[Paper Review] Invariants for critical dimension groups and permutation-Hermite equivalence

David Handelman|arXiv (Cornell University)|Sep 28, 2013

Finite Group Theory Research1 references3 citations

TL;DR

This paper introduces new invariants for permutation-Hermite equivalence of integer matrices—intermediate between Hermite and Smith normal form equivalence—enabling finer classification of dense subgroups of R^n, particularly critical dimension groups. The key contribution is a formula for counting PH-equivalence classes of 3×3 matrices with fixed determinant d, showing asymptotic density behavior and revealing a surprising third difference operator in the count for matrices not equivalent to terminal forms with 1-blocks of size two.

ABSTRACT

Motivated by classification, up to order isomorphism, of some dense subgroups of Euclidean space that are free of minimal rank, we obtain apparently new invariants for an equivalence relation (intermediate between Hermite and Smith) on integer matrices. These then participate in the classification of the dense subgroups. The same equivalence relation has appeared before, in the classification of lattice simplices. We discuss this equivalence relation (called {\it permutation-Hermite}), obtain fairly fine invariants for it, and have density results, and some formulas counting the numbers of equivalence classes for fixed determinant.

Motivation & Objective

• To develop finer invariants than Smith normal form for classifying integer matrices under permutation-Hermite equivalence.
• To classify dense subgroups of R^n that are free of rank n+1 and admit exactly n pure traces—critical dimension groups.
• To characterize almost basic critical groups via the torsion-free rank of G/E(G), where E(G) is the sum of kernels of n−1 pure traces.
• To derive exact counting formulas for PH-equivalence classes of 3×3 matrices with fixed determinant d, especially for square-free d.
• To analyze the natural density and asymptotic behavior of such equivalence classes, revealing unexpected arithmetic structures in the count.

Proposed method

• Define permutation-Hermite equivalence as B ~ B' if there exist U ∈ GL(m,Z) and permutation matrix P such that UBP = B'.
• Construct two families of invariants for PH-equivalence, including the structure of cokernels and invariants derived from pure trace kernels.
• Use the Smith normal form as a coarse invariant and refine it with additional data from the structure of the matrix and its row space.
• Introduce the concept of terminal forms and classify matrices based on whether they are equivalent to a matrix of the form (I_{n−1} | a; 0 | d).
• Apply Dirichlet convolution and multiplicative number-theoretic functions (e.g., φ, J₂, totient) to derive counting formulas for equivalence classes.
• Employ the third difference operator Δ³f_d(−1) to isolate classes not equivalent to matrices with 1-block size two, revealing a deep arithmetic structure.

Experimental results

Research questions

• RQ1 How can we refine the Smith normal form invariant to distinguish PH-equivalent matrices that are not Hermite-equivalent?
• RQ2 What is the exact number of PH-equivalence classes of 3×3 integer matrices with a fixed determinant d, particularly for square-free d?
• RQ3 What is the natural density of PH-equivalence classes that are not equivalent to any matrix with a 1-block of size two in the terminal form?
• RQ4 Why does the third difference operator Δ³f_d(−1) appear in the count of non-terminal-form classes, and what is its arithmetic significance?
• RQ5 Under what conditions is a critical dimension group almost basic, and how does this relate to the structure of G/E(G)?

Key findings

• For square-free d, the number of PH-equivalence classes of 3×3 matrices with |det C| = d is given by a sum of three terms, with the dominant term being (φ * J₂)(d)/6.
• The number of PH-equivalence classes not equivalent to a terminal form with a 1-block of size two is (φ(d)Δ³f_d(−1))/6, where f_d(x) = ∏_{p|d}(x + p).
• When d = pqr (product of three distinct primes), Δ³f_d(−1) = 6, so the number of such classes is exactly φ(d), and the action of Z_d^* on them is regular.
• For d = 30, there are exactly 8 PH-equivalence classes not equivalent to any matrix with a 1-block of size two, corresponding to multiplication of the truncated column by units mod 30.
• The formula derived agrees completely with data from [ALPPT] for d = 30, 42, 70, ..., 210, validating the counting approach.
• As d(m) ranges over square-free integers with ∑_{p|d(m)} 1/p → ∞, the ratio of non-1-block-size-two classes to those with such blocks tends to 1/3, indicating a stable asymptotic density.