤ j ≤ mi − 1. It is easy to see that continuing this process along the successive components in the flag we obtain a basis of Vi and an operator T in L(Vi ) having the given secondary partition on Vi . Taking the direct sum we obtain an operator T on V having the minimal polynomial m(x) and the given primary and secondary partitions. Finally suppose that T, T ′ are two elements in L(V) having the same data. Then the dimension of a primary component Vi equals mi li . (Here li is the largest power of pi (x) dividing the characteristic polynomial.) So by appropriate conjugation by an element of GL(V) we may suppose that both T and T ′ have the same primary components Vi s. So we reduce to the case where mT (x) = mT ′ (x) = p(x)d , where p(x) is a monic irreducible in F[x]. Next by hypothesis T, T ′ have the same secondary partitions. Then we can construct the flags and the bases ej ’s, e′j ’s of V adapted to the respective flags. Then the element g ∈ GL(V), gei 7→ e′i conjugates T into T ′ . This finishes the proof of part A) of theorem 7.1. The proof of part B) can be completed along the same lines using the results in section 6. As for the proof of Theorem 7.3, observe that the data the isomorphism class of (E, α) determines an irreducible polynomial in F[x]. So the proof may be completed along the lines of theorem 7.1. 9. Proof of the Parametrization Theorem 7.2. Let S, T be in the same z−class in L(V). This means that ZL (S)∗ and ZL (T )∗ are conjugate by an element u in GL(V). First we show that this implies that ZL (S) and ZL (T ) are conjugate, in fact by the same element u, in GL(V). This follows from the following lemma. Lemma 9.1. Let T be in L(V). Then ZL (T ) as an F-subalgebra of L(V) and ZL (T )∗ as a subgroup of GL(V) uniquely determine each other. Proof. Indeed ZL (T ) determines ZL (T )∗ as the multiplicative subgroup of its units. Conversely let S be a non-invertible element in ZL (T ). Then mS (x) = xk f (x), with k > 0, and f (0) 6= 0. Let V0 = ker S k , and V1 = ker f (S). So V = V0 ⊕ V1 is a T -invariant decomposition. For any such decomposition, let JV0 ,V1 denote the operator which is identity on V0 , and zero on V1 . Then JV0 ,V1 is in ZL (T ), and S1 = S + JV0 ,V1 is clearly in ZL (T )∗ . Thus ZL (T ) is a linear span of ZL (T )∗ and the operators JV0 ,V1 corresponding to all T -invariant decompositions V = V0 ⊕ V1 . This proves that ZL (T )∗ determines ZL (T ). Remark 9.2. Although the following observation is not needed in the proof that follows, the above lemma raises a question whether ZL (T ) itself is always a linear span of ZL (T )∗ . This is indeed the case if F has more than two elements. For indeed, let S, V0 , and V1 be as in the above proof. Let c be an element in F different from 0 and 1. Define U1 as (S − I)|V0 on V0 , and cS|V1 on V1 . Define U2 as I|V0 on V0 , DYNAMICS OF LINEAR AND AFFINE MAPS 341 and (1 − c)S|V1 on V1 . Then U1 , U2 are in ZL (T )∗ and S = U1 + U2 . Thus in fact an element in ZL (T ) is a sum of at most two elements in ZL (T )∗. Remark 9.3. The restriction that F has more than two elements in the above remark is a genuine one. For example consider an n-dimensional vector space V over F2 , the field with two elements. Assume n ≥ 2. Let T be an operator with mT (x) = xk (x − 1)l , where k ≥ 1, l ≥ 1. Consider the T -invariant decomposition V = V0 ⊕ V1 , where V0 = ker T k , and V1 = ker (T − I)l . Then ZL (T ) = ZL (T |V0 ) × ZL (T |V1 ). Clearly ZL (T |V0 ) = F[T |V0 ], and ZL (T |V1 ) = F[T |V1 ], whereas ZL (T )∗ consists of f (T ) where f (0) = 1. If we take the sum of even number of elements of ZL (T )∗ then we get an operator all of whose eigenvalues are 0. On the other hand if we take the sum of odd number of elements of ZL (T )∗ then we get an operator all of whose eigenvalues ∗ are 1. It follows T cannot be written as a sum of elements of ZL (T ) . In view