RIGID JORDAN TUPLES DAVID P. ROBERTS
Fix for this entire paper an algebraically closed eld E. Consider unordered z-tuples j = fj1 ; : : :; jz g of non-central conjugacy classes in GLn(E), z 0 and n 1 being integers. Impose the determinant condition (0.1) det(j1 ) det(jn ) = 1: Also impose the rigidity condition (0.2) dim(j1 ) + + dim(jz ) = 2(n2 ? 1): We call such fj1; : : :; jz g rigid Jordan z -tuples of rank n. Here we use the word \Jordan" because we will be describing each ji in terms of Jordan canonical forms. Let j1 , : : : ,jz be non-central conjugacy classes in GLn (E) satisfying Conditions 0.1 and 0.2. Let V (j1 ; : : :; jz ) be the subset of j1 jz consisting of matrix tuples (g1; : : :; gz ) with g1 gz = 1 and hg1; : : :; gz i acting irreducibly on E n . The group PGLn(E) acts on V (j1 ; : : :; jn) by simultaneous conjugation. A naive dimension count suggests that there are only nitely many orbits. It is elementary that the number of orbits is independent of the ordering of the ji . A cohomological argument says that in fact the number of orbits is either zero or one [11]. In the latter case we say that fj1; : : :; jz g is realizable. Katz [7] gave an inductive algorithm for determining whether a given j is realizable. He wrote of \a fascinating bestiary waiting to be compiled" [7, page 9]. In this paper we present such a compilation. Thus our subject is the explicit classi cation of realizable rigid Jordan tuples. In x1,2 we present Katz's algorithm in a streamlined form which exploits the fact that rigid Jordan tuples come in families, and it makes sense to talk about realizability at the level of families. In x3,4 we classify realizable families in some detail for low ranks n; we describe the inner structure of some of these families in x5. In x6,7,8 we identify \extremal" realizable families for all ranks n. The mathematical setting of our subject is remarkably elementary. Katz emphasizes in [7] that, after his formalism has been set up, linear algebra plays no role; even the ground eld E enters only through its multiplicative group E , and the formalismmakes sense with E replaced by any abelian group. We emphasize here that only the very simplest parts of Katz's algorithm are needed when one works at the level of families; at this simpler level E does not play a role at all, and the formalism centers on partitions. On the other hand, the interest in our subject is that it forms the combinatorial core of a very rich motivic theory [7, Chapter 8]. Rigid matrix tuples (g1; : : :; gz ) 2 V (j1 ; : : :; jz ) arise as the underlying monodromy representations. Two in nite series of realizable families can reasonably be considered classical. We call these families Hn and Pn, as hypergeometric functions n?1Fn and Pochhammer functions arise respectively as period integrals of the motives. The remaining realizable families give rise to similar functions which have just begun to be studied. Scattered throughout the paper are several remarks very brie y pursuing some of these external considerations, mostly by reference to the literature. 1
2
DAVID P. ROBERTS
Contents
1. Jordan formalism 2. Katz's algorithm 3. n 11: quick look 4. n 6: closer look 5. n 6, z = 3: internal structure 6. Maximal series: H and P 7. Submaximal series: A, B, C, D, E, F, G and I, J K, L, M, N 8. Minimal series: , , , and References
2 3 5 6 7 10 11 14 15
1. Jordan formalism For n a positive integer let Pn be the set of partitions of n. We write out partitions by listing their parts: = f1 ; : : :; `g. Order does not matter, and we usuallyPchoose to list parts in decreasing order. The rank of is simply the number jj = n = k . The maxmult of 2 Pn is simply the largest part m(). The length P of is the number `() of parts. The norm of is the quantity N() = jjjj = 2k . Note that jj jjjj modulo two. We often omit braces and commas when the meaning is clear. Thus P4 = ff1; 1; 1; 1g; f2; 1; 1g; f2; 2g; f3; 1g; f4gg = f1111; 211; 22; 31; 4g. Once below we need to make use of the standard duality t : Pn ! Pn, xing rank n and interchanging maxmult m and length `. Graphically, one can think of t as transpose, e.g. 0 ; 1t A = ; ; = (2; 1; 1): (3; 1)t = @
Algebraically, the dual t of a given partition is the de ned letting tk be the number of parts of of size k. For 2 Pn , de ne J to be the set of formal symbols fa1 1 ; : : :; a` g with ak 2 E . So J is a copy of the `-dimensional torus (E )` modded out by a product of symmetric groups. E.g. J321 is simply a three-dimensional torus, the general element being fa3 ; b2; cg for a unique (a; b; c) 2 E 3. On the other hand J222 is the torus modulo S3 , as the general element can be expressed as fa2; b2; c2g for six di erent choices of the ordered triple (a; b; c). Here the exponents are part of the formalism, and not simply an indication of multiplication. Put a Jn := (1.1) J : `
2Pn
