Cokernels of random matrix products and flag Cohen–Lenstra heuristic

Yifeng Huang

doi:10.1515/forum-2024-0129

Article Open Access

Cokernels of random matrix products and flag Cohen–Lenstra heuristic

Yifeng Huang

Published/Copyright: November 14, 2024

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From the journal Forum Mathematicum Volume 37 Issue 5

Abstract

In [H. H. Nguyen and R. Van Peski, Universality for cokernels of random matrix products, Adv. Math. 438 2024, Paper No. 109451], Nguyen and Van Peski raised the question of whether the surjective flag of ℤ p -modules modeled by cok ⁢ ( M 1 ⁢ … ⁢ M k ) ↠ ⋯ ↠ cok ⁢ ( M 1 ) for independent random matrices M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) satisfies the Cohen–Lenstra heuristic. We answer the question affirmatively when M 1 , … , M k follow the Haar measure, and our proof demonstrates how classical ideas in Cohen–Lenstra heuristic adapt naturally to the flag setting. We also prove an analogue for non-square matrices.

Keywords: Cohen–Lenstra heuristic; flags of abelian groups; random p-adic matrices

MSC 2020: 15B52; 20K01

1 Introduction

We start by introducing our notation for surjective flags of ℤ p -modules, which is our main subject of investigation.

1.1 Notation and terminology

Fix k ∈ ℤ ≥ 1 and a prime p. Let 𝐌𝐨𝐝 ℤ p denote the category of finitely generated ℤ p -modules. We refer to a diagram of surjections

G k ↠ ⋯ ↠ G 1 ( ↠ G 0 := 0 )

in 𝐌𝐨𝐝 ℤ p as a k-surjective flag of (finitely generated) ℤ p -modules. We denote by 𝐅𝐥 k = 𝐅𝐥 k ⁢ ( ℤ p ) the set of k-surjective flags of ℤ p -modules up to isomorphism. For 𝐆 = ( G k ↠ ⋯ ↠ G 1 ) ∈ 𝐅𝐥 k , let Aut ⁡ ( 𝐆 ) denote the automorphism of the flag 𝐆 . See also [9, Definitions 18 and 19].

Let n ∈ ℤ ≥ 1 and let Mat n ⁡ ( ℤ p ) denote the set of n × n matrices over ℤ p . For M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) , consider the flag

cok ⁢ ( M 1 ⁢ … ⁢ M k ) ↠ ⋯ ↠ cok ⁢ ( M 1 ) ,

where the surjection cok ⁢ ( M 1 ⁢ … ⁢ M i + 1 ) = ℤ p n / im ⁡ ( M 1 ⁢ … ⁢ M i + 1 ) ↠ cok ⁢ ( M 1 ⁢ … ⁢ M i ) = ℤ p n / im ⁡ ( M 1 ⁢ … ⁢ M i ) is induced by the inclusion im ⁡ ( M 1 ⁢ … ⁢ M i + 1 ) ⊆ im ⁡ ( M 1 ⁢ … ⁢ M i ) . We denote this flag by 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) .

As Mat n ⁡ ( ℤ p ) ≃ ℤ p n 2 is a compact topological group, there is a unique probability Haar measure on Mat n ⁡ ( ℤ p ) . We refer to a random element of Mat n ⁡ ( ℤ p ) following the Haar measure as a Haar-random matrix in Mat n ⁡ ( ℤ p ) .

1.2 Background

When M 1 , … , M k are random matrices in Mat n ⁡ ( ℤ p ) , we get a probability measure on 𝐅𝐥 k modeled by the flag 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) . In [9], Nguyen and Van Peski initiated the investigation of 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) by studying the joint distribution of cok ⁢ ( M 1 ) , … , cok ⁢ ( M 1 ⁢ … ⁢ M k ) as ℤ p -modules. They proved a universality result in the sense of Wood [12], namely, if the k ⁢ n 2 entries from M 1 , … , M k are independent and each is not too concentrated mod p, then as n → ∞ , the limiting joint distribution of cok ⁢ ( M 1 ) , … , cok ⁢ ( M 1 ⁢ … ⁢ M k ) is insensitive to the exact distributions of these k ⁢ n 2 entries. The limiting joint distribution is also explicitly determined.

As the datum ( cok ⁢ ( M 1 ) , … , cok ⁢ ( M 1 ⁢ … ⁢ M k ) ) is just 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) forgetting the surjections in between, it is natural to expect that the above results are explained by a universal distribution on the level of the flag 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) ∈ 𝐅𝐥 k . Nguyen and Van Peski [9, Section 10] asked if the above results lift to 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) ; they defined the conjectured limiting distribution in [9, Theorem 1.3].

1.3 Our result

We answer their question affirmatively when M 1 , … , M k are distributed independently and follow the Haar measure. In this case, we also obtain the exact distribution of 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) for each fixed n. For G ∈ 𝐌𝐨𝐝 ℤ p , let r ⁢ ( G ) := dim 𝔽 p ⁡ G / p ⁢ G be the rank of G.

Theorem 1.1.

