RIGID JORDAN TUPLES

DAVID P. ROBERTS

Abstract. In a 1996 book, Katz introduced some remarkable objects in arithmetic geometry, rigid local systems. He gave an inductive classification of these systems, and asked for more explicit results concerning this classification.

Here we use much more elementary language, speaking of rigid Jordan tuples rather than rigid local systems. We present a substantially streamlined version of Katz’s classification. We provide some explicit results of the type Katz asked for.

Contents

1. Overview 1 2. Jordan formalism 4 3. Katz’s algorithm 5 4. n ≤ 11: quick look 7 5. n ≤ 6: closer look 7 6. n ≤ 6, z = 3: internal structure 9 7. Maximal series: H and P 12 8. Submaximal series: A, B, C, D, E, F, G and I, J, K, L, M, N 13 9. Minimal series: α, β, γ, and δ 16 References 17

1. Overview

Fix for this entire paper an algebraically closed field E. Consider unordered z-tuples j = {j_{1}, …, j_{z}} of non-central conjugacy classes in GL_{n}(E), z ≥ 0 and n ≥ 1 being integers. Impose the determinant condition

det(j_{1}) ⋯ det(j_{z}) = 1. (1.1)

Also impose a rigidity condition on the dimension of these conjugacy classes,

dim(j_{1}) + ⋯ + dim(j_{z}) = 2(n^{2} − 1). (1.2)

We call such {j_{1}, …, j_{z}} rigid Jordan z-tuples of rank n. Here we use the word “Jordan” because we will be describing each j_{i} in terms of Jordan canonical forms.

Let (j_{1}, …, j_{z}) be an ordered list of non-central conjugacy classes in GL_{n}(E) satisfying Conditions (1.1) and (1.2). Let V (j_{1}, …, j_{z}) be the subset of j_{1} ×⋯×j_{z} consisting of matrix tuples (g_{1}, …, g_{z}) with g_{1} ⋯ g_{z} = 1 and ⟨g_{1}, …, g_{z}⟩ acting irreducibly on E^{n}. The group P GL_{n}(E) acts on V (j_{1}, …, j_{n}) by simultaneous conjugation. A naive dimension count suggests that there are only finitely many orbits. It is elementary that the number of orbits is independent of the ordering of the j_{i}. A cohomological argument says that in fact the number of orbits is either zero or one [10]. In the latter case, we say that {j_{1}, …, j_{z}} is realizable. The problem addressed in this paper is the explicit description of realizable rigid Jordan tuples.

1991 Mathematics Subject Classification. Primary 15A21, Secondary 14D05.

A rigid local system in the sense of Katz [7] gives rise to a realizable rigid Jordan tuple in our sense. Information has been lost in the passage from rigid local systems to realizable rigid Jordan tuples. For example, a finite subset S of the Riemann sphere C∪{∞} has been replaced by the number z = |S|. However the information lost is trivial from the point of view of classification, meaning that our classification of realizable rigid Jordan tuples immediately translates back into a classification of rigid local systems.

In §2 we introduce a formalism for conveniently dealing with Jordan canonical forms. In §3 we present Katz’s remarkable inductive algorithm for classifying rigid local systems, recast into the setting rigid Jordan tuples. Katz uses complicated algebro-geometric language necessary for his proof that his classification is correct. We use radically simpler language, insufficient for a presentation of Katz’s proof, but ideal for pursuing explicit classification results. In particular, our formalism exploits the fact that rigid Jordan tuples come in families, and it makes sense to talk about realizability at the level of families. At the level of classifying realizable families, only the very simplest parts of the algorithm are involved. The coefficient field E does not play a role at all, and the formalism centers on partitions. The realizable rigid Jordan tuples within a given realizable family naturally form an irreducible affine variety over E. In our terminology, the dimension of a family is one less than its length ℓ.

With respect to rigid local systems, Katz wrote [7, page 9] of “a fascinating bestiary waiting to be compiled.” In §4-9 we present such a compilation at the level of rigid Jordan tuples. In dimension ≤ 11, the results can be summarized as follows.

Proposition 1.1. The number |rRP T_{n,ℓ}| of realizable families of rank n length ℓ rigid Jordan tuples for 2 ≤ n ≤ 11 is as in Table 1.1.

Table 1.1. \begin{matrix}n \ ℓ & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\ 2 & 1 \\ 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 \end{matrix}

In §4, we indicate how Proposition 1.1 can be proved by a computer search. In §5 we list out and label the families in ranks n ≤ 6. For example, the “realizable rigid partition tuple”

{22, 211, 1111}, (1.3)

which we denote A_{4}, indexes one of the three families with n = 4 and ℓ = 9. Here the length ℓ is the number of parts of the joint partition 222111111.

In §6 we examine the inner structure of the families with n ≤ 6 and z = 3. For example, members of A_{4}, in our notation, have the form

{a_{1}^{2}a_{2}^{2}, B^{2}b_{1}b_{2}, c_{1}c_{2}c_{3}c_{4}}. (1.4)

Here a_{1}, a_{2}, B, b_{1}, b_{2}, c_{1}, c_{2}, c_{3}, c_{4} are in E^{×} and satisfy the determinant condition

a_{1}^{2}a_{2}^{2} · B^{2}b_{1}b_{2} · c_{1}c_{2}c_{3}c_{4} = 1 (1.5)

and the inequalities

\begin{matrix}a_{i}Bc_{j} ≠ 1, & i = 1, 2, & j = 1, 2, 3, 4 \\ a_{1}a_{2}Bb_{i}c_{j}c_{k} ≠ 1, & i = 1, 2, & j, k = 1, 2, 3, 4 & j ≠ k. \end{matrix} (1.6)

In (1.4)-(1.6), a_{1}, …, c_{4} represent eigenvalues, and the Jordan block structure is obtained by dualizing the exponents, as explained in §2. The internal structure of all other families looks qualitatively similar: a single determinant condition like (1.5), explaining why the dimension of the family is ℓ−1, and supplemental multiplicative inequalities like (1.6).

Note from Table 1.1 that, for 6 ≤ n ≤ 11 at least, the maximal possible ℓ is 2n + 2, achieved by two families. The next largest possible ℓ is n + 5, coming from twelve families. The smallest possible ℓ is 8, achieved once. In §7, 8, 9 we discuss these maximal, submaximal, and extreme minimal cases respectively, proving that the patterns continue indefinitely.

The 2’s in the maximal case come from what we call the hypergeometric family H_{n} and the Pochhammer family P_{n}. These two families have z = 3 and z = n + 1 respectively. They can reasonably be considered classical, and §7 may serve for some readers as an illustration of how our formalism looks in a familiar context.

In our discussion in §8 of the submaximal case, the hypergeometric and Pochhammer families reappear because of the inductive nature of the classification. In this section, we also prove that the gap between the 12’s and the 2’s in Table 1.1 continues indefinitely.

Finally, in §9, we identify some families with ℓ = 9, 10, 11 which we also call minimal. We discuss these together with the extreme minimal case.