If j 2 J we say that is the centralizer partition of j. Let g 2 GLn (E) be a matrix. For a 2 E , let na be the dimension of the generalized eigenspace of a, i.e. the dimension of the kernel of (g ? aIn )n . Let a be the partition of na giving the sizes of the Jordan blocks belonging to a. Put a = ta . The theory of Jordan canonical forms identi es the set of conjugacy classes in the group GLn(E) with the set Jn , via [g] = fa1 1 ; : : :; a` g: `
RIGID JORDAN TUPLES
3
Here, for a given a 2 E , the exponents on a all together form a . The largest part of a is denoted d([g]; a), and called the drop of [g] with respect to a. In the Jordan formalism just set up, the natural action of E on conjugacy classes is given by the formula (1.2) afa1 1 ; : : :; a` g = f(aa1 )1 ; : : :; (aa`) g: Similarly, det fa1 1 ; : : :; a` g = a1 1 a` : (1.3) expresses the determinant of a conjugacy class. Since the determinant condition 0.1 plays a central role in this paper, so does 1.3. The centralizer in GLn (E) of a matrix g with class (a1 1 ; : : :; a` ) has dimension `
`
`
`
`
(1.4)
centdim(fa1 1 ; : : :; a` g) = `
` X
k=1
2k = jjjj
Q
In fact if g 2 J is semisimple then the centralizer has the form k GL (E). The dimension of the conjugacy class is (1.5) dim(fa1 1 ; : : :; a` g) = n2 ? jjjj Since the rigidity condition 0.2 plays a central role in this paper, so does 1.4 and 1.5. k
`
2. Katz's algorithm The notation set up in the previous section will generally be used henceforth with an appended index. Thus i = ff i; 1g; : : :; i;` g now typically denotes a single partition of n. Similarly ji typically denotes a single class in GLn (E). By replacing each ji by the partition i indexing its component (1.1), one associates to a rigid Jordan tuple fj1; : : :; jz g a rigid partition tuple f1 ; : : :; z g. Throughout this paper we systematically use the following abbreviations r realizable R rigid J Jordan T tuple p plausible P partition. Thus z Tn denotes the set of rigid Jordan z-tuples of rank n. Similarly RJT = ` RJ RJ z Tn is the set of all rigid Jordan tuples. We will be focused on a diagram rRJT pRJT RJT (2.1) # # # rRPT pRPT RPT In this diagram, each vertical arrow can be viewed as the passage to connected components. The word \family" will be used as a synonym for \rigid partition triple". Thus our subject is the explicit description of rRJT and we focus mostly on the explicit description of the set rRPT of realizable families. For = f1; : : :; z g with i in Pn de ne its joint partition to be a a = 1 z 2 Pzn: (2.2) Whether or not is in RPT depends only on , as the rigidity condition 0.2 becomes, via 1.4 and 1.5, (2.3) jjjj = (z ? 2)n2 + 2: i
4
DAVID P. ROBERTS
Write = f1 ; 2; 3; g with i i+1 . We say that 2 RPz Tn is plausible i n = 1 or (2.4) 1 + 2 + + n?1 (z ? 2)n < 1 + 2 + + n?1 + n : Let j 2 RJz Tn . We say that j is plausible if its centralizer partition tuple 2 RPz Tn is plausible and moreover (2.5) If a1 az = 1; then d(j1; a1) + + d(jz ; az ) (z ? 2)n: We denote the set of plausible rigid Jordan z-tuples of rank n by pRJz Tnz . It is a relatively elementary fact that rRJT pRJT. Similarly rRPT pRPT. The remaining theoretical issue is to distinguish realizability from mere plausibilty. Here one proceeds inductively on n. The base of the induction is the case n = 1. This n = 1 case is quite degenerate: RJT1 has one element ffgg which is plausible and moreover realizable; similarly RPT1 has one element ffgg which is plausible and realizable. On the partition level the induction is very simple. A marking v on 2 pRPz Tn is a part vi in each i such that the sum d(v) = v1 + + vz ? (z ? 2)n is positive. The derivative of with respect to v is denoted @v . Here (@v )i is obtained from i by replacing one vi in i by vi ? d(v) and dropping scalar partitions. The Katz classi cation is the following. Let 2 pRPT . Let v be a marking on . Then @v realizable (2.6) implies realizable. Since derivation reduces rank by at least one, this statement is indeed an e ective classi cation: after enough derivations one has reached either a non-plausible tuple or a tuple already known to be realizable. At this level of partition tuples, one has a canonical marking vmax. Here vmax;i is just the largest part of i . Here is a sample computation: 721 221 21 11 5 311 2 111 1 11 1 = 631 ! ! ! (2.7) ! : 7111 2111 111 11 55 Each arrow indicates maximal derivation, the superscript indicating the associated drop in rank. All the displayed partition triples are plausible, and the last one is realizable; so the computation shows that is realizable. On the Jordan-level the induction is more complicated, as one has to keep track of changing eigenvalues. Let j 2 J with 2 pRPT. A marking V on j is an element Vi = avi of each ji such that 1) each vi maximal for ai ; 2) v is a marking on . Automatically (V ) := a1 az v is not one by 2.4. Write ji = fai ; resti g. De ne the derivative of j with respect to the marking V to be @V j = f(@V j)1 ; : : :; (@V j)z g with (2.8) (@V j)i = ai f((V )=ai)v ?d(v) ; resti g The rigidity (0.2) and determinant (0.1) conditions are satis ed, so that @V j 2 RJTn?d(v); it may or may not be plausible. The Katz classi cation here says Let j 2 pRJT . Let V be a marking for j . Then @V j realizable (2.9) implies j realizable. Illustrations of 2.9 are given in Sections 5,6. Katz's statement of his algorithm is more complicated than the statement we have given here. Here are four salient points, which should guide the reader in checking that i