Fix k ∈ Z ≥ 1 . Let M 1 , … , M k ∈ Mat n ⁡ ( Z p ) be independent and Haar-random, and fix

𝐆 = ( G k ↠ ⋯ ↠ G 1 ) ∈ 𝐅𝐥 k

such that | G k | < ∞ . Then, for n ≥ r ⁢ ( G k ) ,

(1.1) Prob M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) ⁡ ( 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) ≃ 𝐆 ) = 1 | Aut ⁡ ( 𝐆 ) | ⁢ ( ∏ i = n - r ⁢ ( G k ) + 1 n ( 1 - p - i ) ) ⁢ ( ∏ i = 1 n ( 1 - p - i ) ) k .

In particular, when n → ∞ ,

lim n → ∞ ⁡ Prob M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) ⁡ ( 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) ≃ 𝐆 ) = 1 | Aut ⁡ ( 𝐆 ) | ⁢ ( ∏ i = 1 ∞ ( 1 - p - i ) ) k .

Remark.

It is clear that if | G k | = ∞ or n < r ⁢ ( G k ) , then the probability in (1.1) is zero: G k = cok ⁢ ( M 1 ⁢ … ⁢ M k ) must have rank at most n, and | G k | = ∞ only happens when at least one of det ⁡ ( M 1 ) , … , det ⁡ ( M k ) is zero, which happens with probability zero.

The flag 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) is probably the finest datum one could get from a chain of matrices ( M 1 , … , M k ) .^[1] For example, cok ⁢ ( M i ⁢ M i + 1 ⁢ … ⁢ M j ) for 1 ≤ i ≤ j ≤ k is isomorphic to the kernel of G j ↠ G i - 1 , a concatenation of several surjections from 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) . The distribution of 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) thus encodes the joint distribution of all cok ⁢ ( M i ⁢ M i + 1 ⁢ … ⁢ M j ) . We note that the joint distribution of certain subsets of these cokernels have natural connections to Hall algebras (see [9, p. 66]).

An important purpose of the paper is to demonstrate that some classical ideas to study the Cohen–Lenstra heuristic [3, 6] adapt to the (apparently highly refined) flag setting nicely. This point will be evident once we set up the language in Section 2.1. The ease to work with flags will be showcased in our proof of Theorem 1.1; for example, no knowledge about | Aut ⁡ ( 𝐆 ) | is required.^[2] In light of its simplicity, we conjecture that a suitable combination of our method and the general machinery of Sawin and Wood [10] would yield a universality version of Theorem 1.1.

1.4 Further applications

A slight modification of the proof of Theorem 1.1 implies an analogue for non-square matrices.

Theorem 1.2.

Fix k ∈ Z ≥ 1 , and u 1 , … , u k ∈ Z ≥ 0 . For 1 ≤ i ≤ k , let M i ∈ Mat ( n + u i - 1 ) × ( n + u i ) ⁡ ( Z p ) , where u 0 := 0 , and assume M 1 , … , M k are independent and Haar-random. Fix G = ( G k ↠ ⋯ ↠ G 0 = 0 ) ∈ Fl k . Then

(1.2) Prob M 1 , … , M k ⁡ ( 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) ≃ 𝐆 ) = ∏ j = 1 k ( | G j | | G j - 1 | ) - u j | Aut ⁡ ( 𝐆 ) | ⁢ ( ∏ i = n - r ⁢ ( G k ) + 1 n ( 1 - p - i ) ) ⁢ ( ∏ j = 1 k ∏ i = 1 n ( 1 - p - i - u j ) )

if | G k | < ∞ and r ⁢ ( G k ) ≤ n , and zero otherwise. In particular, when n → ∞ ,

(1.3) lim n → ∞ ⁡ Prob M 1 , … , M k ⁡ ( 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) ≃ 𝐆 ) = ∏ j = 1 k ( | G j | | G j - 1 | ) - u j | Aut ⁡ ( 𝐆 ) | ⁢ ∏ j = 1 k ∏ i = 1 ∞ ( 1 - p - i - u j )

if | G k | < ∞ , and zero otherwise.

Remark.

It is less obvious a priori why | cok ⁢ ( M 1 ⁢ … ⁢ M k ) | < ∞ with probability one, since we cannot use determinants anymore. This will be a consequence of our proof.

The non-flag case (i.e., k = 1 ) was proved in [12], together with the universality result. We conjecture that (1.3) holds if M 1 , … , M k are independent and ε-balanced in the definition of [12].

As an application of the non-square analogue, we construct a k-parameter family of deformations of the “Cohen–Lenstra probability measure” in (1.1). For any flag 𝐆 = ( G k ↠ ⋯ ↠ G 0 = 0 ) ∈ 𝐅𝐥 k such that | G k | < ∞ , we let n 1 ⁢ ( 𝐆 ) , … , n k ⁢ ( 𝐆 ) ∈ ℤ ≥ 0 be defined by

p n i ⁢ ( 𝐆 ) := | G i | | G i - 1 | .

Definition 1.3.