of the lemma we can assume that ZL (S) and ZL (T ) are conjugate by an element u in GL(V). Replacing S by uSu−1 we may assume that ZL (S) = ZL (T ) Let C be the center of ZL (T ). By the Frobenius’ bicommutant theorem, we have C = F[S] = F[T ]. It is important to note that C does not determine T . However every element of C leaves every T −invariant (or S−invariant) subspace invariant. Let pi (x) be the primes associated to T, and V = ⊕Vi the corresponding primary decomposition. Let W be a T -invariant subspace of Vi such that the pair (W, T |W ) is dynamically equivalent to (F[x]/(pi (x)d ), µx ). Then Wj = ker pi (x)j , 0 ≤ j ≤ d, are precisely all the T -invariant subspaces of W. Since a subspace of V is T -invariant iff it is S-invariant, it follows that Wj ’s are precisely also all the S-invariant subspaces of W. It follows that mS|W (x) must be of the form q(x)e where q(x) is a monic irreducible polynomial in F(x). Next note that the same q(x) works for every T -invariant subspace U such that the pair (W, T |W ) is dynamically equivalent to (F[x]/(pi (x)e ), µx ) for some e. For there exists an operator A in ZL (T ) = ZL (S) which maps W onto U equivariantly with the action of S. It follows that P V = ⊕Vi is also a primary decomposition with respect to S. So in particular, n = i ni , dim Vi = ni is a well-defined choice of a primary partition of n. Now restrict the action of ZL (S) = ZL (T ) to Vi . For the same reason we see that the refined flag, and in particular the secondary partitions are well-defined invariants of ZL (S) = ZL (T ), which are independent of the choices of a T with the property C = F[T ]. Finally considering the action of ZL (S) = ZL (T ) on Vdi /Vdi−1 we see that the simple field extension Ei = F[x]/(pi (x)) is a well-defined invariant of ZL (S) = ZL (T ). Conversely, given the the primary and secondary partitions and the field extensions Ei ’s of appropriate degree there clearly exists an operator having this data, and its orbit class is uniquely determined. This finishes the proof of the theorem 7.2 in the linear case. Using the results of section 6, the proof can be extended to the affine case. 10. Generating Functions for z-classes. Let D be a collection of extension fields of finite degree of a field F with the property that D contains only finitely many extensions of a given degree. Significantly, this property is automatically satisfied for the collection of all extension fields in the following cases: 1) F algebraically closed, 2)F = R, 3) F = a local field, 4) F = a finite field. A case of arithmetic interest is 5) F = Q, S = a finite set of primes, and D = the collection of all extension fields obtained by adjoining all n-th roots of all primes in S. From the parametrization 342 R. S. KULKARNI theorem 7.2, it follows that for any such collection D, and for a fixed n, there are only finitely many z-classes of linear maps on an n-dimensional vector space over F with the extension fields in D. So one can form a generating function n ZF,D (x) = Σ∞ n=0 z(n)x . As is expected from the parametrization theorems, these functions are closely related to the generating functions for partitions. One may also consider the restricted generating functions which enumerate the z-classes of dynamically semi-simple operators, or cyclic operators. In both cases the secondary partitions have simple types. (Recall that a pair (V, T ) is cyclic if there exists a vector v such that V = F[T ]v. Clearly (V, T ) is cyclic iff deg mT (x) = dim V.) We denote the corresponding generating functions by ZF,D,s (x) and ZF,D,c (x) respectively. Let Πn denote the set of all partitions of n, and p(n) the cardinality of Πn . A partition π of n with signature (1a1 2a2 . . . nan ) is the partition in which i occurs ai times, so n = Σi ai i. n Let f (x) = 1 + Σ∞ n=1 b(n)x be a formal power series. To f (x) we associate