i
i
RIGID JORDAN TUPLES
5
our restatement is correct. First, Katz carries out as much as possible without imposing the rigidity condition 0.2. Second, Katz's notation is purposely highly redundant as he emphasizes; for our purposes this redundancy is not useful and so of his (r; m; e; E) we use only (r; e) (his r being our n). Third, Katz works with nite subsets D of some algebraically closed eld K. A direct translation to language close to ours would be to identify D [ f1g with f1; : : :; z g and work with ordered z-tuples and allow scalar classes. Our passage to unordered tuples, disallowed scalar classes, and thus perhaps decreasing z is highly unnatural geometrically; however it is natural from the limited point of view of our classi cation questions. Fourth our operation of derivation involves middle-tensoring, middle-convoluting, and middle-tensoring again, this being essentially one iteration of Step II-Step VI of his algorithm [7, 6.4.1]. One advantage of combining these operations is that the elements of Katz's set D and the extra point 1 are treated on the same footing, the distinction thus disappearing in our formalism. 3. n 11: quick look The reader can quickly check that using 0.2, 1.4, and 1.5, that RPT1 = fH1g H1 = fg RPT2 = fH2g where H2 = f11; 11; 11g RPT3 = fH3; P3g H3 = f21; 111; 111g P3 = f21; 21; 21;21g: Also in these cases the plausibility condition 2.4 and the realization condition 2.9 are satis ed so that rRPTn = pRPTn = RPTn. Note that H1, with z = 0 is anomolous; all other members of RPT have z 3. Note that it would also be natural to give a given element of rRPT several names, e.g. H1 = P1 and H2 = P2 . The case n = 4 is quite manageable by hand too, but at some point soon thereafter it is only reasonable to use a computer. In fact we have implemented two reasonable approaches. One, following our main text, lists out pRPT and then selects rRPT from these. The other, more in line with the proof of Theorem 7.1, starts from H1 and at every step takes all legal antiderivatives; this method never sees non-realizable tuples. Either way, the code required is modest (about 40 lines). Similarly the run-time needed for the following table is modest too (about one hour) Table 3.1. The non-zero cardinalities jrRPTn;`j for 2 n 11
n n ` 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 21 3 2 4 1 3 2 5 1 2 6 2 6 1 4 9 12 2 7 1 3 12 14 12 2 8 1 5 19 32 25 12 2 9 1 6 24 47 53 12 12 2 10 1 7 33 84 96 65 6 12 2 11 1 7 42 106 143 96 32 12 2
`
Here rRPTn = rRPTn;` where a component in rRPTn;` has dimension ` ? 1.
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DAVID P. ROBERTS
Table 3.1 provides a good guide to the rest of the paper. x4 and 5 look more closely at cases with n 6. x6 says that the \maximal" diagonal of 2's continues inde nitely. x7 says that both the \submaximal" diagonal of 12's and the gap between the maximal and submaximal diagonals continues inde nitely. x8 says that the column of 1's continues inde nitely, and identi es three other \minimal" series as well. 4. n 6: closer look Of course the programs giving Table 3.1 give rRPTn;`, not just jrRPTn;`j: Table 4.1. The elements of rRPTn;`, n 5, and their maximal derivatives. ` H1 6 H2 8 H3 8 P3 10 H4 10 P4 9 A4 9 B4 9 I4 8 4;2 12 H5 12 P5 10 A5 10 B5 10 C5 10 I5 10 J5 10 M5 9 5 9 q5 8 5
1
2
3
11 21 21 31 31 22 211 31 31 41 41 32 311 32 41 41 41 221 41 32
11 111 21 1111 31 211 211 31 22 11111 41 221 221 2111 32 41 41 221 32 32
11 111 21 1111 31 1111 211 22 22 11111 41 11111 2111 2111 311 221 41 221 32 32
@max
21 31
31
211 22 41
41 41
311 221 32 32 221 32
H1 H2 H1 H3 H1 H3 H2 H2 P3 H4 H1 A4 H3 H3 H2 P3 H2 B4 P3 P3
In [7], page 165, Katz gave examples due to Deligne of plausible but non-realizable rigid Jordan tuples. These examples all have rank seven. It is natural to ask whether there are examples of lower rank. The next two sections show that there are no examples in ranks n 3, but then examples in rank n = 4 belonging to the series A4 and B4 . One can also ask for the rst examples of plausible but non-realizable rigid partition tuples. In fact for n 5 there are none. For n = 6 there is one: f51; 33; 33; 3111g is plausible but its maximal derivative f31; 31; 31;1111g is not. Finally one can ask for the rst examples of plausible but non-realizable rigid Paritition tuples, in the context z = 3. There are none for n 6. For n = 7 there is one: f331; 331; 31111g is plausible but its maximal derivative f311; 311; 11111g is not. Deligne's plausible but non-realizable rigid Jordan tuples belong to this family. Remarks. Table 4.1 and 4.2 are useful in that they aid in working explicitly with examples. For example, consider the 120 element group A~5 . It has a presentation A~5 = hg1 ; g2; g3; z jg1g2 g3 = 1; g12 = g23 = g35 = z; z 2 = 1; z centrali
RIGID JORDAN TUPLES
7
Table 4.2. The elements of rRPT6;` and their maximal derivatives.