For k , n ∈ ℤ ≥ 1 , t 1 , … , t k ∈ [ 0 , p ) , we define a measure P n , ( t 1 , … , t k ) on 𝐅𝐥 k by

(1.4) P n , ( t 1 , … , t k ) ⁢ ( 𝐆 ) := ∏ j = 1 k t j n j ⁢ ( 𝐆 ) | Aut ⁡ ( 𝐆 ) | ⁢ ( ∏ i = n - r ⁢ ( G k ) + 1 n ( 1 - p - i ) ) ⁢ ( ∏ j = 1 k ∏ i = 1 n ( 1 - p - i ⁢ t j ) )

if | G k | < ∞ and r ⁢ ( G k ) ≤ n , and zero otherwise. Similarly, define

P ∞ , ( t 1 , … , t k ) ⁢ ( 𝐆 ) := ∏ j = 1 k t j n j ⁢ ( 𝐆 ) | Aut ⁡ ( 𝐆 ) | ⁢ ( ∏ j = 1 k ∏ i = 1 ∞ ( 1 - p - i ⁢ t j ) )

if | G k | < ∞ , and zero otherwise.

When t i = 1 , the measure (1.4) reduces to (1.1); when t i = p - u i , u i ∈ ℤ ≥ 0 , the measure (1.4) reduces to (1.2). This means that in these cases, (1.4) comes from a random matrix model. A brief argument in Section 5 implies:

Corollary 1.4.

P n , ( t 1 , … , t k ) and P ∞ , ( t 1 , … , t k ) are probability measures on Fl k .

When k = 1 , Definition 1.3 is precisely the measure considered by Fulman and Kaplan [7]. The joint distribution of ( G 1 , … , G k ) for 𝐆 distributed according to (1.4) is expressible in terms of Hall–Littlewood polynomials and can be extracted from [11], see Proposition 5.1.

Part of the content of Corollary 1.4 is that the normalizing constant in Definition 1.3 is correct. For example, P ∞ , ( t 1 , … , t k ) being a probability measure is equivalent to the formal identity

(1.5) ∑ 𝐆 | G k | < ∞ ∏ j = 1 k t j n j ⁢ ( 𝐆 ) | Aut ⁡ ( 𝐆 ) | = ∏ j = 1 k ∏ i = 1 ∞ 1 1 - p - i ⁢ t j ∈ ℂ ⁢ [ [ t 1 , … , t k ] ] .

Its function field analogue naturally connects to matrices in a parabolic subalgebra; see Section 5.1.

Deformations of the Cohen–Lenstra measure are already interesting without motivations from random matrix models; see [4, 7] and [9, Section 7] for the intrinsic study of some probability measures of this type, and note the ubiquity of Hall–Littlewood polynomials. But furthermore, such deformations also arise as predicted distributions of arithmetic objects: for example [3], if p is odd and we take k = 1 , u = u 1 , G = 𝐆 = G 1 in (1.3), then the resulting probability mass function, proportional to 1 / ( | G | u ⁢ | Aut ⁡ ( G ) | ) , predicts the distribution of the p-part of the class group of a random quadratic extension of ℚ ( u = 0 for imaginary, u = 1 for real). We propose the following question to conclude the introduction.

Question 1.5.

Are there arithmetic settings that naturally produce a flag of finite abelian groups? Is the distribution of its p-part predicted by one of the distributions above?

2 Preliminaries

2.1 Dictionary of flags

Given n , k ∈ ℤ ≥ 1 and a surjective flag 𝐆 = ( G k ↠ ⋯ ↠ G 1 ) ∈ 𝐅𝐥 k , a surjection from ℤ p n to 𝐆 simply refers to a surjective ℤ p -linear map f k : ℤ p n ↠ G k . Any such surjection f k induces a chain of surjections

ℤ p n ⁢ ↠ f k ⁢ G k ↠ ⋯ ↠ G 1 ↠ G 0 = 0 ,

which induces a surjection f i : ℤ p n ↠ G i for 0 ≤ i ≤ k by composition. We can equivalently think of a surjection f : ℤ p n ↠ 𝐆 as a collection ( f i : ℤ p n ↠ G i ) i , but keeping in mind that f k determines the rest.

By a k-injective flag in ℤ p n , we mean a tower of ℤ p -submodules of ℤ p n :

F k ⊆ F k - 1 ⊆ ⋯ ⊆ F 1 ⊆ F 0 := ℤ p n .

We denote by ℱ ⁢ l k ⁢ ( ℤ p n ) the set of k-injective flags in ℤ p n .

There is a one-to-one correspondence between ℱ ⁢ l k ⁢ ( ℤ p n ) and the set of k-surjective flags 𝐆 together with a surjection ℤ p n ↠ 𝐆 . To ℱ = ( F k ⊆ ⋯ ⊆ F 0 = ℤ p n ) , we associate ℤ p n ↠ 𝐆 = ( G k ↠ ⋯ ↠ G 0 = 0 ) , where G i = ℤ p n / F i for 0 ≤ i ≤ k and the maps are the natural quotient maps. Conversely, given f : ℤ p n ↠ 𝐆 , we recover ℱ by F i = ker ⁡ ( f i : ℤ p n ↠ G i ) for 0 ≤ i ≤ k . We introduce the natural notation 𝐆 = ℤ p n / ℱ and ℱ = 𝑘𝑒𝑟 ( f : ℤ p n ↠ 𝐆 ) .