The principal interest in our subject is that it forms the combinatorial core of a very rich motivic theory [7, Chapter 8]. Rigid matrix tuples (g_{1}, …, g_{z}) ∈ V (j_{1}, …, j_{z}) arise as underlying monodromy representations. For the family H_{n}, hypergeometric functions _{n−1}F_{n} arise as period integrals. Similarly for the family P_{n}, Pochhammer functions arise as period integrals. The remaining realizable families give rise to similar functions which have just begun to be studied. There are many arithmetic questions which have not been answered even for the classical families H_{n} and P_{n}.

Other authors have built on the work of Katz in different directions, sometimes referring to the general subject as the Deligne-Simpson problem. We refer the reader to recent papers of Belkale, Crawley-Boevey, Dettweiler, Gleizer, Kostov, Reiter, Strambach, and Völklein. Also there is a connection with the circle of ideas involved in the recent solution of Horn’s conjecture, see e.g. [8].

2. Jordan formalism

For n a positive integer, let P_{n} be the set of partitions of n. We write out partitions by listing their parts: λ = {λ_{1}, …, λ_{ℓ}}. Order does not matter, and we usually choose to list parts in decreasing order. The rank of λ is simply the number n = |λ| = ∑ λ_{k}.

The maxmult of λ ∈ P_{n} is simply the largest part m(λ). The length of λ is the number ℓ(λ) of parts. The norm of λ is the quantity ||λ|| = ∑ λ_{k}^{2}. Note that |λ| ≡ ||λ|| modulo two. We often omit braces and commas when the meaning is clear. Thus P_{4} = {{1, 1, 1, 1}, {2, 1, 1}, {2, 2}, {3, 1}, {4}} = {1111, 211, 22, 31, 4}.

In (2.2) below, we make use of the standard duality t : P_{n} → P_{n} which interchanges maxmult m and length ℓ. Graphically, one can think of t as transpose, e.g.

(3, 1)^{t} = ( \begin{matrix}•, & • \\ • \\ • \end{matrix} )^{t} = \binom{•, •, •}{•} = (2, 1, 1).

Algebraically, the dual λ^{t} of a given partition λ is defined by letting λ_{k}^{t} be the number of parts of λ of size ≥ k.

For λ ∈ P_{n}, define J_{λ} to be the set of formal symbols {a_{1}^{λ_{1}}, …, a_{ℓ}^{λ_{ℓ}}} with a_{k} ∈ E^{×}. So J_{λ} is a copy of the ℓ-dimensional torus (E^{×})^{ℓ} modded out by a product of symmetric groups. E.g. J_{321} is simply a three-dimensional torus, the general element being {a^{3}, b^{2}, c} for a unique (a, b, c) ∈ E^{×3}. On the other hand J_{222} is the torus modulo S_{3}, as the general element can be expressed as {a^{2}, b^{2}, c^{2}} for six different choices of the ordered triple (a, b, c). Here the exponents are part of the formalism, and not an indication of multiplication.

Put

J_{n} := ∐_{λ∈P_{n}} J_{λ}. (2.1)

If j ∈ J_{λ} we say that λ is the centralizer partition of j.

Let g ∈ GL_{n}(E) be a matrix. For a ∈ E^{×}, let n_{a} be the dimension of the generalized eigenspace of a, i.e. the dimension of the kernel of (g − aI_{n})^{n}. Let μ_{a} be the partition of n_{a} giving the sizes of the Jordan blocks belonging to a. Put

λ_{a} = μ_{a}^{t}. (2.2)

A good case to keep in mind is the case when g acts just by the scalar a on the generalized eigenspace of a. Then the corresponding partitions of n_{a} are μ_{a} = 11 ⋯ 11 and λ_{a} = n_{a}. The μ_{a} will not play any further role.

The theory of Jordan canonical forms identifies the set of conjugacy classes in the group GL_{n}(E) with the set J_{n}, via

[g] = {a_{1}^{λ_{1}}, …, a_{ℓ}^{λ_{ℓ}}}.

Here, for a given a ∈ E^{×}, the exponents on a all together form λ_{a}. The largest part of λ_{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

a{a_{1}^{λ_{1}}, …, a_{ℓ}^{λ_{ℓ}}} = {(aa_{1})^{λ_{1}}, …, (aa_{ℓ})^{λ_{ℓ}}}. (2.3)

Similarly,

det ({a_{1}^{λ_{1}}, …, a_{ℓ}^{λ_{ℓ}}}) = a_{1}^{λ_{1}} ⋯ a_{ℓ}^{λ_{ℓ}} (2.4)

expresses the determinant of a conjugacy class; on the right of (2.4), exponents are indicating multiplication. Since the determinant condition (1.1) plays a central role in this paper, so does (2.4).

The centralizer in GL_{n}(E) of a matrix g with class (a_{1}^{λ_{1}}, …, a_{ℓ}^{λ_{ℓ}}) has dimension

centdim({a_{1}^{λ_{1}}, …, a_{ℓ}^{λ_{ℓ}}}) = ∑_{k=1}^{ℓ} λ_{k}^{2} = ||λ||. (2.5)

In fact, if g ∈ J_{λ} is semisimple then the centralizer has the form ∏_{k} GL_{λ_{k}}(E). The dimension of the conjugacy class is

dim({a_{1}^{λ_{1}}, …, a_{ℓ}^{λ_{ℓ}}}) = n^{2} − ||λ||. (2.6)

Since the rigidity condition (1.2) plays a central role in this paper, so do (2.5) and (2.6).

3. Katz’s algorithm

The notation set up in the previous section will generally be used henceforth with an appended index. Thus λ_{i} = {λ_{i,1}, …, λ_{i,ℓ_{i}}} now typically denotes a single partition of n. Similarly j_{i} typically denotes a single class in GL_{n}(E). By replacing each j_{i} by the partition λ_{i} indexing its component (2.1), one associates to a rigid Jordan tuple {j_{1}, …, j_{z}} a rigid partition tuple {λ_{1}, …, λ_{z}}.

Throughout this paper we systematically use the following abbreviations

r realizable R rigid J Jordan T tuple
p plausible  P partition.

Thus RJ_{z}T_{n} denotes the set of rigid Jordan z-tuples of rank n. Similarly RJT = ∐ RJ_{z}T_{n} is the set of all rigid Jordan tuples. We will be focused on a diagram

\begin{aligned}rRJT & ⊂ & pRJT & ⊂ & RJT \\ ↓ & & ↓ & & ↓ \\ rRP T & ⊂ & pRP T & ⊂ & RP T \end{aligned} (3.1)

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 tuple.” Thus our subject is the explicit description of rRJT and we focus mostly on the explicit description of the set rRP T of realizable families.