a new n power series, P(f (x)) = 1 + Σ∞ n=1 c(n)x . Here c(n) is a sum Σπ∈Πn cπ , where cπ = a1 a2 ai n Πi=1 b(i) , if π has signature (1 2 . . . nan ). Notice that the well-known Eulerian n generating function for partitions P (x) = 1 + Σ∞ n=1 p(n)x is P(g(x)) where g(x) = n ∞ Σn=0 x is the geometric series. The Absolute Case: Here F is algebraically closed. Here D consists of a single element, namely F itself, and we omit its mention. First consider the easy cases of i) semisimple operators, or ii) cyclic operators. In both cases, the secondary partitions are uniquely determined. Let the primary partition of an operator T be π : n = P r ni i=1 ni . If T is semisimple then the secondary partitions of ni s have signatures (1 ). 1 If T is cyclic then the secondary partitions of ni s have signatures (ni ). So Zs (x) = Zc (x) = P (x). On the other hand consider the case of all z-classes. Let T be the operator whose primary partition has signature (1a1 2a2 . . . nan ). Then the number of secondary partitions associated with this partition is p(1)a1 p(2)a2 . . . p(n)an . It follows that Z(x) = P(P (x)) = Π∞ k=1 1 . 1 − p(k)xk Theorem 10.1. Z(x) is a meromorphic function on the unit disc, and it cannot be extended beyond the unit disc. √ Proof. A simple estimate for p(n) is p(n) ≤ eK n , for K > 0, cf. [3], ch. VII, 1 section 3. It follows that p(n) n tends to 1 as n tends to infinity. So for |x| < 1, Z(x) − 1 2kπi defines a meromorphic function on the unit disc. Its poles are at x = p(n) n e n , for n = 1, 2, . . . and 0 ≤ k ≤ n. So it also follows that the function cannot be extended meromorphically beyond the unit disc. DYNAMICS OF LINEAR AND AFFINE MAPS 343 It appears that this type of generating function has not appeared in number theory before. Notice that if we consider more generally the case of an arbitrary field F, but restrict to D = {F}, then we get the same generating functions. General Case: Let E be an element of D, and [E; F] = m. Clearly the contribution to ZD (x) coming from E is Z(xm ). We denote this contribution by ZF,E (x). Clearly ZF,D (x) = ΠE∈D ZF,E (x). Since we have assumed that D contains only finitely many extensions of a given degree, this product is well-defined. Two notable cases are i) F = R, the field of real numbers, and ii) F = Fq , the finite field with q elements, and D consists of all extensions of finite degree. Then ZR,D (x) = Z(x)Z(x2 ). n ZFq ,D (x) = Π∞ n=1 Z(x ). REFERENCES [1] A. Borel, Linear algebraic groups, second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991. xii+288 pp. ISBN: 0-387-97370-2. [2] A. Borel, The work of Chevalley in Lie groups and algebraic groups, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), 1–22, Manoj Prakashan, Madras, 1991. [3] K. Chandrasekharan, Arithmetic Functions, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 167, Springer Verlag, 1970. [4] C. Chevalley, A new kind of relationship between matrices, Amer. J. Math., 65 (1943), pp. 521–531. [5] E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. xii+429 pp. [6] R. Gouraige, z-Classes of Elements in Central Simple Algebras, thesis, City University of New York (2006). [7] M. W. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra, Pure and Applied Mathematics, Vol. 60. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. xi+358 pp. [8] N. Jacobson, Basic algebra, I. Second edition. W. H. Freeman and Company, New York, 1985. xviii+499 pp. ISBN: 0-7167-1480-9. [9] R. S. Kulkarni, Dynamical Types, and Conjugacy Classes of Centralizers in Groups, J. Ramanujan Math. Soc., 22 (2007), pp. 35–56. [10] G. Pilz, Near-rings, the theory and its applications, North-Holland Mathematics Studies, No. 23. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. xiv+393 pp. [11] J. H. M. Wedderburn, The canonical form of a matrix, Ann. of Math. (2), 39 (1938), pp. 178– 180. 344 R. S. KULKARNI