Given matrices M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) , we define an injective flag by F i = im ⁡ ( M 1 ⁢ … ⁢ M i ) ⊆ ℤ p n for 1 ≤ i ≤ k . We denote this flag by 𝑖𝑚 ⁢ ( M 1 , … , M k ) . We canonically have 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) ≃ ℤ p n / 𝑖𝑚 ⁢ ( M 1 , … , M k ) .

3 Proof of Theorem 1.1

Fix 𝐆 = ( G k ↠ ⋯ ↠ G 1 ) ∈ 𝐅𝐥 k with | G k | < ∞ . Let r := r ⁢ ( G k ) , and fix n ≥ r . We separate the major steps of proving Theorem 1.1 into the following lemmas.

Lemma 3.1.

The number of k-injective flags F in Z p n such that Z p n / F ≃ G is

| G k | n | Aut ⁡ ( 𝐆 ) | ⁢ ∏ i = n - r ⁢ ( G k ) + 1 n ( 1 - p - i ) .

Proof.

Fix a copy of 𝐆 , and let Surj ⁢ ( ℤ p n , 𝐆 ) denote the set of surjections from ℤ p n to 𝐆 , which is nothing but the set of surjections ℤ p n ↠ G k . By Nakayama’s lemma,

| Surj ⁢ ( ℤ p n , 𝐆 ) | = | G k | n ⁢ ∏ i = n - r + 1 n ( 1 - p - i ) .

Let Aut ⁡ ( 𝐆 ) act on Surj ⁢ ( ℤ p n , 𝐆 ) by composition. As usual, the action is free: if σ = ( σ i ) ∈ Aut ⁡ ( 𝐆 ) with σ i ∈ Aut ⁡ ( G i ) is such that σ i ∘ f i = f i for all i, then since f i is surjective, we must have σ i = id . As a consequence, the orbit space has cardinality given by

| Surj ⁢ ( ℤ p n , 𝐆 ) Aut ⁡ ( 𝐆 ) | = | Surj ⁢ ( ℤ p n , 𝐆 ) | | Aut ⁡ ( 𝐆 ) | = | G k | n | Aut ⁡ ( 𝐆 ) | ⁢ ∏ i = n - r + 1 n ( 1 - p - i ) .

Finally, it is easy to verify that the orbit space above is in a canonical bijection with the set of k-injective flags ℱ with ℤ p n / ℱ ≃ 𝐆 . The conclusion then follows. ∎

Lemma 3.2.

Fix a k-injective flag F such that Z p n / F ≃ G . If M 1 , … , M k ∈ Mat n ⁡ ( Z p ) are independent and Haar-random, then

Prob M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) ⁡ ( 𝑖𝑚 ⁢ ( M 1 , … , M k ) = ℱ ) = | G k | - n ⁢ ( ∏ i = 1 n ( 1 - p - i ) ) k .

Proof.

Let ℱ = ( F k ⊆ … ⊆ F 1 ⊆ ℤ p n ) . Since | G k | = | ℤ p n / F k | < ∞ , every F i is a free module of rank n. We first pick M 1 with the condition that im ⁡ ( M 1 ) = F 1 . This means two things:

M 1 : ℤ p n → ℤ p n has image in F 1 .
The induced ℤ p -linear map M 1 : ℤ p n → F 1 is surjective.

The probability that im ⁡ ( M 1 ) ⊆ F 1 is the probability that every column of M 1 lies in F 1 . Hence we obtain that Prob M 1 ∈ Mat n ⁡ ( ℤ p ) ⁡ ( im ⁡ ( M 1 ) ⊆ F 1 ) is | ℤ p n / F 1 | - n = | G 1 | - n . By Nakayama’s lemma, since F 1 is of rank n, the probability that a Haar-random linear map ℤ p n → F 1 be surjective is ∏ i = 1 n ( 1 - p - i ) . As a result,

Prob M 1 ∈ Mat n ⁡ ( ℤ p ) ⁡ ( im ⁡ ( M 1 ) = F 1 ) = | G 1 | - n ⁢ ∏ i = 1 n ( 1 - p - i ) .

Now we fix M 1 and pick M 2 with the condition that im ⁡ ( M 1 ⁢ M 2 ) = F 2 . We note that M 1 : ℤ p n → F 1 is an isomorphism because it is a surjective map between rank n free modules over ℤ p . Therefore, M 1 induces an isomorphism of flags from M 1 - 1 ⁢ ( F 2 ) ⊆ ℤ p n to F 2 ⊆ F 1 . Thus, im ⁡ ( M 1 ⁢ M 2 ) = F 2 if and only if im ⁡ ( M 2 ) = M 1 - 1 ⁢ ( F 2 ) ⊆ ℤ p n . By the same argument as above, and the fact that | ℤ p n / M 1 - 1 ⁢ ( F 2 ) | = | F 1 / F 2 | = | G 2 | / | G 1 | , we get

Prob M 2 ∈ Mat n ⁡ ( ℤ p ) ⁡ ( im ⁡ ( M 1 ⁢ M 2 ) = F 2 ∣ im ⁡ ( M 1 ) = F 1 ) = ( | G 2 | | G 1 | ) - n ⁢ ∏ i = 1 n ( 1 - p - i ) .