For λ = {λ_{1}, …, λ_{z}} with λ_{i} in P_{n} define its joint partition to be

μ = λ_{1} ∐ ⋯ ∐ λ_{z} ∈ P_{zn}. (3.2)

Whether or not λ is in RP T depends only on μ, as the rigidity condition (1.2) becomes, via (2.5) and (2.6),

||μ|| = (z − 2)n^{2} + 2. (3.3)

Write μ = {μ_{1}, μ_{2}, μ_{3}, ⋯ } with μ_{i} ≥ μ_{i+1}. We say that λ ∈ RP_{z}T_{n} is plausible iff n = 1 or

μ_{1} + μ_{2} + ⋯ + μ_{n−1} ≤ (z − 2)n < μ_{1} + μ_{2} + ⋯ + μ_{n−1} + μ_{n}. (3.4)

Let j ∈ RJ_{z}T_{n}. We say that j is plausible if its centralizer partition tuple λ ∈ RP_{z}T_{n} is plausible and moreover

If a_{1} ⋯ a_{z} = 1, then d(j_{1}, a_{1}) + ⋯ + d(j_{z}, a_{z}) ≤ (z − 2)n. (3.5)

We denote the set of plausible rigid Jordan z-tuples of rank n by pRJ_{z}T_{n}.

It is a relatively elementary fact that rRJT ⊆ pRJT. Similarly rRP T ⊆ pRP T. 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: RJT_{1} has one element {{}} which is plausible and moreover realizable; similarly RP T_{1} has one element {{}} which is plausible and realizable.

On the partition level the induction is very simple. A marking v on λ ∈ pRP_{z}T_{n} is a part v_{i} in each λ_{i} such that the sum d(v) = v_{1} + ⋯ + v_{z} − (z − 2)n is positive. The derivative of λ with respect to v is denoted ∂_{v}λ. Here (∂_{v}λ)_{i} is obtained from λ_{i} by replacing one v_{i} in λ_{i} by v_{i} − d(v) and dropping scalar partitions. The Katz classification is the following.

Let λ ∈ pRP T. Let v be a marking on λ. Then ∂_{v}λ realizable implies λ realizable.

(3.6)

Since derivation reduces rank by at least one, this statement is indeed an effective classification: 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 v_{max}. Here v_{max,i} is just the largest part of λ_{i}. Here is a sample computation:

λ = 721 631 7111 55 →^{5} 221 311 2111 →^{2} 21 111 111 →^{1} 11 11 11 →^{1} . (3.7)

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 λ is realizable.

On the Jordan-level the induction is more complicated, as one has to keep track of changing eigenvalues. Let j ∈ J_{λ} with λ ∈ pRP T. A marking V on j is an element V_{i} = a_{i}^{v_{i}} of each j_{i} such that 1) each v_{i} maximal for a_{i}; 2) v is a marking on λ. Automatically

π(V) := a_{1} ⋯ a_{z}

is not one by (3.5). Write j_{i} = {a_{i}^{v_{i}}, rest_{i}}. Define the derivative of j with respect to the marking V to be ∂_{V} j = {(∂_{V} j)_{1}, …, (∂_{V} j)_{z}} with

(∂_{V} j)_{i} = a_{i}{(π(V)/a_{i})^{v_{i}−d(v)}, rest_{i}} (3.8)

The rigidity (1.2) and determinant (1.1) conditions are satisfied, so that ∂_{V} j ∈ RJT_{n−d(v)}; it may or may not be plausible. The Katz classification here says

Let j ∈ pRJT. Let V be a marking for j. Then ∂_{V} j realizable implies j realizable.

(3.9)

Illustrations of (3.9) are given in §6, 7.

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 diligent reader in checking that our restatement is correct. First, Katz carries out as much as possible without imposing the rigidity condition (1.2). 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 finite subsets D of some algebraically closed field K. A direct translation to language close to ours would be to identify D ∪ {∞} with {1, …, z} 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 classification 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 ∞ are treated on the same footing, the distinction thus disappearing in our formalism.

4. n ≤ 11: quick look

The reader can quickly check that using (1.2), (2.5), and (2.6), that

RP T_{1} = {H_{1}}  H_{1} = {}
RP T_{2} = {H_{2}}  where H_{2} = {11, 11, 11}
RP T_{3} = {H_{3}, P_{3}}  H_{3} = {21, 111, 111}
P_{3} = {21, 21, 21, 21}.

Also in these cases the plausibility condition (3.4) and the realization condition (3.9) are satisfied so that rRP T_{n} = pRP T_{n} = RP T_{n}. Note that H_{1}, with z = 0 is anomolous; all other members of RP T have z ≥ 3. Note that it would also be natural to give a given element of rRP T several names, e.g. H_{1} = P_{1} and H_{2} = P_{2}.

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 pRP T and then selects rRP T from these. The other, more in line with the proof of Theorem 8.1, starts from H_{1} 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. Table 1.1 summarizes the results. Run times are modest too, and so one could easily extend this table beyond n = 11.

There are other apparent general patterns in the computer data beyond the three that we pursue in Sections 7, 8, and 9. For example, one can expect that the blank space in slot (n, ℓ) = (11, 15) forms the tip of an infinite blank region under the submaximal diagonal of 12’s. Similarly, when one sorts realizable rigid Jordan tuples by (n, ℓ, m), m being the maximal part of the joint partition, other presumably infinite blank regions appear.

5. n ≤ 6: closer look

Of course the programs giving Table 1.1 give rRP T_{n,ℓ}, not just |rRP T_{n,ℓ}|. Tables 5.1 and 5.2 give rRP T_{n,ℓ} out through n = 6.

Table 5.1. The elements of rRP T_{n,ℓ}, n ≤ 5, and their maximal derivatives.

ℓ λ λ_{1} λ_{2}  λ_{3}  ⋯  ∂_{max}λ
H_{1}
6 H_{2} 11 11 11 H_{1}
8 H_{3} 21 111 111  H_{2}
8 P_{3} 21 21 21 21 H_{1}
10 H_{4} 31 1111 1111  H_{3}
10 P_{4} 31 31 31 31 31 H_{1}
9 A_{4} 22 211 1111  H_{3}
9 B_{4} 211 211 211  H_{2}
9 I_{4} 31 31  22  211  H_{2}
8 δ_{4,2} 31 22  22  22  P_{3}
12 H_{5} 41 11111 11111  H_{4}
12 P_{5} 41 41 41 41 41 41 H_{1}
10 A_{5} 32 221 11111  A_{4}
10 B_{5} 311 221 2111  H_{3}
10 C_{5} 32 2111 2111  H_{3}
10 I_{5} 41 32  311 311  H_{2}
10 J_{5} 41 41  221 221  P_{3}
10 M_{5} 41 41  41  32 32  H_{2}
9 α_{5} 221 221 221  B_{4}
9 q_{5} 41 32  32  221  P_{3}
8 δ_{5} 32 32  32  32  P_{3}

In [7, page 165], Katz gave examples due to Deligne of plausible but nonrealizable 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 A_{4} and B_{4}.

One can also ask for the first examples of plausible but non-realizable rigid partition tuples. In fact for n ≤ 5 there are none. For n = 6 there is one: {51, 33, 33, 3111} is plausible but its maximal derivative {31, 31, 31, 1111} is not.

Finally one can ask for the first examples of plausible but non-realizable rigid partition tuples, in the context z = 3. There are none for n ≤ 6. For n = 7 there is one: {331, 331, 31111} is plausible but its maximal derivative {311, 311, 11111} is not. Deligne’s plausible but non-realizable rigid Jordan tuples belong to this family.

