Diagonal conjugates in products of two divisibility chains

Let a and b be divisibility chains in a domain R (a i ∣ a j and b i ∣ b j for i ≤ j) with nonzero entries, write A = diagonal a and B = diagonal b, and let σ and τ be invertible matrices. If every entry in row i of σ A τ B is divisible by a i b i, that is σ A τ B = A B γ for an integral γ, then A⁻¹ σ A is integral: a i ∣ σ i j * a j for all i and j.

Over a principal ideal domain this says, in terms of lattices, that among the lattices L with A B Rⁿ ⊆ L ⊆ Rⁿ, Rⁿ / L of type a and L / A B Rⁿ of type b, the only one is A Rⁿ (the left coset σ A determines L = σ A Rⁿ). Over ℤ it says that in the Hecke ring of (SL_n(ℤ), Δ) the double coset T(a b) occurs in T(a) · T(b) with multiplicity one: the leading term of the triangular expansion of a product of diagonal double cosets, on which Shimura's polynomial presentation of the p-local Hecke ring rests.

The proof is first given over a local domain, by induction on the size of the matrices. The last row of σ is primitive, so it has a unit entry, and the bottom row of σ A τ B shows that a_last ∣ σ_last,k a_k for every k. These two facts complete the last row of σ to an invertible matrix v with A⁻¹ v A integral, and replacing σ by σ v⁻¹ (and τ by A⁻¹ v A τ) makes the last row of σ that of the identity. The same move on τ, with b in place of a, does the same to τ, and the upper-left blocks then satisfy the hypothesis one size down. The general domain follows by localizing at every maximal ideal.

Main results

• Matrix.dvd_apply_mul_of_dvd_mul_diagonal_mul_diagonal_apply: if a, b are divisibility chains of nonzero elements of a domain, σ, τ are invertible and a i * b i divides every entry in row i of σ * diagonal a * τ * diagonal b, then a i ∣ σ i j * a j.

References

• G. Shimura, Introduction to the arithmetic theory of automorphic functions, §3.2, Theorem 3.20, whose triangular expansion this leading term enters.

theorem Matrix.dvd_apply_mul_of_dvd_mul_diagonal_mul_diagonal_apply {R : Type u_1} [CommRing R] [IsDomain R] {m : ℕ} {a b : Fin m → R} (ha : ∀ (i j : Fin m), i ≤ j → a i ∣ a j) (hb : ∀ (i j : Fin m), i ≤ j → b i ∣ b j) (ha₀ : ∀ (i : Fin m), a i ≠ 0) (hb₀ : ∀ (i : Fin m), b i ≠ 0) {σ τ : Matrix (Fin m) (Fin m) R} (hσ : IsUnit σ.det) (hτ : IsUnit τ.det) (h : ∀ (i j : Fin m), a i * b i ∣ (σ * diagonal a * τ * diagonal b) i j) (i j : Fin m) : a i ∣ σ i j * a j

Diagonal conjugates in a product of two divisibility chains. Let a and b be divisibility chains of nonzero elements of a domain R, and let σ and τ be invertible matrices. If a i * b i divides every entry in row i of σ * diagonal a * τ * diagonal b — that is, σ A τ B = A B γ for some matrix γ — then a i ∣ σ i j * a j for all i and j, that is, diagonal a⁻¹ * σ * diagonal a is integral.

Over ℤ this is the statement that the double coset T(a b) occurs in T(a) · T(b) with multiplicity one.