Repeating the argument and multiplying all of the above probabilities together, we conclude that

Prob M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) ⁡ ( 𝑖𝑚 ⁢ ( M 1 , … , M k ) = ℱ ) = | G 1 | - n ⁢ ∏ i = 1 n ( 1 - p - i ) ⋅ ( | G 2 | | G 1 | ) - n ⁢ ∏ i = 1 n ( 1 - p - i ) ⁢ ⋯ ⁢ ( | G k | | G k - 1 | ) - n ⁢ ∏ i = 1 n ( 1 - p - i )

= | G k | - n ⁢ ( ∏ i = 1 n ( 1 - p - i ) ) k . ∎

Proof of Theorem 1.1.

Since 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) ≃ ℤ p n / 𝑖𝑚 ⁢ ( M 1 , … , M k ) , we have

Prob M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) ⁡ ( 𝐜𝐨𝐤 ⁢ ( M 1 , … , M k ) ≃ 𝐆 ) = ∑ ℱ ∈ ℱ ⁢ l k ⁢ ( ℤ p n ) ℤ p n / ℱ ≃ 𝐆 Prob M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) ⁡ ( 𝑖𝑚 ⁢ ( M 1 , … , M k ) = ℱ ) .

Since the probability in Lemma 3.2 depends only on 𝐆 but not on ℱ , the proof is complete by multiplying the results of Lemmas 3.1 and 3.2. ∎

4 Proof of Theorem 1.2

We state and prove a convenient lemma first.

Lemma 4.1.

Let A , B , C be free modules over Z p of finite ranks a , b , c , and suppose f : B ↠ A is a surjective linear map. Then for a Haar-random linear map g : C → B , we have

Prob g ∈ Hom ⁡ ( C , B ) ⁡ ( im ⁡ ( f ⁢ g ) = A ) = ∏ i = c - a + 1 c ( 1 - p - i ) .

Proof.

Without loss of generality, we may assume A = ℤ p a , B = ℤ p b , C = ℤ p c and f : ℤ p b → ℤ p a is the projection to the first a coordinates. Write g ∈ Mat b × c ⁡ ( ℤ p ) as

g = [ g 1 g 2 ] ,

where g 1 ∈ Mat a × c ⁡ ( ℤ p ) and g 2 ∈ Mat ( b - a ) × c ⁡ ( ℤ p ) . Then f ⁢ g = g 1 . Thus, the probability that fg be surjective is the probability that g 1 be surjective, which is ∏ i = c - a + 1 c ( 1 - p - i ) by Nakayama’s lemma. ∎

To prove Theorem 1.2, we follow the same argument as Theorem 1.1, except that we need to prove a more general version of Lemma 3.2. Again, we fix 𝐆 = ( G k ↠ ⋯ ↠ G 1 ) ∈ 𝐅𝐥 k , but we do not assume | G k | < ∞ . Fix ℱ ∈ ℱ ⁢ l k ⁢ ( ℤ p n ) such that ℤ p n / ℱ ≃ 𝐆 . For 1 ≤ i ≤ k , let M i ∈ Mat ( n + u i - 1 ) × ( n + u i ) ⁡ ( ℤ p ) , where u 0 := 0 , and assume M 1 , … , M k are independent and Haar-random.

Lemma 4.2.

In the setting above, we have

(4.1) Prob M 1 , … , M k ⁡ ( 𝑖𝑚 ⁢ ( M 1 , … , M k ) = ℱ ) = | G k | - n ⁢ ∏ j = 1 k ( | G j | | G j - 1 | ) - u j ⋅ ∏ j = 1 k ∏ i = 1 n ( 1 - p - i - u j )

if | G k | < ∞ , and zero otherwise.

Proof.

Let ℱ = ( F k ⊆ ⋯ ⊆ F 1 ⊆ ℤ p n ) . We first pick M 1 ∈ Mat n × ( n + u 1 ) with im ⁡ ( M 1 ) = F 1 . If | G 1 | = ∞ , then we have | ℤ p n / F 1 | = ∞ , so the probability that each column of M 1 be in F 1 is zero. Therefore, we may assume | G 1 | < ∞ from now on. As a result, F 1 is free of rank n. By the similar argument in the proof of Lemma 3.2, we get

(4.2) Prob M 1 ∈ Mat n × ( n + u 1 ) ⁡ ( ℤ p ) ⁡ ( im ⁡ ( M 1 ) = F 1 ) = | G 1 | - ( n + u 1 ) ⁢ ∏ i = u 1 + 1 n + u 1 ( 1 - p - i ) = | G 1 | - ( n + u 1 ) ⁢ ∏ i = 1 n ( 1 - p - i - u 1 ) .