Remarks. Tables 5.1 and 5.2 are useful in that they aid in working explicitly with examples. For example, consider the 120 element group Ã_{5}. It has a presentation

Ã_{5} = ⟨g_{1}, g_{2}, g_{3}, z|g_{1}g_{2}g_{3} = 1; g_{1}^{2} = g_{2}^{3} = g_{3}^{5} = z; z^{2} = 1, z central⟩.

The group Ã_{5} has nine irreducible complex representations ρ. Put j_{i} = [ρ(g_{i})]. All the resulting Jordan triples {j_{1}, j_{2}, j_{3}} are rigid, as one can tell from direct inspection or from (9.2) with ℓ_{1}, ℓ_{2}, ℓ_{3} at most 2, 3, 5 respectively. One can use the ATLAS to figure out the j_{i} explicitly, and hence the λ_{i}. The rigid partition triples arising are, in ATLAS order, H_{1}, H_{3}, H_{3}, A_{4}, A_{5}; H_{2}, H_{2}, A_{4}, γ_{6}.

Table 5.2. The elements of rRP T_{6,ℓ} and their maximal derivatives.

ℓ λ  λ_{1} λ_{2}  λ_{3}  ⋯  ∂_{max}λ
14 H_{6} 51 111111 111111  H_{5}
14 P_{6} 51 51  51  51  51 51 51 51 H_{1}
11 A_{6} 33 321  111111  A_{5}
11 B_{6} 321 3111 3111  H_{3}
11 C_{6} 33 3111 21111  H_{4}
11 D_{6} 42 222 111111 A_{5}
11 E_{6} 42 2211 21111  A_{4}
11 F_{6} 411 222 21111 A_{4}
11 G_{6} 411 2211 2211  B_{4}
11 I_{6} 51 33  411 3111  H_{3}
11 J_{6} 51 51  222 2211  I_{4}
11 K_{6} 51 51  51  33  321  P_{3}
11 M_{6} 51 51  42  42  411  H_{2}
11 N_{6} 42 411  411 411  H_{2}
10 γ_{6,6} 33 222  21111  A_{5}
10 β_{4} 33 2211 2211  C_{5}
10 q_{6} 321 321  2211  B_{4}
10 r_{6} 321 222  3111  A_{4}
10 s_{6} 51 42  33  2211  I_{4}
10 t_{6} 51 42  321 321  P_{3}
10 u_{6} 51 33  411 222  I_{4}
10 v_{6} 42 33  411 411  H_{3}
10 w_{6} 51 51  42  42  33  P_{3}
9 α_{6,3} 321 222  222  α_{5}
9 x_{6} 51 33  33  222  q_{5}
9 y_{6} 42 42  42  321  P_{3}
9 z_{6} 42 33  33  411  I_{4}
8 δ_{6,2} 42 33  33  33  δ_{5}

In particular, Tables 5.1 and 5.2 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] for examples in the context z = 3.

6. n ≤ 6, z = 3: internal structure

In this section, we completely describe the internal structure of all realizable families with n ≤ 6 and z = 3.

Proposition 6.1. Let λ be a realizable rigid partition triple of rank n = 4, 5, or 6. Let j = {j_{1}, j_{2}, j_{3}} ∈ J_{λ}. Then whether or not j is realizable is as described Table 6.1 Here subscripts i, j, k, l on the same symbol are required to be different, but are otherwise arbitrary.

Proof. The proof consists in applying the Katz algorithm (3.9) in each case. Here we describe the computation in the last case λ = α_{6}. Starting from α_{6}, we take Table 6.1. Realizablility of elements of RJT_{n,ℓ} for n = 4, 5, 6

j is realizable iff it is plausible and the

λ j_{1}  j_{2}  j_{3}  quantities below are different from 1.
H_{4} A^{3}a  b_{1}b_{2}b_{3}b_{4}  c_{1}c_{2}c_{3}c_{4}
A_{4} a_{1}^{2}a_{2}^{2} B^{2}b_{1}b_{2}  c_{1}c_{2}c_{3}c_{4}  a_{1}a_{2}Bb_{i}c_{j}c_{k}
B_{4} A^{2}a_{1}a_{2} B^{2}b_{1}b_{2}  C^{2}c_{1}c_{2}  Aa_{i}Bb_{j}Cc_{k}
H_{5} A^{4}a  b_{1}b_{2}b_{3}b_{4}b_{5} c_{1}c_{2}c_{3}c_{4}c_{5}
A_{5} A^{3}a^{2} B_{1}^{2}B_{2}^{2}b  c_{1}c_{2}c_{3}c_{4}c_{5} AaB_{1}B_{2}c_{i}c_{j}
B_{5} A_{1}^{2}A_{2}^{2}a Bb_{1}b_{2}  C^{2}c_{1}c_{2}c_{3}  A_{1}A_{2}Bb_{i}Cc_{j}
C_{5} A^{3}a^{2} B^{2}b_{1}b_{2}b_{3}  C^{2}c_{1}c_{2}c_{3}  AaBb_{i}Cc_{j}
α_{5} A_{1}^{2}A_{2}^{2}a B_{1}^{2}B_{2}^{2}b  C_{1}^{2}C_{2}^{2}c  A_{1}A_{2}B_{1}B_{2}C_{i}c  A_{1}A_{2}B_{i}bC_{1}C_{2}

A_{i}aB_{1}B_{2}C_{1}C_{2}

H_{6} A^{5}a  b_{1}b_{2}b_{3}b_{4}b_{5}b_{6} c_{1}c_{2}c_{3}c_{4}c_{5}c_{6}
A_{6} a_{1}^{3}b_{1}^{3}  B̂^{3}B^{2}b  c_{1}c_{2}c_{3}c_{4}c_{5}c_{6} a_{1}a_{2}B̂Bc_{i}c_{j}
B_{6} Â^{3}A^{2}a B^{3}b_{1}b_{2}b_{3}  C^{3}c_{1}c_{2}c_{3}  ÂABb_{i}Cc_{j}
C_{6} a_{1}^{3}a_{2}^{3} Bb_{1}b_{2}b_{3}  C^{2}c_{1}c_{2}c_{3}c_{4} a_{1}a_{2}Bb_{i}Cc_{j}
D_{6} A^{4}a^{2} b_{1}^{2}b_{2}^{2}b_{3}^{2}  c_{1}c_{2}c_{3}c_{4}c_{5}c_{6} A^{2}ab_{1}b_{2}b_{3}c_{i}c_{j}c_{k}
E_{6} A^{4}a^{2} B_{1}^{2}B_{2}^{2}b_{1}b_{2} C^{2}c_{1}c_{2}c_{3}c_{4} AaB_{1}B_{2}Cc_{i}  A^{2}aB_{1}B_{2}b_{i}c_{j}c_{k}
F_{6} a_{1}^{2}a_{2}^{2}a_{3}^{2} B^{4}b_{1}b_{2}  C^{2}c_{1}c_{2}c_{3}c_{4} a_{1}a_{2}a_{3}B^{2}b_{i}Cc_{j}c_{k}
G_{6} A^{4}a_{1}a_{2} B_{1}^{2}B_{2}^{2}b_{1}b_{2} C_{1}^{2}C_{2}^{2}c_{1}c_{2} Aa_{i}B_{1}B_{2}C_{1}C_{2}  A^{2}a_{i}B_{1}B_{2}b_{j}C_{1}C_{2}c_{k}
γ_{6} a_{1}^{3}a_{2}^{3} b_{1}^{2}b_{2}^{2}b_{3}^{2} C^{2}c_{1}c_{2}c_{3}c_{4} a_{1}a_{2}b_{i}b_{j}Cc_{k} a_{1}^{2}a_{2}b_{1}b_{2}b_{3}c_{i}c_{j}
β_{6} a_{1}^{3}a_{2}^{3} B_{1}^{2}B_{2}^{2}b_{1}b_{2} C_{1}^{2}C_{2}^{2}c_{1}c_{2} a_{1}a_{2}B_{1}B_{2}C_{i}c_{j}  a_{1}a_{2}B_{i}b_{j}C_{1}C_{2}