` 14 14 11 11 11 11 11 11 11 11 11 11 11 11 10 10 10 10 10 10 10 10 10 9 9 9 9 8
H6 P6 A6 B6 C6 D6 E6 F6 G6 I6 J6 K6 M6 N6
6;6 4 q6 r6 s6 t6 u6 v6 w6 6;3 x6 y6 z6 6;2
1 51 51 33 321 33 42 42 411 411 51 51 51 51 42 33 33 321 321 51 51 51 42 51 321 51 42 42 42
2 111111 51 321 3111 3111 222 2211 222 2211 33 51 51 51 411 222 2211 321 222 42 42 33 33 51 222 33 42 33 33
3 111111 51 51 111111 3111 21111 111111 21111 21111 2211 411 3111 222 2211 51 33 42 42 411 411 21111 2211 2211 3111 33 2211 321 321 411 222 411 411 42 42 222 33 222 42 321 33 411 33 33
51
321 411
33
51
51
51
@max H5 H1 A5 H3 H4 A5 A4 A4 B4 H3 I4 P3 H2 H2 A5 C5 B4 A4 I4 P3 I4 H3 P3 5 q5 P3 I4 5
The group A~5 has nine irreducible complex representations . Put ji = [(gi )]. All the resulting Jordan triples fj1 ; j2; j3g are rigid (as one can tell from 8.2 with `1 , `2 , `3 at most 2; 3; 5 respectively). One can use the ATLAS to gure out the ji explicitly, and hence the i . The rigid partition triples arising are, in ATLAS order, H1, H3, H3 , A4 , A5 ; H2, H2 , A4 , 6 . In particular, Tables 4.1 and 4.2 single out certain covers of the projective line as having a motivic interpretation and, as a consequence, good reduction even at primes dividing the order of the monodromy group. See [9] and [10] for examples in the context z = 3. 5. n 6, z = 3: internal structure Sections 3-8 take place mostly at the partition level. In fact only in this section and the next do we work at the Jordan level at all. This section gives a much better idea of the general case.
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DAVID P. ROBERTS
Proposition 5.1. Let be a realizable rigid partition triple of rank n = 4, 5, or 6. Let j = fj1; j2; j3g 2 J . Then whether or not j is rigid is described on the following table:
j is realizable i it is j1 j2 j3 plausible and the quantities below 6= 1. H4 A3 a b1b2 b3b4 c1 c2 c3c4 A4 a21a22 B 2 b1b2 c1 c2 c3c4 a1 a2Bbi cj ck : B4 A2 a1a2 B 2 b1b2 C 2 c1c2 Aai Bbj Cck 4 H5 A a b1b2 b3b4 b5 c1 c2 c3c4 c5 A5 A3 a2 B12 B22 b c1 c2 c3c4 c5 AaB1 B2 ci cj B5 A21 A22a Bb1 b2 C 2 c1c2 c3 A1 A2Bbi Ccj 3 2 2 C5 A a B b1b2 b3 C 2 c1c2 c3 AaBbi Ccj 5 A21 A22a B12 B22 b C12 C22c A1 A2B1 B2 Cic A1A2 Bi bC1 C2 AiaB1 B2 C1C2 5 H6 A a b1b2 b3b4 b5b6 c1 c2 c3c4 c5c6 ^ i cj A6 a31b31 B^ 3 B 2 b c1 c2 c3c4 c5c6 a1 a2BBc 3 2 3 3 ^ i Ccj B6 A^ A a B b1b2 b3 C c1c2 c3 AABb C6 a31a32 Bb1 b2b3 C 2 c1c2 c3c4 a1 a2Bbi Ccj D6 A4 a2 b21b22 b23 c1 c2 c3c4 c5c6 A2ab1 b2b3 cicj ck 4 2 2 2 2 E6 A a B1 B2 b1 b2 C c1c2 c3c4 AaB1 B2 Cci A2aB1 B2 bicj ck F6 a21a22 a23 B 4 b1b2 C 2 c1c2 c3c4 a1a2 a3B 2 biCcj ck 4 2 2 2 2 G6 A a1a2 B1 B2 b1 b2 C1 C2 c1 c2 Aai B1 B2 C1C2 A2ai B1 B2 bj C1C2ck
6 a31a32 b21b22 b23 C 2 c1c2 c3c4 a1 a2bi bj Cck a21a2 b1b2 b3ci cj 3 3 2 2 6 a1a2 B1 B2 b1 b2 C12 C22c1 c2 a1 a2B1 B2 Ci cj a1a2 Bi bj C1 C2 a21a2 B1 B2 bi C1C2cj ^ 2 ci ^ k a1a2 a3BBbC q6 a21a22 a23 B^ 3 B 2 b C 3 c1c2 c3 ai aj BBCc 3 2 3 2 3 ^ BBCc ^ i ^ ^ ^ ^ ^ ^ Aa C c1c2 c3 AABBCci AABbCci r6 A A a B B b 2 2 2 2 2 2 3 2 2 ^ ^ ^ ^ 2 6 a1a2 a3 b1b2 b3 Ccc ai aj bk bl CC a1a2 a3b1 b2b3 C C a1a2 a3b1 b2b3 CC Here subscripts i, j , k, l on the same symbol are required to be di erent, but are otherwise arbitrary. Proof. The proof consists in applying the Katz algorithm 2.9 in each case. Here we describe the computation in the last case = 6. Starting from 6, we take three derivatives (2.8) successively, as indicated by the !d ; here d indicates the drop in rank. In between, as indicated by =, we simplify by scalar multiplication (1.2). The s at each derivation step indicate the choice of marking; the s on the next line indicate the slots corresponding to the previous s, to increase readability. In this computation, exponents are part of the Jordan formalism, except for the very last line where they simply indicate multiplication.