Now we fix M 1 and pick M 2 ∈ Mat ( n + u 1 ) × ( n + u 2 ) ⁡ ( ℤ p ) with the condition that im ⁡ ( M 1 ⁢ M 2 ) = F 2 , which is equivalent to im ⁡ ( M 2 ) ⊆ M 1 - 1 ⁢ ( F 2 ) . The third isomorphism theorem applied to the surjection M 1 : ℤ p n + u 1 → F 1 gives an isomorphism ℤ p n + u 1 / M 1 - 1 ⁢ ( F 2 ) ≃ F 1 / F 2 , so

| ℤ p n + u 1 / M 1 - 1 ⁢ ( F 2 ) | = | F 1 / F 2 | = | G 2 | | G 1 | .

Thus,

(4.3) Prob M 2 ∈ Mat ( n + u 1 ) × ( n + u 2 ) ⁡ ( ℤ p ) ⁡ ( im ⁡ ( M 1 ⁢ M 2 ) ⊆ F 2 ∣ im ⁡ ( M 1 ) = F 1 ) = ( | G 2 | | G 1 | ) - ( n + u 2 )

if | G 2 | < ∞ , and zero otherwise. So again, we may assume | G 2 | < ∞ from now on.

We now find the probability that im ⁡ ( M 1 ⁢ M 2 ) = F 2 conditioned on im ⁡ ( M 1 ⁢ M 2 ) ⊆ F 2 , so M 2 is a Haar-random linear map in Hom ⁡ ( ℤ p n + u 2 , M 1 - 1 ⁢ ( F 2 ) ) . Note that we are in the setting of Lemma 4.1 with A = F 2 , B = M 1 - 1 ⁢ ( F 2 ) , C = ℤ p n + u 2 , f = M 1 : M 1 - 1 ⁢ ( F 2 ) ↠ F 2 , and g = M 2 : ℤ p n + u 2 → M 1 - 1 ⁢ ( F 2 ) . The condition that im ⁡ ( M 1 ⁢ M 2 ) = F 2 is equivalent to im ⁡ ( f ⁢ g ) = A . Since | ℤ p n / F 2 | = | G 2 | < ∞ , A is free of rank n. Since | ℤ p n + u 1 / B | = | G 2 | / | G 1 | < ∞ , B is free of rank n + u 1 . By Lemma 4.1 with a = n and c = n + u 2 , we get

Prob M 2 ∈ Mat ( n + u 1 ) × ( n + u 2 ) ⁡ ( ℤ p ) ⁡ ( im ⁡ ( M 1 ⁢ M 2 ) = F 2 ∣ im ⁡ ( M 1 ) = F 1 , im ⁡ ( M 1 ⁢ M 2 ) ⊆ F 2 ) = ∏ i = u 2 + 1 n + u 2 ( 1 - p - i ) .

Combined with (4.3), we get

(4.4) Prob M 2 ∈ Mat ( n + u 1 ) × ( n + u 2 ) ⁡ ( ℤ p ) ⁡ ( im ⁡ ( M 1 ⁢ M 2 ) = F 2 ∣ im ⁡ ( M 1 ) = F 1 ) = ( | G 2 | | G 1 | ) - ( n + u 2 ) ⁢ ∏ i = 1 n ( 1 - p - i - u 2 ) .

Repeating the argument inductively and multiplying the probabilities in formulas (4.2), (4.4), and so on, the desired formula (4.1) follows, along with the fact that each G i must be finite in order for the probability to be nonzero. ∎

Proof of Theorem 1.2.

Multiply the results of Lemma 3.1 and Lemma 4.2. ∎

5 Probability measures on flags

Here we prove Corollary 1.4 as an immediate consequence of Theorem 1.2.

Proof of Corollary 1.4.

It is clear that P n , ( t 1 , … , t k ) is a nonnegative measure, so it suffices to show that

∑ 𝐆 ∈ 𝐅𝐥 k P n , ( t 1 , … , t k ) ⁢ ( 𝐆 ) = 1 .

We note that the measure in (1.2) is precisely P n , ( p - u 1 , … , p - u k ) . In particular, Theorem 1.2 implies that P n , ( p - u 1 , … , p - u k ) is a probability measure for every u 1 , … , u k ∈ ℤ ≥ 0 , so the formal power series ∑ 𝐆 ∈ 𝐅𝐥 k P n , ( t 1 , … , t k ) ⁢ ( 𝐆 ) in t 1 , … , t k must be 1, completing the proof.

The case of P ∞ , ( t 1 , … , t k ) is similar. ∎

Write 𝐆 = ( G k ↠ ⋯ ↠ G 1 ) . We compute the joint distribution of ( G 1 , … , G k ) if 𝐆 is distributed according to (1.4). For partitions λ ( 1 ) , … , λ ( k ) , we write 𝐆 ∼ ( λ ( 1 ) , … , λ ( k ) ) if the type of G i is λ ( i ) .

Proposition 5.1.