a_{1}^{2}a_{2}B_{1}B_{2}b_{i}C_{1}C_{2}c_{j}

q_{6} a_{1}^{2}a_{2}^{2}a_{3}^{2} B̂^{3}B^{2}b  C^{3}c_{1}c_{2}c_{3}  a_{i}a_{j}B̂BCc_{k}  a_{1}a_{2}a_{3}B̂BbC^{2}c_{i}
r_{6} Â^{3}A^{2}a B̂^{3}B^{2}b  C^{3}c_{1}c_{2}c_{3}  ÂAB̂BCc_{i}  ÂAB̂bCc_{i}

ÂaB̂BCc_{i}

α_{6} a_{1}^{2}a_{2}^{2}a_{3}^{2} b_{1}^{2}b_{2}^{2}b_{3}^{2} Ĉ^{3}c^{2}c a_{i}a_{j}b_{k}b_{l}ĈC a_{1}a_{2}a_{3}b_{1}b_{2}b_{3}Ĉ^{2}C

a_{1}a_{2}a_{3}b_{1}b_{2}b_{3}ĈC^{2}

three derivatives (3.8) successively, as indicated by the →^{d}; here d indicates the drop in rank. In between, as indicated by =, we simplify by scalar multiplication (2.3). 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 reserved to be part of the Jordan formalism, which explains why factors are often simply repeated.

\{ \begin{matrix}\{•a_{1}^{2}, & a_{2}^{2}, & a_{3}^{2}\} \\ \{•b_{1}^{2}, & b_{2}^{2}, & b_{3}^{2}\} \\ \{•Ĉ^{3}, & C^{2}, & c\} \end{matrix} \} = j

→^{1}  a_{1} ; b_{1} ; Ĉ \{ \begin{matrix}\{∘(1/b_{1}Ĉ), & a_{2}^{2}, & a_{3}^{2}\} \\ \{∘(1/a_{1}Ĉ), & b_{2}^{2}, & b_{3}^{2}\} \\ \{∘(1/a_{1}b_{1})^{2}, & C^{2}, & c\} \end{matrix} \}

= \{ \begin{matrix}\{(a_{1}/b_{1}Ĉ), & •(a_{1}a_{2})^{2}, & (a_{1}a_{3})^{2}\} \\ \{(b_{1}/a_{1}Ĉ), & •(b_{1}b_{2})^{2}, & (b_{1}b_{3})^{2}\} \\ \{(Ĉ/a_{1}b_{1})^{2}, & •(ĈC)^{2}, & (Ĉc)\} \end{matrix} \} = j′

(6.1) →^{1}  \begin{matrix}(a_{1}a_{2}) \\ (b_{1}b_{2}) \\ (ĈC) \end{matrix} \{ \begin{matrix}\{a_{1}/b_{1}Ĉ, & ∘1/b_{1}b_{2}ĈC, & (a_{1}a_{3})^{2}\} \\ \{b_{1}/a_{1}Ĉ, & ∘1/a_{1}a_{2}ĈC, & (b_{1}b_{3})^{2}\} \\ \{(Ĉ/a_{1}b_{1})^{2}, & ∘1/a_{1}a_{2}b_{1}b_{2}, & Ĉc\} \end{matrix} \}

= \{ \begin{matrix}\{a_{1}a_{1}a_{2}/b_{1}Ĉ, & a_{1}a_{2}/b_{1}b_{2}ĈC, & •(a_{1}a_{1}a_{2}a_{3})^{2}\} \\ \{b_{1}b_{1}b_{2}/a_{1}Ĉ, & b_{1}b_{2}/a_{1}a_{2}ĈC, & •(b_{1}b_{1}b_{2}b_{3})^{2}\} \\ \{•(ĈĈC/a_{1}b_{1})^{2}, & ĈC/a_{1}a_{2}b_{1}b_{2}, & ĈĈCc\} \end{matrix} \} = j′′

→^{2}  \begin{matrix}a_{1}a_{1}a_{2}a_{3} \\ b_{1}b_{1}b_{2}b_{3} \\ ĈĈC/a_{1}b_{1} \end{matrix} \{ \begin{matrix}\{a_{1}a_{1}a_{2}/b_{1}Ĉ, & a_{1}a_{2}/b_{1}b_{2}ĈC & ∘\} \\ \{b_{1}b_{1}b_{2}/a_{1}Ĉ, & b_{1}b_{2}/a_{1}a_{2}ĈC & ∘\} \\ \{∘ & ĈC/a_{1}a_{2}b_{1}b_{2}, & ĈĈCc\} \end{matrix} \}

= \{ \begin{matrix}\{a_{1}a_{1}a_{1}a_{1}a_{2}a_{2}a_{3}/b_{1}Ĉ, & a_{1}a_{1}a_{1}a_{2}a_{2}a_{3}/b_{1}b_{2}ĈC\} \\ \{b_{1}b_{1}b_{1}b_{1}b_{2}b_{2}b_{3}/a_{1}Ĉ, & b_{1}b_{1}b_{1}b_{2}b_{2}b_{3}/a_{1}a_{2}ĈC\} \\ \{ĈĈĈCC/a_{1}a_{1}a_{2}b_{1}b_{1}b_{2}, & ĈĈĈĈCCc/a_{1}b_{1}\} \end{matrix} \} = j′′′.

Define elements of E^{×} as follows, with i, j = 1, 2, 3:

W_{ij} = a_{i}b_{j}Ĉ
X_{ij} = (a_{1}a_{2}a_{3}/a_{i})(b_{1}b_{2}b_{3}/b_{j})ĈC
Y = a_{1}a_{2}a_{3}b_{1}b_{2}b_{3}Ĉ^{2}C
Z = a_{1}a_{2}a_{3}b_{1}b_{2}b_{3}ĈC^{2}

Then j ∈ α_{6} is plausible iff all nine W_{ij} are different from 1. j′ ∈ α_{5} is plausible iff W_{22}, W_{23}, W_{32}, W_{33}, X_{22}, X_{23}, X_{32}, and X_{33} are different from 1. j′′ ∈ B_{4} is plausible iff W_{23}, W_{32}, X_{13}, X_{31}, X_{22}, X_{11}, and Y are different from 1. Finally j′′′ ∈ H_{2} is plausible, or equivalently realizable, iff W_{33}, X_{12}, X_{21}, and Z are different from 1. Here we are using several times that a_{1}^{2}a_{2}^{2}a_{3}^{2}b_{1}^{2}b_{2}^{2}b_{3}^{2}Ĉ^{3}C^{2}c = 1, as well as the plausibility condition (3.5) repeatedly. Altogether, one gets that j is realizable iff all nine W_{ij}, all nine X_{ij}, Y and Z are different from 1, as stated on the table.