8 fa2; a2; a2g 9 < 21 22 32 = b1 ; b2 ; b 3 g = j : f fC^ 3; C 2; cg ; 8 9 ^ a1 < f(1=b1C); a22 ; a23 g = 1 ^ ! b1 : f(1=a1C); b22; b23 g ; 2 ^ C f(1=a1b1) ; C 2; cg
RIGID JORDAN TUPLES
= (5.1)
!1 =
!2 =
8 f(a =b C); 9 < 1 1 ^ (a1a2)22; (a1a3)22g = 0 ^ (b1 b2) ; (b1 b3) g ; = j 1=a1 C); : ff((bC=a ^ 1 b1)2 ; (CC) ^ 2 ; (Cc) ^ g 8 ^ (a1 a3)2 g 9 ^ (a1 a2) < fa1=b1C; 1=b1b2CC; = ^ ^ (b1 b2) : fb1=a1C; 1=a1a2CC; (b1 b3)2 g ; ^ ^ b )2 ; 1=a a2b1 b2; ^g (CC) f(C=a Cc 8 fa a a =b C;^ 1 1 a a =b b 1CC; 9 (a1a1 a2a3 )2 g = < 112 1 1 2 1 2^ ^ ^ fb1b1 b2=a1C; b1 b2=a1a2 CC; (b1 b1b2 b3)2 g ; = j 00 : f 2 ^ 1b1) ; CC=a ^ 1a2 b1b2 ; ^ g (C^ CC=a C^ CCc 8 9 ^ ^ a1a2 =b1b2CC g = a1 a1a2 a3 < fa1a1 a2=b1 C; ^ ^ b1b2=a1 a2CC b1b1 b2b3 g fb1b1 b2=a1C; : ^ ^ ^ 1a2b1 b2; C^ CCc ^ g; C CC=a1b1 f CC=a 8 fa4a2a =b C;^ ^ g9 a31 a22a3 =b1b2CC < 41 22 3 1 = 3 b2b3=a1 a2CC ^ ^ g = j 000 f b b b =a C; b 3 1 1 2 1 2 : fC^ 3C 2=a2a b2b ; C^4C 2c=a b g ; 1 2 1 2
1 1
9
De ne elements of E as follows: Wij = ai bj C^ ^ Xij = (a1 a2a3 =ai )(b1b2 b3=bj )CC Y = a1 a2a3 b1b2 b3C^ 2C ^ 2 Z = a1 a2a3 b1b2 b3CC Then j 2 6 is plausible i all nine Wij are di erent from 1. j 0 2 5 is plausible i W22, W23 , W32, W33, X22 , X23 , X32, and X33 are di erent from 1. j 00 2 B4 is plausible i W23 , W32 , X13 , X31 , X22, X11, and Y are di erent from 1. Finally j 000 2 H2 is plausible, or equivalently realizable, i W33 , X12 , X21, and Z are di erent from 1. Here we are using several times that a21a22 a23 b21b22b23 C^ 3C 2c = 1. Altogether, one gets that j is realizable i all nine Wij , all nine Xij , Y and Z are di erent from one, as stated on the table. One can reinterpret Proposition 5.1 by thinking of A, a, b1 , b2, : : : not as elements of E , but rather as coordinate functions on a torus T covering J 6 . Then Proposition 5.1 gives de ning equations for the degeneracy locus T 1 T corresponding to non-realizable triples. An attribute of the proof is that it does not exploit the fact that T 1 is stable under the natural action of S3 S3 S1 on T. Rather, computations such as 5.1 break such symmetries, since they involve arbitrary choices of markings. For example, in the case of 6, the nine divisors (Xij ) are permuted transitively by S3 S3 S1 . But in 5.1, four of these divisors appear at the j 0 level, three more at the j 00 level, and the nal two at the j 000 level. It would be desirable to be able to formulate Katz's classi cation in non-inductive terms, preferably in a way that was fully invariant under the action of these Weyl groups. Remarks. The dual of a Jordan class ji 2 Jn is obtained by replacing each eigenvalue by its inverse. Call a realizable rigid Jordan tuple j = fj1; : : :; jz g strictly self-dual if each ji is self-dual. This notion play a fundamental role in computing monodromy groups, for example. The table in Proposition 5.1 interacts in interesting ways with this notion; for example many of the families do not have strictly self-dual members for E of characteristic 2 as all singletons are forced to be 1, often contradicting plausibility.
