Generalized Dedekind’s theorem and its application to integer group determinants

For a finite group $G$, let $x_{g}$ be an indeterminate for each $g\in G$ and let $\mathbb{Z}[x_{g}]$ be the multivariate polynomial ring in $x_{g}$ over $\mathbb{Z}$. The group matrix $M_{G}(x_{g})$ and the group determinant $\Theta_{G}(x_{g})$ of $G$ were defined by Dedekind as follows:

When the elements of $G$ are reordered arbitrarily, the group matrix $M$ formed according to this reordering is of the form $M=P^{-1}M_{G}(x_{g})P$, where $P$ is an appropriate permutation matrix. Thus, $\Theta_{G}(x_{g})$ is invariant under any reordering of the elements of $G$. For a finite group $G$, let $\widehat{G}$ be a complete set of representatives of the equivalence classes of irreducible representations of $G$ over $\mathbb{C}$. Around 1880, for the case that $G$ is abelian, Dedekind gave the irreducible factorization of $\Theta_{G}(x_{g})$ over $\mathbb{C}$: Let $G$ be a finite abelian group. Then

This is called Dedekind’s theorem. In 1896, Frobenius [3] gave the irreducible factorization of $\Theta_{G}(x_{g})$ over $\mathbb{C}$ for any finite group: Let $G$ be a finite group. Then

This is the most well known generalization of Dedekind’s theorem. This generalization is obtained from the decomposition of the regular representation $L$ of $G$ as a direct sum of irreducible representations and the expression $M_{G}(x_{g})=\sum_{g\in G}x_{g}L(g)$. Frobenius created the character theory of finite groups in the process of obtaining the irreducible factorization. For the history on the theory, see, e.g., [2, 4, 5, 6, 20]. On the other hand, another generalization of Dedekind’s theorem was given in [21]: Let $G$ be a finite abelian group and let $H$ be a subgroup of $G$. For every $h\in H$, there exists a homogeneous polynomial $A_{h}\in\mathbb{C}[x_{g}]$ satisfying $\deg{A_{h}}=\left|G/H\right|$ and

If $H=G$, then we can take $A_{h}=x_{h}$ for each $h\in H$. This generalization shows that the group determinant of an abelian group can be written by the group determinant of any subgroup. Let ${\rm C}_{n}=\left\{\overline{0},\overline{1},\ldots,\overline{n-1}\right\}$ be the cyclic group of order $n$. The matrix $M_{{\rm C}_{n}}(x_{g})$ is similar to the circulant matrix of order $n$. That is, the circulant determinant is a special case of the group determinant. For the circulant determinant, Laquer [12, Theorem 2] gave the following factorization in 1980: Let $n=rs$, where $r$ and $s$ are relatively prime, and let $x_{j}:=x_{\overline{j-1}}$ for any $1\leq j\leq n$. Then

where $\zeta_{s}$ is a primitive $s$-th root of unity. We call this theorem Laquer’s theorem. In recently, Laquer’s theorem was generalized as follows [24, Theorem 1.1]: Let $G=H\times K$ be a direct product of finite abelian groups. Then we have

In this paper, we give a refinement of (1), which is a generalization of (2).

For any $f(x_{g})\in\mathbb{Z}[x_{g}]$, we denote by $f(x_{g})_{h}$ the sum of all monomials $cx_{g_{1}}x_{g_{2}}\cdots x_{g_{k}}$ in $f(x_{g})$ satisfying $g_{1}g_{2}\cdots g_{k}=h$. The following theorem gives another expression for $z_{h}$ in Theorem 1.1. When calculating $z_{h}$, the expression for $z_{h}$ in the following theorem might be more useful than one in Theorem 1.1 (see Example 2.3).

Theorem 1.1 is a refinement of (1) since $\left\{\chi|_{H}\mid\chi\in X\right\}=\widehat{H}$ holds and we can take $A_{h}=z_{h}$ in (1). Note that for a finite abelian group $G$ and any subgroup $K$, there exists a subgroup $H$ of $G$ satisfying $K\cong G/H$. That is, Theorem 1.1 implies that $\Theta_{G}(x_{g})$ can also be expressed as a product of the group determinants of any subgroup. Thus, Theorem 1.1 derives (2). We apply Theorems 1.1 and 1.2 to the study of the integer group determinant.

A group determinant called an integer group determinant when its variables are integers. For a finite group $G$, let

It immediately follows from $M_{G}(x_{g})=\sum_{g\in G}x_{g}L(g)$ that $S(G)$ is a monoid. Determining $S(G)$ is an open problem. For the $G={\rm C}_{n}$ cases, determining $S(G)$ is called Olga Taussky-Todd’s circulant problem since Olga Taussky-Todd suggested it at the meeting of the American Mathematical Society in Hayward, California [14]. Even Olga Taussky-Todd’s circulant problem remains as an open problem.