Fix partitions λ ( 1 ) , … , λ ( k ) . For P n , ( t 1 , … , t k ) ⁢ ( G ) as in (1.4), the quantity

∑ 𝐆 ∼ ( λ ( 1 ) , … , λ ( k ) ) P n , ( t 1 , … , t k ) ⁢ ( 𝐆 )

is given by [11, p. 19, Proposition 2.6] with q = 0 , a j ( i ) = t i ⁢ p - j , b = ( 1 , p - 1 , … , p - ( n - 1 ) ) , and t = 1 p in their notation.

Proof.

If t i = 1 , by Theorem 1.1, the quantity in question is the probability that cok ⁢ ( M 1 ⁢ … ⁢ M i ) is of type λ ( i ) for all i, where M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) are independent and Haar-random. This is given by [11, p. 27, Corollary 3.4] with N i = ∞ in their notation.

To go from the t i = 1 case to the general case, we notice that if we fix ∅ = λ ( 0 ) , λ ( 1 ) , … , λ ( k ) and letting n i = | λ ( i ) | - | λ ( i - 1 ) | for 1 ≤ i ≤ k , then from (1.4), we have

∑ 𝐆 ∼ ( λ ( 1 ) , … , λ ( k ) ) P n , ( t 1 , … , t k ) ⁢ ( 𝐆 ) ∑ 𝐆 ∼ ( λ ( 1 ) , … , λ ( k ) ) P n , ( 1 , … , 1 ) ⁢ ( 𝐆 ) = ∏ j = 1 k ( t j n j ⁢ ∏ i = 1 n 1 - p - i ⁢ t j 1 - p - i ) .

Combining this with the t i = 1 case above finally gives the desired formula. ∎

5.1 Relation to matrices over finite fields

We give a direct proof of Proposition 5.2, a function field analogue of (1.5), by establishing (5.2) that connects it to counting matrices over finite fields. Let 𝔽 q be the finite field with q elements, and let R denote the power series ring 𝔽 q ⁢ [ [ T ] ] . We can similarly define 𝐅𝐥 k ⁢ ( R ) to be the set of k-surjective flags of R-modules. Given 1 ≤ i ≤ k and any flag 𝐆 = ( G k ↠ ⋯ ↠ G 0 = 0 ) ∈ 𝐅𝐥 k ( R ) such that dim 𝔽 q ⁡ G k < ∞ , let n i ⁢ ( 𝐆 ) := dim 𝔽 q ⁡ G i - dim 𝔽 q ⁡ G i - 1 . Define the flag Cohen–Lenstra series of R as

(5.1) Z ^ R ⁢ ( t 1 , … , t k ) := ∑ 𝐆 ∈ 𝐅𝐥 k ⁢ ( R ) dim ⁡ G k < ∞ ∏ j = 1 k t j n j ⁢ ( 𝐆 ) | Aut ⁡ ( 𝐆 ) | ∈ ℚ ⁢ [ [ t 1 , … , t k ] ] .

When k = 1 , this construction is precisely the Cohen–Lenstra series defined in [8].

Proposition 5.2.

We have

Z ^ 𝔽 q ⁢ [ [ T ] ] ⁢ ( t 1 , … , t k ) = ∏ j = 1 k ∏ i = 1 ∞ 1 1 - q - i ⁢ t j .

Proof.

To specify a k-surjective flag 𝐆 of 𝔽 q ⁢ [ [ T ] ] -modules with given n i ⁢ ( 𝐆 ) = n i , it suffices to specify the k-surjective flag of underlying 𝔽 q -vector spaces

V k ⁢ ↠ ⁢ ⋯ ⁢ ↠ ⁢ V 1 ⁢ ↠ ⁢ V 0 = 0 ,

together with 𝔽 q ⁢ [ [ T ] ] -module structures on V i that are compatible with the flag. Up to isomorphism of flags of 𝔽 q -vector spaces, we may assume V i = 𝔽 q n 1 + … + n i , and the surjection ϕ i is the projection to the first n 1 + … + n i coordinates. Then the compatible 𝔽 q ⁢ [ [ T ] ] -module structures on V i are determined by a nilpotent endomorphism M on V k (as the multiplication by T map), such that M factors through an endomorphism on V i for each i. If W i := ker ⁡ ( V k ↠ V i ) for 0 ≤ i ≤ k , then this simply means M ⁢ W i ⊆ W i for 0 ≤ i ≤ k . Since W i is the span of the last n i + 1 + … + n k basis vectors in V k = 𝔽 q n 1 + … + n k , this happens if and only if M is block-lower-triangular with respect to the block sizes n 1 , … , n k in both rows and columns. We denote the set of such block-lower-triangular matrices by Mat n 1 , … , n k ⁡ ( 𝔽 q ) .