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DAVID P. ROBERTS
6. Maximal series: H and P The Hypergeometric series. For positive integers n, de ne a partition triple
8 n ? 1; 1 9 = < Hn = : 1; : : :; 1 ; : 1; : : :; 1
Each Hn is rigid and plausible. H1 is realizable. The unique derivative of Hn is Hn?1. So each Hn is realizable by induction. The general member of the family JH has the form 8 An?1; a 9 < = j = : b1 ; ; b n ; : (6.1) c1 ; ; c n Here we regard all variables but a as freely chosen from E . The variable a is then determined by the condition 0.1. For n > 1, the Jordan triple j is plausible i Abi cj 6= 1 for all i; j 2 f1; : : :; ng. With respect to the marking (A; bn; cn), the derivative of j is 8 9 8 9 A < (1=bncn )n?2; a Abn cn < (1=bncn )n?2; a = = bn : (1=Acn)0 ; b1; : : :; bn?1 1 : cb11;; :: :: :;:; cbnn??11 ; cn (1=Abn)0 ; c1; : : :; cn?1 ; 1 8 An?2a0 9 < = = : b1 : : :bn?1 ; c1 : : :cn?1 with a0 = Aabn cn. Here indicates modi cation by a rank one twist, which changes neither plausibility or realizability. By induction, if j is plausible then j is realizable, in this classical case. The Pochhammer series. For positive integers n, de ne a partition (n + 1)-tuple Pn = f(n ? 1)1; : : :; (n ? 1)1g Each Pn is rigid and plausible, with P2 = H2 and P1 = H1. The derivatives of Pn are (6.2) @(n?1;:::;n?1)Pn = H1 (6.3) @(n?1;:::;n?1;1)Pn = Pn?1: So realizability of Pn follows either directly from 6.2 or by induction from 6.3. The general member of the family JP has the form 8 An?1a 9 > < 1. 1 > = .. j=> (6.4) : An?1an+1 > ; n+1 subject to Condition 0.1. The Jordan tuple j is plausible i (6.5) := A1 An+1 6= 1 (6.6) ai =Ai 6= 1 (i = 1; : : :; n + 1) both hold. One can check, in two ways, that here too plausibility implies realizability. Remarks. There is an enormous literature on mathematics related to Hn or Pn. Particularly motivic references are [6] in the hypergeometric case and [2] in the Pochhammer case. A recent paper presenting Hn and Pn in tandem in the context of covers of the projective line is [12]. n
n
RIGID JORDAN TUPLES
11
In general, given a realizable rigid Jordan tuple j = fj1; : : :; jz g one can ask for a matrix tuple g = (g1 ; : : :; gz ) 2 V (j1 ; : : :; jz ) representing it. In the hypergeometric and Pochhammer cases this problem has a uniform solution (see e.g. [12]). We restate the solution for Hn here in our language. By twisting, one can take A = 1 in 6.1. De ne f(x) = det(x ? j2 ) = (x ? b1 ) (x ? bn) = xn + f1xn?1 + + fn?1 x + fn h(x) = det(x ? j3 ) = (x ? c1 ) (x ? cn) = xn + h1xn?1 + + hn?1x + hn De ne e0 = 1=(fnhn ) and ei = hi =(fn hn) ? fn?i =fn for i = 1; : : :; n ? 1. De ne matrices by following the pattern evident in the case n = 4: (6.7) 0 0 0 0 0 ?f 1 0 ?h 1 0 0 1 1 e0 0 0 0 4 1 B B B C C C: 1 0 0 ? f e 1 0 0 ? h 3 1 2 B B B C C g1 = @ e 0 1 0 A g2 = @ 0 1 0 ?f A g3 = @ ?h 00 01 10 C A 2 2 3 e3 0 0 1 ?h4 0 0 0 0 0 1 ?f1 n Then g1g2 g3 = 1. The group hg1; g2; g3i acts irreducibly on E i no root of f is the inverse of a root of h. 7. Submaximal series: A, B , C , D, E , F , G and I , J K , L, M , N In this section and the next we work only at the partition level; thus in these sections, we are only classifying realizable families, not examining the internal structure of a given family. Call a realizable rigid partition tuple f1 ; : : :; z g 2 rRPT submaximal if its length ` and its rank n satisfy the inequality n + 5 ` < 2n + 2. In this section we classify submaximal partition tuples, restricting to n 6 to avoid degeneracies. On Table 7.1 we give 12 partition tuples in each rRPTn;n+5. In each block the top line refers to n even and the bottom line refers to n odd. Thus our notation places these tuples in 11 full series and two half series Geven and Lodd . For typographical reasons, we use the following abbreviations on Table 7.1: u, v, w, x, y, and z respectively stand for n=2 + 3=2, n=2 + 1, n=2 + 1=2, n=2, n=2 ? 