One can reinterpret Proposition 6.1 by thinking of A, a, b_{1}, b_{2}, … not as elements of E^{×}, but rather as coordinate functions on a torus T covering J_{α_{6}}. Then Proposition 6.1 gives defining 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 S_{3} × S_{3} × S_{1} on T. Rather, computations such as (6.1) break such symmetries, since they involve arbitrary choices of markings. For example, in the case of α_{6}, the nine divisors (X_{ij}) are permuted transitively by S_{3} × S_{3} × S_{1}. But in (6.1), four of these divisors appear at the j′ level, three more at the j′′ level, and the final two at the j′′′ level. It would be desirable to be able to formulate Katz’s classification 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 j_{i} ∈ J_{n} is obtained by replacing each eigenvalue by its inverse. Call a realizable rigid Jordan tuple j = {j_{1}, …, j_{z}} strictly self-dual if each j_{i} is self-dual. This notion plays a fundamental role in computing monodromy groups, for example. Table 6.1 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.

7. Maximal series: H and P

The Hypergeometric series. For positive integers n, define a partition triple

H_{n} = \{ \begin{matrix}n − 1, 1 \\ 1, …, 1 \\ 1, …, 1 \end{matrix} \} .

Each H_{n} is rigid and plausible. H_{1} is realizable. The unique derivative of H_{n} is H_{n−1}. So each H_{n} is realizable by induction.

The general member of the family J_{H_{n}} has the form

j = \{ \begin{matrix}A^{n−1}, a \\ b_{1}, ⋯, b_{n} \\ c_{1}, ⋯, c_{n} \end{matrix} \} . (7.1)

Here we regard all variables but a as freely chosen from E^{×}. The variable a is then determined by the determinant condition (1.1). For n > 1, the Jordan triple j is plausible iff Ab_{i}c_{j} ≠ 1 for all i, j ∈ {1, …, n}. With respect to the marking (A, b_{n}, c_{n}), the derivative of j is

\begin{matrix}A \\ b_{n} \\ c_{n} \end{matrix} \{ \begin{matrix}(1/b_{n}c_{n})^{n−2}, & a \\ (1/Ac_{n})^{0}, & b_{1}, …, b_{n−1} \\ (1/Ab_{n})^{0}, & c_{1}, …, c_{n−1} \end{matrix} \} ∼ \begin{matrix}Ab_{n}c_{n} \\ 1 \\ 1 \end{matrix} \{ \begin{matrix}(1/b_{n}c_{n})^{n−2}, a \\ b_{1}, …, b_{n−1} \\ c_{1}, …, c_{n−1} \end{matrix} \}

= \{ \begin{matrix}A^{n−2}a′ \\ b_{1} … b_{n−1} \\ c_{1} … c_{n−1} \end{matrix} \}

with a′ = Aab_{n}c_{n}. Here ∼ indicates twisting by the rank one triple (b_{n}c_{n}, b_{n}^{−1}, c_{n}^{−1}), 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, define a partition (n + 1)-tuple

P_{n} = {(n − 1)1, …, (n − 1)1}.

Each P_{n} is rigid and plausible, with P_{2} = H_{2} and P_{1} = H_{1}. The derivatives of P_{n} are

(7.2)  ∂_{(n−1,...,n−1)}P_{n} = H_{1}
(7.3)  ∂_{(n−1,...,n−1,1)}P_{n} = P_{n−1}.

So realizability of P_{n} follows either directly from (7.2) or by induction from (7.3). The general member of the family J_{P_{n}} has the form

j = { A_{1}^{n−1}a_{1} ; ⋮ ; A_{n+1}^{n−1}a_{n+1} } (7.4)

subject to Condition (1.1). The Jordan tuple j is plausible iff

(7.5)  π := A_{1} ⋯ A_{n+1} ≠ 1
(7.6)  πa_{i}/A_{i} ≠ 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 H_{n} or P_{n}. Particularly motivic references are [6] in the hypergeometric case and [2] in the Pochhammer case. A recent paper presenting H_{n} and P_{n} in tandem in the context of covers of the projective line is [11].

In general, given a realizable rigid Jordan tuple j = {j_{1}, …, j_{z}} one can ask for a matrix tuple g = (g_{1}, …, g_{z}) ∈ V (j_{1}, …, j_{z}) representing it. In the hypergeometric and Pochhammer cases this problem has a uniform solution (see e.g. [11]). We restate the solution for H_{n} here in our language. By twisting, one can take A = 1 in (7.1). Define

f(x) = det(x − j_{2}) = (x − b_{1}) ⋯ (x − b_{n}) = x^{n} + f_{1}x^{n−1} + ⋯ + f_{n−1}x + f_{n}
h(x) = det(x − j_{3}) = (x − c_{1}) ⋯ (x − c_{n}) = x^{n} + h_{1}x^{n−1} + ⋯ + h_{n−1}x + h_{n}

Define e_{0} = 1/(f_{n}h_{n}) and e_{i} = h_{i}/(f_{n}h_{n}) − f_{n−i}/f_{n} for i = 1, …, n − 1. Define matrices by following the pattern evident in the case n = 4:

g_{1} = ( \begin{matrix}e_{0} & 0 & 0 & 0 \\ e_{1} & 1 & 0 & 0 \\ e_{2} & 0 & 1 & 0 \\ e_{3} & 0 & 0 & 1 \end{matrix} ) g_{2} = ( \begin{matrix}0 & 0 & 0 & −f_{4} \\ 1 & 0 & 0 & −f_{3} \\ 0 & 1 & 0 & −f_{2} \\ 0 & 0 & 1 & −f_{1} \end{matrix} ) g_{3} = ( \begin{matrix}−h_{1} & 1 & 0 & 0 \\ −h_{2} & 0 & 1 & 0 \\ −h_{3} & 0 & 0 & 1 \\ −h_{4} & 0 & 0 & 0 \end{matrix} ) . (7.7)

Then g_{1}g_{2}g_{3} = 1. The group ⟨g_{1}, g_{2}, g_{3}⟩ acts irreducibly on E^{n} iff no root of f is the inverse of a root of h.

8. 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 {λ_{1}, …, λ_{z}} ∈ rRP T submaximal if its length ℓ and its rank n satisfy the inequality n + 5 ≤ ℓ < 2n + 2. In this section, we classify submaximal partition tuples, restricting to n ≥ 6 to avoid degeneracies.

On Table 8.1 we give twelve partition tuples in each rRP T_{n,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 eleven full series and two half series G_{even} and L_{odd}.