Let GL n 1 , … , n k ⁡ ( 𝔽 q ) and Nilp n 1 , … , n k ⁢ ( 𝔽 q ) be the set of invertible and nilpotent matrices in Mat n 1 , … , n k ⁡ ( 𝔽 q ) , respectively. By the discussion above, k-surjective flags of 𝔽 q ⁢ [ [ T ] ] -modules with a given dimension vector ( n i ) are parametrized by Nilp n 1 , … , n k ⁢ ( 𝔽 q ) . The group GL n 1 , … , n k ⁡ ( 𝔽 q ) acts on Nilp n 1 , … , n k ⁢ ( 𝔽 q ) by conjugation. The orbits correspond to isomorphism classes of flags of 𝔽 q ⁢ [ [ T ] ] -modules, and the stabilizers correspond to automorphisms of flags of 𝔽 q ⁢ [ [ T ] ] -modules. By a standard argument involving the orbit-stabilizer theorem, we get

(5.2) Z ^ 𝔽 q ⁢ [ [ T ] ] ⁢ ( t 1 , … , t k ) = ∑ n 1 , … , n k ≥ 0 | Nilp n 1 , … , n k ⁢ ( 𝔽 q ) | | GL n 1 , … , n k ⁡ ( 𝔽 q ) | ⁢ t 1 n 1 ⁢ … ⁢ t k n k .

Since a matrix in Mat n 1 , … , n k ⁡ ( 𝔽 q ) is invertible respectively nilpotent if and only if every diagonal block is invertible respectively nilpotent, the generating series simplifies to

(5.3) Z ^ 𝔽 q ⁢ [ [ T ] ] ⁢ ( t 1 , … , t k ) = ∏ j = 1 k ∑ n j ≥ 0 | Nilp n j ⁢ ( 𝔽 q ) | | GL n j ⁡ ( 𝔽 q ) | ⁢ t j n j .

But it is well known that the sum equals

∏ i = 1 ∞ 1 1 - q - i ⁢ t j ;

for instance, we may use the theorem | Nilp n ⁢ ( 𝔽 q ) | = q n 2 - n of Fine and Herstein [5], and then apply an identity of Euler [1, equation (2.2.5)]. This completes the proof. ∎

Remark.

Of course, Mat n 1 , … , n k ⁡ ( 𝔽 q ) and GL n 1 , … , n k ⁡ ( 𝔽 q ) are nothing but the parabolic subalgebra and the parabolic subgroup of a suitable partial flag variety of type A. As is made apparent by equation (5.2), the construction (5.1) fits naturally into the study of commuting varieties of parabolic subalgebras; we refer the readers to [2] for a geometric aspect of such research.

Communicated by Jan Bruinier

Acknowledgements

The author thanks Roger Van Peski for fruitful discussions and comments on earlier drafts.

References

[1] G. E. Andrews, The Theory of Partitions, Cambridge Math. Libr., Cambridge University, Cambridge, 1998. Search in Google Scholar

[2] M. Bulois and L. Evain, Nested punctual Hilbert schemes and commuting varieties of parabolic subalgebras, J. Lie Theory 26 (2016), no. 2, 497–533. Search in Google Scholar

[3] H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups of number fields, Number Theory, Noordwijkerhout 1983, Lecture Notes in Math. 1068, Springer, Berlin (1984), 33–62. 10.1007/BFb0099440Search in Google Scholar

[4] C. Delaunay and F. Jouhet, p ℓ -torsion points in finite abelian groups and combinatorial identities, Adv. Math. 258 (2014), 13–45. 10.1016/j.aim.2014.02.033Search in Google Scholar

[5] N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, Illinois J. Math. 2 (1958), 499–504. 10.1215/ijm/1255454112Search in Google Scholar

[6] E. Friedman and L. C. Washington, On the distribution of divisor class groups of curves over a finite field, Théorie des nombres, de Gruyter, Berlin (1989), 227–239. 10.1515/9783110852790.227Search in Google Scholar

[7] J. Fulman and N. Kaplan, Random partitions and Cohen–Lenstra heuristics, Ann. Comb. 23 (2019), no. 2, 295–315. 10.1007/s00026-019-00425-ySearch in Google Scholar

[8] Y. Huang, Mutually annihilating matrices, and a Cohen–Lenstra series for the nodal singularity, J. Algebra 619 (2023), 26–50. 10.1016/j.jalgebra.2022.11.021Search in Google Scholar

[9] H. H. Nguyen and R. Van Peski, Universality for cokernels of random matrix products, Adv. Math. 438 (2024), Paper No. 109451. 10.1016/j.aim.2023.109451Search in Google Scholar

[10] W. Sawin and M. Matchett Wood, The moment problem for random objects in a category, preprint (2022), https://arxiv.org/abs/2210.06279. Search in Google Scholar

[11] R. Van Peski, Limits and fluctuations of p-adic random matrix products, Selecta Math. (N. S.) 27 (2021), no. 5, Paper No. 98. 10.1007/s00029-021-00709-3Search in Google Scholar

[12] M. M. Wood, Random integral matrices and the Cohen–Lenstra heuristics, Amer. J. Math. 141 (2019), no. 2, 383–398. 10.1353/ajm.2019.0008Search in Google Scholar

Received: 2024-07-23

Revised: 2024-10-12

Published Online: 2024-11-14

Published in Print: 2025-06-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/forum-2024-0129

Keywords for this article

Cohen–Lenstra heuristic; flags of abelian groups; random p-adic matrices

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