1=2, and n=2 ? 1. For n even only the integers v, x, and z appear; for n odd only the integers u, w, and y appear. Note that series A, B, and C share a joint partition (2.2) and so do series D, E, F, and G: 8 xxxz1 1 Series ABC, n even > < wyyy1 1 Series n odd = (n ? 2)2 2111111 Series ABC, (7.1) DEFG, > : (n ? 2)2 211111 Series DEF, n noddeven The rigidity condition 0.2 is quickly veri ed in every case. For example, in Series ABC, n even one has 3x2 + z 2 + (n + 1)12 = 3( n2 )2 + ( n2 ? 1)2 + (n + 1) = 34 n2 + ( 41 n2 ? n + 1) + (n + 1) = n2 + 2 Similarly plausibility condition 2.4 is quickly veri ed in all cases. Table 7.1 gives all possible derivatives Ym 2 RPTm of each Xn 2 rRPTn. Here those with m = n ? 1 are placed in Column 1, those with m = n ? 2 are placed in Column 2,
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DAVID P. ROBERTS
Table 7.1. The twelve tuples in rRPTn;n+5 and their derivatives
1; (i )
2
3
z=3 An xx wy Bn xz1 yy1 Cn xx wy Dn (n ? 2)2 (n ? 2)2 En (n ? 2)2 (n ? 2)2 Fn (n ? 2)11 (n ? 2)11 Gn (n ? 2)11
xz1 yy1 x1 1 w1 1 x1 1 y1 1 2 2 2 21 2 211 2 2111 2 2 2 21 2 211
1 1 1 1 x1 1 y1 1 z1 1 y1 1 2 2111111 2 211111 2 21111 2 2111 2 21111 2 2111 2 211
A A B; C B; C C C E D; E D; E E F F; G F
z=4 In (n ? 1)1 (n ? 1)1 Jn (n ? 1)1 (n ? 1)1
xx wy (n ? 1)1 (n ? 1)1
x1 1 w1 1 2 2 2 21
I I J J
z = dn=2e + 1 (L; N) Kn (n ? 1)1 xz1 (n ? 1)1 wy Ln (n ? 1)1 yy1 Mn (n ? 2)2 (n ? 2)11 (n ? 2)2 (n ? 1)1 Nn (n ? 2)2 (n ? 2)11 (n ? 2)2 (n ? 1)1
(4 )
v1 1 w1 1 2 211 2 21
1
2
D D E E F F G
J J
z = dn=2e + 2 (K; M) xx K wy yy1 (n ? 1)1 (n ? 1)1 (n ? 2)11 (n ? 2)11
(n ? 1)1 (n ? 1)1 (n ? 2)11 (n ? 2)11
K M M N N
>2
H x ; Hv Hx Hv H w ; Hy
Hv Hw
P v ; Px P u ; Pw ; Py M M N N
Pw H2 H2 H2 H2
and any that remain are placed in Column >2. For example, consider Bn for n even; the four blocks of 7.2 account for the four derivatives of Bn printed on Table 7.1.
RIGID JORDAN TUPLES
13
Bn = xz1 x11 1 x11 1 zz1 z11 1 x 1 1 = Bn?1 (7.2)
Bn = xz1 x11 1 x11 1 xz z11 1 z11 1 = Cn?1 Bn = xz1 x11 1 x11 1 z1 11 1 11 1 = Hx
Bn = xz1 x11 1 x11 1 x 1 111 1 111 1 = Hx+1 The fact that our notation places the tuples in eleven full series and two half series has some arti cial aspects; for example, Fodd is just as tightly tied to Geven as it is to Feven . On the other hand, the unions A, BC, DE, FG, I, J, KL, M, and N are stable under drop-1 and drop-2 derivation. Proposition 7.1. For n 6, the set rRPTn;n+5 contains exactly the twelve tuples listed in 7.1. Also rRPTn;` is empty for n + 5 < ` < 2n + 2. Sketch of proof. Let rRPTn; =
2a n+2
`=n+5
rRPTn;`:
One can check, without using any classi cation results, that rRJTn; is closed under derivation. One then has to check that if one starts with H1 and successively antidi erentiates, keeping only realizable families with ` n + 5, one gets only the families listed in Table 7.1. Here is a sample of the computations needed. Consider antiderivatives of H2 of rank m+2. They must consist of a copies of m11, (3 ? a) copies of (m + 1)1, and x copies of m2, for some 0 a 3 and x 2 Z0. The rigidity condition 0.2 for forces a(m2 + 12 + 12) + (3 ? a)((m + 1)2 + 12) + x(m2 + 22 ) = (x + 1)(m + 2)2 + 2 which simpli es to 2m(2x ? m + a ? 1) = 0. For a = 0; 1; 2; 3, this gives Modd , Meven , Nodd , and Neven , respectively, and nothing more. Remarks. Okubo, Yokoyama, Haraoka and others have studied the di erential equations indexed by = f1; : : :; z g 2 rRPT such that all the i but perhaps one have the form (mi ; 1; : : :; 1). They prove that the possibilities are as follows: z=3 z>3 Our Their Our Their n notation Notation notation Notation Arbitrary Hn In Pn In Even ( 4) Bn IIn In IIn Odd ( 5) Bn IIIn In IIIn 6 F6 IV6 N6 IV6 See [3] for emphasis on di erential equations; see [4] and [5] for explicit matrix formulas analogous to 6.7.
