[Paper Review] Integer group determinants for abelian groups of order 16

Yuka Yamaguchi, Naoya Yamaguchi|arXiv (Cornell University)|Nov 27, 2022

graph theory and CDMA systems4 citations

TL;DR

This paper resolves the integer group determinant problem for the abelian group $\mathbb{C}_4 \times \mathbb{C}_2^2$, the last unsolved abelian group of order 16. Using algebraic number theory and determinant decomposition techniques, the authors fully characterize the set $S(G)$ of all possible integer group determinant values as a union of arithmetic progressions and scaled prime-related forms, completing a long-standing classification problem in group determinant theory.

ABSTRACT

For any positive integer $n$, let ${ m C}_{n}$ be the cyclic group of order $n$. We determine all possible values of the integer group determinant of ${ m C}_{4} imes { m C}_{2}^{2}$, which is the only unsolved abelian group of order $16$.

Motivation & Objective

• To determine the complete set $S(G)$ of all possible integer group determinant values for the abelian group $G = \mathbb{C}_4 \times \mathbb{C}_2^2$, which was the last unsolved abelian group of order 16.
• To extend the classification of integer group determinants beyond previously solved cases, particularly for groups of order 16.
• To resolve a longstanding open problem initiated by Olga Taussky-Todd on circulant determinants and generalized to finite groups.
• To provide a precise arithmetic description of the image of the integer group determinant map for this group using number-theoretic sets and modular forms.

Proposed method

• Employed the group determinant construction $\det(x_{gh^{-1}})_{g,h \in G}$ for $G = \mathbb{C}_4 \times \mathbb{C}_2^2$, with integer-valued variables $x_g$.
• Used recursive decomposition of the group determinant via subgroups, particularly leveraging the isomorphism $\mathbb{C}_4 \times \mathbb{C}_2 \cong \mathbb{C}_4 \times \mathbb{C}_2$ and known results on $\mathbb{C}_4$ and $\mathbb{C}_2$ determinants.
• Applied Corollary 2.1 to decompose $D_4$ determinants into products of $D_2$ and complex $D_2$ forms, enabling algebraic simplification.
• Used the structure of $\mathbb{C}_4 \times \mathbb{C}_2^2$ as a product to define $D_{4\times 2\times 2}$ and reduce the problem to evaluating symmetric polynomial expressions.
• Defined key number-theoretic sets: $P_r$ (primes $\equiv r \pmod{8}$), $P'$ (primes $\equiv 1 \pmod{8}$ with $a+b \equiv \pm 3 \pmod{8}$), and sets $A$, $B$, $C$, $D$ to describe the image of the determinant map.
• Constructed explicit integer-valued variable assignments to realize each class of determinant value, proving membership in $S(G)$ via direct determinant computation.

Experimental results

Research questions

• RQ1 What is the complete set of integer values that can arise as the determinant of a group matrix over $\mathbb{C}_4 \times \mathbb{C}_2^2$ with integer entries?
• RQ2 How does the structure of $\mathbb{C}_4 \times \mathbb{C}_2^2$ as a product of cyclic groups influence the image of its integer group determinant?
• RQ3 Can the integer group determinant of this group be fully described using arithmetic progressions and prime-related forms, as in prior cases?
• RQ4 What role do primes congruent to 1 or 5 modulo 8 play in the determinant image, and how do they interact with powers of 2?
• RQ5 Is there a complete inclusion chain of determinant sets across groups of order 16, and does $S(\mathbb{C}_4 \times \mathbb{C}_2^2)$ fit strictly between $S(\mathbb{C}_2^4)$ and $S(\mathbb{C}_4^2)$?

Key findings

• The set of all integer group determinant values for $\mathbb{C}_4 \times \mathbb{C}_2^2$ is exactly $S(G) = \{16m+1,\ 2^{16}(4m+1),\ 2^{16}(8m+3),\ 2^{17}p(2m+1),\ 2^{18}m \mid m \in \mathbb{Z},\ p \in P_5\} \cup \{2^{16}m \mid m \in A \cup B\}$, where $P_5$ is the set of primes $\equiv 5 \pmod{8}$.
• The value $2^{16}(4m+1)$ is achieved via a symmetric assignment of variables with constant blocks, reducing the determinant to a product of $D_4$ forms.
• The value $2^{17}p(2m+1)$ is realized for any prime $p \equiv 5 \pmod{8}$ by constructing a variable assignment that exploits the sum of squares representation of $2p$ as a sum of two squares.
• The set $A = \{(8k-3)(8l+3) \mid k,l \in \mathbb{Z}\}$ and $B = \{p(8m-1) \mid p \in P', m \in \mathbb{Z}\}$ parameterize additional determinant values of the form $2^{16}m$.
• The inclusion $S(\mathbb{C}_2^4) \subsetneq S(\mathbb{C}_4 \times \mathbb{C}_2^2) \subsetneq S(\mathbb{C}_4^2) \subsetneq S(\mathbb{C}_8 \times \mathbb{C}_2) \subsetneq S(D_{16}) \subsetneq S(\mathbb{C}_{16})$ holds, confirming a strict hierarchy of determinant images.
• The result completes the classification of integer group determinants for all abelian groups of order 16, resolving the final open case in this family.