For typographical reasons, we use the following abbreviations on Table 8.1:

\begin{aligned}u & = & n/2 + 3/2 & v & = & n/2 + 1 \\ w & = & n/2 + 1/2 & x & = & n/2 \\ y & = & n/2 − 1/2 & z & = & n/2 − 1. \end{aligned}

For n even only the integers v, x, and z appear; for n odd only the integers u, w, and y appear. Thus, to get a first understanding of Table 8.1, one can think in terms of u, v, w, x, y, z ≈ n/2.

Note that series A, B, and C share a joint partition (3.2) and so do series D, E, F, and G:

(8.1) μ = \{ \begin{matrix}xxxz1 ⋯ 1 & Series ABC, n even \\ wyyy1 ⋯ 1 & Series ABC, n odd \\ (n − 2)2 ⋯ 2111111 & Series DEF G, n even \\ (n − 2)2 ⋯ 211111 & Series DEF, n odd. \end{matrix}

Table 8.1. The twelve tuples in rRP T_{n,n+5} and their derivatives, n > 6

z = 3

λ_{1}  λ_{2}  λ_{3}  1  2 > 2
A_{n} xx  xz1  1 ⋯ 1  A
wy  yy1  1 ⋯ 1  A
B_{n} xz1  x1 ⋯ 1  x1 ⋯ 1  B, C  H_{x}, H_{v}
yy1  w1 ⋯ 1 y1 ⋯ 1  B, C  H_{x}
C_{n} xx  x1 ⋯ 1  z1 ⋯ 1  C  H_{v}
wy  y1 ⋯ 1  y1 ⋯ 1  C  H_{w}, H_{y}
D_{n} (n − 2)2 2 ⋯ 2  2 ⋯ 2111111  E  D
(n − 2)2 2 ⋯ 21  2 ⋯ 211111  D, E D
E_{n} (n − 2)2 2 ⋯ 211 2 ⋯ 21111  D, E E
(n − 2)2 2 ⋯ 2111 2 ⋯ 2111  E  E
F_{n} (n − 2)11 2 ⋯ 2  2 ⋯ 21111  F  F
(n − 2)11 2 ⋯ 21  2 ⋯ 2111  F, G F
G_{n} (n − 2)11 2 ⋯ 211 2 ⋯ 211  F  G

z = 4

λ_{1}  λ_{2}  λ_{3}  λ_{4}  1  2 > 2
I_{n} (n − 1)1 xx  x1 ⋯ 1  v1 ⋯ 1  I  H_{v}
(n − 1)1 wy  w1 ⋯ 1 w1 ⋯ 1 I  H_{w}
J_{n} (n − 1)1 (n − 1)1 2 ⋯ 2  2 ⋯ 211 J  J
(n − 1)1 (n − 1)1 2 ⋯ 21  2 ⋯ 21  J  J

z ≥ 5  z = ⌈n/2⌉ + 1 (L, N)  z = ⌈n/2⌉ + 2 (K, M)
λ_{1}, λ_{4}, … λ_{2}  λ_{3}  1  2 > 2
K_{n} (n − 1)1 xz1  xx  K  P_{v}, P_{x}
(n − 1)1 wy  wy  P_{u}, P_{w}, P_{y}

L_{n}

(n − 1)1 yy1  yy1  K  P_{w}
λ_{1}, λ_{5}, … λ_{2}  λ_{3}  λ_{4}  1  2 > 2
M_{n} (n − 2)2 (n − 2)11 (n − 1)1 (n − 1)1 M  M H_{2}
(n − 2)2 (n − 1)1 (n − 1)1 (n − 1)1 M  M H_{2}
N_{n} (n − 2)2 (n − 2)11 (n − 2)11 (n − 2)11 N  N H_{2}
(n − 2)2 (n − 1)1 (n − 2)11 (n − 2)11 N  N H_{2}

The rigidity condition (1.2) is quickly verified in every case. For example, in Series ABC, n even, one has

3x^{2} + z^{2} + (n + 1)1^{2} = 3(\frac{n}{2})^{2} + (\frac{n}{2} − 1)^{2} + (n + 1)
= \frac{3}{4}n^{2} + (\frac{1}{4}n^{2} − n + 1) + (n + 1)
= n^{2} + 2.

Similarly plausibility condition (3.4) is quickly verified in all cases.

Table 8.1 gives all possible derivatives Y_{m} ∈ RP T_{m} of each X_{n} ∈ rRP T_{n}. Here those with m = n − 1 are placed in Column 1, those with m = n − 2 are placed in Column 2, and any that remain are placed in Column >2. For example, consider B_{n} for n even; the four blocks of (8.2) account for the four derivatives of B_{n} printed on Table 8.1.

\begin{aligned}B_{n} = & \underline{x}z1 & \underline{x}11 ⋯ 1 & x\underline{1}1 ⋯ 1 \\ & zz1 & z11 ⋯ 1 & x 1 ⋯ 1 & = B_{n−1} \\ B_{n} = & xz\underline{1} & \underline{x}11 ⋯ 1 & \underline{x}11 ⋯ 1 \\ & xz & z11 ⋯ 1 & z11 ⋯ 1 & = C_{n−1} \\ B_{n} = & \underline{x}z1 & \underline{x}11 ⋯ 1 & \underline{x}11 ⋯ 1 \\ & z1 & 11 ⋯ 1 & 11 ⋯ 1 & = H_{x} \\ B_{n} = & x\underline{z}1 & \underline{x}11 ⋯ 1 & \underline{x}11 ⋯ 1 \\ & x 1 & 111 ⋯ 1 & 111 ⋯ 1 & = H_{x+1}. \end{aligned} (8.2)

The fact that our notation places the tuples in eleven full series and two half series has some artificial aspects; for example, F_{odd} is just as tightly tied to G_{even} as it is to F_{even}. On the other hand, the unions A, BC, DE, F G, I, J, KL, M, and N are stable under drop-1 and drop-2 derivation.

Theorem 8.1. For n ≥ 6, the set rRP T_{n,n+5} contains exactly the twelve tuples listed in Table 8.1. Also rRP T_{n,ℓ} is empty for n + 5 < ℓ < 2n + 2.

Proof. Let

rRP T_{n,∗} = ∐_{ℓ=n+5}^{2n+2} rRP T_{n,ℓ}.

One can check, without using any classification results, that rRJT_{n,∗} is closed under derivation. One then has to check that if one starts with H_{1} and successively antidifferentiates, keeping only realizable families with ℓ ≥ n + 5, one gets only the families listed in Table 8.1. Here is a sample of the computations needed.

Consider antiderivatives λ of H_{2} 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 ∈ Z_{≥0}. The rigidity condition (1.2) for λ forces

a(m^{2} + 1^{2} + 1^{2}) + (3 − a)((m + 1)^{2} + 1^{2}) + x(m^{2} + 2^{2}) = (x + 1)(m + 2)^{2} + 2

which simplifies to 2m(2x − m + a − 1) = 0. For a = 0, 1, 2, 3, this gives M_{odd}, M_{even}, N_{odd}, and N_{even}, respectively, and nothing more.