For $S({\rm C}_{n})$, the following relation is known [10, Lemma 3.6]: Let $n,q\geq 1$. If $q\mid n$, then

From Theorems 1.1 and 1.2, we obtain a generalization of (3).

Corollary 1.3 is generalized as follows.

For some types of groups, the problem was solved in [1, 10, 12, 13, 14, 15, 18, 19, 25, 28]. As a result, for every group $G$ of order at most $15$, $S(G)$ is determined (see [15, 19]). For the groups of order $16$, the complete descriptions of $S(G)$ were obtained for ${\rm D}_{16}$ [1, Theorem 5.3], ${\rm C}_{16}$ [25] and ${\rm C}_{2}^{4}$ [28], where ${\rm D}_{n}$ denotes the dihedral group of order $n$.

Laquer [12] determined $S({\rm C}_{2p})$, where $p$ is an odd prime, by using Laquer’s theorem which provides an expression for the integer circulant determinant of ${\rm C}_{2p}$ as a product of two integer circulant determinants of ${\rm C}_{p}$. In [28], $S({\rm C}_{2}^{4})$ is determined by using (2) which provides an expression for the integer group determinant of ${\rm C}_{2}^{n}$ as a product of two integer group determinants of ${\rm C}_{2}^{n-1}$. We can generalize these approaches by using Theorem 1.1 to determine $S(G)$ for any abelian groups. There are fourteen groups of order $16$ up to isomorphism [7, 30], and five of them are abelian. The unsolved abelian groups of order $16$ are ${\rm C}_{8}\times{\rm C}_{2}$, ${\rm C}_{4}^{2}$ and ${\rm C}_{4}\times{\rm C}_{2}^{2}$. By applying Theorem 1.1, we determine $S({\rm C}_{8}\times{\rm C}_{2})$.

The remaining two abelian groups could also be solved by using Theorem 1.1. (While this paper under review, it have been solved in [26, 29]. Also, as for non-abelian groups of order 16, ${\rm D}_{8}\times{\rm C}_{2}$, ${\rm Q}_{8}\times{\rm C}_{2}$ [16, Theorems 3.1 and 4.1], ${\rm Q}_{16}$ [17] and ${\rm C}_{2}^{2}\rtimes{\rm C}_{4}$ [27] have been solved, where ${\rm Q}_{n}$ denotes the generalized quaternion group of order $n$.)

Pinner and Smyth [19, p.427] noted the following inclusion relations for all groups of order $8$:

From preceding results and Theorem 1.5, we have

Determining the integer group determinants aims to investigate the structure of a group by means of the group determinant. It is expected that individual new results will help us understand more about groups.

This paper is organized as follows. In Section 2, we prove Theorems 1.1 and 1.2. In Sections 3 and 4, we prove Theorems 1.4 and 1.5, respectively.

For a finite group $G$, let $x_{g}$ be an indeterminate for each $g\in G$, let $\mathbb{C}[x_{g}]$ be the multivariate polynomial ring in $x_{g}$ over $\mathbb{C}$, let $\mathbb{C}G$ the group algebra of $G$ over $\mathbb{C}$, and let $\mathbb{C}[x_{g}]G:=\mathbb{C}[x_{g}]\otimes\mathbb{C}G=\left\{\sum_{g\in G}A_{g}g\mid A_{g}\in\mathbb{C}[x_{g}]\right\}$ be the group algebra of $G$ over $\mathbb{C}[x_{g}]$. Also, for a finite abelian group $G$ and a subgroup $H$ of $G$, let

It is easily verified that $\widehat{G}_{H}=\left\{\varphi\circ\pi\>|\>\varphi\in\widehat{G/H}\right\}$, where $\pi:G\rightarrow G/H$ is the canonical homomorphism. To prove Theorem 1.1, we use the following lemma.

From Theorems 1.1 and 1.2, we can take $z_{h}\in\mathbb{Z}[x_{g}]$ satisfying $\Theta_{G}(x_{g})=\Theta_{H}(z_{h})$. Thus, Corollary 1.3 is obtained.

The lower bound in Theorem 1.4 is derived from [23, Lemma 3.2]. Also, the upper bound immediately follows from the following lemma essentially provided in [22, Theorem 1.4].

The proof in [22] is not concise. We give a brief proof of Lemma 3.1. For the purpose, we use the following identity [9, p. 82, Theorem 2.6]; see also [8, 11]: Given the block matrix $M$ of the form

where the matrices $M_{ij}$ are pairwise commuting of size $m\times m$ then

In this section, by applying Theorem 1.1, we determine $S({\rm C}_{8}\times{\rm C}_{2})$.