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DAVID P. ROBERTS
Simpson has approached our topic from a Hodge-theoretic viewpoint [11]. In our language, one of his classi cation results is the following. The only partition tuples = f1 ; : : :; z g 2 rRPT with some i = (1; : : :; 1) are the tuples Hn, An, and D6 . 8. Minimal series: , , , and Call a partition tuple f1; : : :; z g 2 RPT minimal if the corresponding length tuple f`1; : : :; `z g satis es (`1 ; : : :; `z ) := `1 + + `1 ? (z ? 2) 0: (8.1) 1 z In this section we classify minimal rigid partition tuples. For n k positive integers de ne partition triples as follows: n=k is the unique partition with k parts and all members dn=ke or bn=kc. Thus, for example, 13 below is f13=2; 13=3; 13=6g = f76; 544; 322222g. If k divides n then n=k := fn=k + 1; n=k; n=k; n=k ? 1g: Thus 12;2 below is f12;2; 12=3; 12=6g = f75; 444; 222222g. Put k (n) = 1 if k divides n and k (n) = 0 else. Proposition 8.1. For n > 6 the set RPTn;min contains exactly the 4 + 4(n) + 26(n) elements listed in the following table.
d [n] 3 0 1 2 4 0
n;3 n n n;2 n;4 1 n 2; 3 n 6 0
n;2
n;3
n;6 1
n 2; 3; 4; 5 n 2 0 n;2 1 n
= = = = = = = = = = = = = =
Derivatives
1 fn=3; n=3; n=3g n?1 fn=3; n=3; n=3g n?1;3 fn=3; n=3; n=3g n?1 fn=2; n=4; n=4g n?1 fn=2; n=4; n=4g n?1 fn=2; n=4; n=4g n?1;2; n?1;4 fn=2; n=4; n=4g n?1 fn=2; n=3; n=6g
n?1 fn=2; n=3; n=6g
n?1 fn=2; n=3; n=6g
n?1 fn=2; n=3; n=6g
n?1;2; n?1;3; n?1;6 fn=2; n=3; n=6g
n?1 fn=2; n=2; n=2; n=2g n?1 fn=2; n=2; n=2; n=2g n?1;2
Partition tuples
2 n?2 n?2
n?2 n?2
All these rigid partition triples are realizable. Sketch of proof. The last statement follows by induction from the table of derivatives
given. As to the completeness of the table, in general let = f1 ; : : :; z g 2 RPTn with corresponding length tuple f`1 ; : : :; `z g. For each i one has jjijj n2 =`i with equality i i = n=` . So one has X (8.2) (z ? 2)n2 + (`1 ; : : :; `z )n2 jjijj = (z ? 2)n2 + 2; i
i
here the inequality is the de nition of while the equality is Condition 0.2. Suppose is minimal, meaning that (`1 ; : : :; `z ) 0. Then 8.2 is extremely restrictive and one can conclude by elementary arguments: the cases with > 0 yield nothing with n > 6; the remaining cases f`1 ; : : :; `z g = f3; 3; 3g, f2; 4; 4g, f2; 3; 6g, and f2; 2; 2; 2g yield the series , , , and respectively.
RIGID JORDAN TUPLES
15
The four series just discussed are naturally associated with the four rank two Dynkin diagrams: ( ; ; ; ) $ (A2 ; B2; G2; A1 A1 ); this connection partially explains our choice of notation. References [1] F. Beukers and G. Heckman. Monodromy for the hypergeometric function n Fn?1 , Inv. Math. 95 (1989) 325{354. [2] P. Deligne and G. Mostow. Monodromy of hypergeometric functions and non-lattice integral monodromy. Publ. Math. IHES 63 (1986), 5-89. [3] Y. Haraoka. Canonical forms of di erential equations free from accessory parameters, SIAM J. Math Anal 25 (1994) no. 4, 1203-1226. [4] . Monodromy representations of systems of di erential equations free from accessory parameters. SIAM J. Math. Anal. 25 (1994), no. 6, 1595{1621. [5] . Monodromy of an Okubo system with non-semisimple exponents. Funkcial. Ekvac. 40 (1997), no. 3, 435{457. [6] N. M. Katz. Exponential Sums and Di erential Equations, Annals of Mathematics Study 124 Princeton University Press (1990). . Rigid Local Systems, Annals of Mathematics Study 138, Princeton University Press (1996). [7] [8] N. Koblitz. The number of points on certain families of hypersurfaces over nite elds, Comp. Math. 48 (1983) 3{23. [9] D. P. Roberts. Linear rigidity and singular primes for three point covers, in preparation. [10] . Number elds with discriminant 2a 3b: Examples from three point covers, Preprint (1998) [11] C. T. Simpson, Products of matrices, Canadian Mathematical Society conference proceedings 12, 1991. [12] H. Volklein. Rigid generators of classical groups. Math. Ann. 311 (1998), no. 1, 1-58. Department of Mathematics, Hill Center, Rutgers University, New Brunswick, NJ 08903
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