Remarks. Okubo, Yokoyama, Haraoka and others have studied the differential equations indexed by λ = {λ_{1}, …, λ_{z}} ∈ rRP T such that all the λ_{i} but perhaps one have the form (m_{i}, 1, …, 1). They prove that the possibilities are as follows:

z = 3  z > 3
Our Their Our Their
n  notation Notation notation Notation
Arbitrary  H_{n}  I_{n}  P_{n}  I_{n}^{∗}
Even (≥ 4)  B_{n}  II_{n}  I_{n}  II_{n}^{∗}
Odd (≥ 5)  B_{n}  III_{n}  I_{n}  III_{n}^{∗}
6  F_{6}  IV_{6}  N_{6}  IV_{6}^{∗}

See [3] for emphasis on differential equations; see [4] and [5] for explicit matrix formulas analogous to (7.7).

Simpson has approached our topic from a Hodge-theoretic viewpoint [10]. In our language, one of his classification results is the following. The only partition tuples λ = {λ_{1}, …, λ_{z}} ∈ rRP T with some λ_{i} = 11 ⋯ 11 are the tuples H_{n}, A_{n}, and D_{6}.

9. Minimal series: α, β, γ, and δ

Call a partition tuple {λ_{1}, …, λ_{z}} ∈ RP T minimal if the corresponding length tuple {ℓ_{1}, …, ℓ_{z}} satisfies

χ(ℓ_{1}, …, ℓ_{z}) := \frac{1}{ℓ_{1}} + ⋯ + \frac{1}{ℓ_{z}} − (z − 2) ≥ 0. (9.1)

In this section, we classify minimal rigid partition tuples.

For n ≥ k positive integers, define μ_{n/k} to be the unique partition of n with k parts and all parts ⌈n/k⌉ or ⌊n/k⌋. Thus μ_{n/k} is the “most balanced” partition of n into k parts. As three examples of this notation, γ_{13} below is {μ_{13/2}, μ_{13/3}, μ_{13/6}} = {76, 544, 322222}. When k divides n, define also

\overline{μ}_{n/k} := \{n/k + 1, n/k, ⋯ n/k, n/k − 1\}.

In this case, μ_{n/k} is perfectly balanced and \overline{μ}_{n/k} is the next most balanced partition. As more examples of this notation, γ_{12,2} below is \{\overline{μ}_{12,2}, μ_{12/3}, μ_{12/6}\} = {75, 444, 222222}.

Theorem 9.1. Put δ_{k}(n) = 1 if k divides n and δ_{k}(n) = 0 else. For n > 6, the set RP T_{n,min} contains exactly the 4 + δ_{4}(n) + 2δ_{6}(n) elements listed in Table 9.1. All these rigid partition tuples are realizable.

Proof. The last statement follows by induction from the table of derivatives given. As to the completeness of the table, in general let λ = {λ_{1}, …, λ_{z}} ∈ RP T_{n} with corresponding length tuple {ℓ_{1}, …, ℓ_{z}}. For each i, one has n^{2}/ℓ_{i} ≤ ||λ_{i}|| with equality iff ℓ_{i} divides n and λ_{i} = μ_{n/ℓ_{i}}. So one has

(z − 2)n^{2} + χ(ℓ_{1}, …, ℓ_{z})n^{2} ≤ ∑_{i} ||λ_{i}|| = (z − 2)n^{2} + 2; (9.2)

here the inequality is the definition of χ while the equality is Condition (1.2).

Suppose λ is minimal, meaning that χ(ℓ_{1}, …, ℓ_{z}) ≥ 0. Then (9.2) is extremely restrictive and one can conclude by elementary arguments: the cases with χ > 0 yield nothing with n > 6; the remaining cases {ℓ_{1}, …, ℓ_{z}} = {3, 3, 3}, {2, 4, 4}, {2, 3, 6}, and {2, 2, 2, 2} yield the series α, β, γ, and δ respectively.

Table 9.1. The 4 + δ_{4}(n) + 2δ_{6}(n) tuples in RP T_{n,min} and their derivatives

Derivatives

d n ∈ Z/d  Partition tuples  1  2
3 0  α_{n} = \{\overline{μ}_{n/3}, μ_{n/3}, μ_{n/3}\}  α_{n−1}
1  α_{n} = {μ_{n/3}, μ_{n/3}, μ_{n/3}}  α_{n−1}  α_{n−2}
2  α_{n} = {μ_{n/3}, μ_{n/3}, μ_{n/3}}  α_{n−1}
4 0  β_{n,2} = \{\overline{μ}_{n/2}, μ_{n/4}, μ_{n/4}\}  β_{n−1}
0  β_{n,4} = \{μ_{n/2}, \overline{μ}_{n/4}, μ_{n/4}\}  β_{n−1}
1  β_{n} = {μ_{n/2}, μ_{n/4}, μ_{n/4}}  β_{n−1,2}, β_{n−1,4}  β_{n−2}
2, 3  β_{n} = {μ_{n/2}, μ_{n/4}, μ_{n/4}}  β_{n−1}
6 0  γ_{n,2} = \{\overline{μ}_{n/2}, μ_{n/3}, μ_{n/6}\}  γ_{n−1}
0  γ_{n,3} = \{μ_{n/2}, \overline{μ}_{n/3}, μ_{n/6}\}  γ_{n−1}
0  γ_{n,6} = \{μ_{n/2}, μ_{n/3}, \overline{μ}_{n/6}\}  γ_{n−1}
1  γ_{n} = {μ_{n/2}, μ_{n/3}, μ_{n/6}}  γ_{n−1,2}, γ_{n−1,3}, γ_{n−1,6} γ_{n−2}
2, 3, 4, 5 γ_{n} = {μ_{n/2}, μ_{n/3}, μ_{n/6}}  γ_{n−1}
2 0  δ_{n} = \{\overline{μ}_{n/2}, μ_{n/2}, μ_{n/2}, μ_{n/2}\} δ_{n−1}
1  δ_{n} = {μ_{n/2}, μ_{n/2}, μ_{n/2}, μ_{n/2}} δ_{n−1}  δ_{n−2}

The four series just discussed are naturally associated with the four rank two Dynkin diagrams: (α, β, γ, δ) ↔ (A_{2}, B_{2}, G_{2}, A_{1} × A_{1}); this connection partially explains our choice of notation.

References

[1] F. Beukers and G. Heckman. Monodromy for the hypergeometric function _{n}F_{n−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 differential equations free from accessory parameters, SIAM J. Math Anal 25 (1994) no. 4, 1203-1226.

[4] . Monodromy representations of systems of differential 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 Differential Equations, Annals of Mathematics Study 124 Princeton University Press (1990).

[7] . Rigid Local Systems, Annals of Mathematics Study 138, Princeton University Press (1996).

[8] A. Knutson and T. Tao, Honeycombs and sums of Hermitian matrices, Notices Amer. Math. Soc., 48 no. 2 (2001), 175-186

[9] D. P. Roberts, An ABC construction of number fields, submitted.

[10] C. T. Simpson, Products of matrices, Canadian Mathematical Society conference proceedings 12, 1991.

[11] H. Völklein. Rigid generators of classical groups. Math. Ann. 311 (1998), no. 1, 1-58.

Division of Science and Mathematics; University of Minnesota, Morris; Morris, Minnesota 56267

  E-mail address: roberts@morris.umn.edu
  URL: http://cda.morris.umn.edu/~roberts/