The Rational Projective Determinant

For even \(n\), the rational projective determinant is a canonically defined function \(\mathrm{Pdet} : \mathrm{Mat}(n;\mathbb{Q}) \to \mathbb{Z}\) introduced by John F. R. Duncan in his paper Arithmetic Groups and the Affine E8 Dynkin Diagram. In this note we define the rational projective determinant and investigate some of its elementary properties. As an aside, this paper by Duncan is a must read.

Definition

Suppose \(A = (a_{ij}) \in \mathrm{Mat}(n; \mathbb{Q})\) for some positive, even \(n\). Since each \(a_{ij}\) is rational we can write \(a_{ij} = b_{ij}/c_{ij}\) for some \(b_{ij}, c_{ij} \in \mathbb{Z}\). Define \(\alpha_A \in \mathbb{Q}^\times\) by

\begin{equation*} \alpha_A = \left| \frac{\mathrm{lcm}\{c_{ij}\}}{\mathrm{gcd}\{b_{ij}\}} \right| \end{equation*}

Then \(\alpha_A\) is the smallest positive element of \(\mathbb{Q}^\times\) such that \(\alpha_A a_{ij} \in \mathbb{Z}\) for each \(1 \leq i,j \leq n\). With this notation, we define the projective determinant function \(\mathrm{Pdet} : \mathrm{Mat}(n;\mathbb{Q}) \to \mathbb{Z}\) by the formula

\begin{equation*} \mathrm{Pdet}(A) := \begin{cases} \det(\alpha_A A) = \alpha_A^n\det(A) & \text{ if } A \ne 0 \\ 0 & \text{ if } A = 0 \end{cases} \end{equation*}

Since \(\alpha_A A\) has integer entries by definition it is clear that \(\mathrm{Pdet}\) takes values in \(\mathbb{Z}\). ∎

Elementary properties.

By combining the next few propositions, it will follow that \(\mathrm{Pdet}\) induces a well-defined function on lattices commensurable to a fixed lattice in \(\mathbb{R}^n\). See also this note on Conway’s big picture.

Proposition 1. \(\mathrm{Pdet}\) descends to a well-defined function

\begin{equation*} \mathrm{Pdet} : \mathrm{PGL}(2;\mathbb{Q}) \longrightarrow \mathbb{Z} \end{equation*}

Proof: Fix any \(A \in \mathrm{Mat}(n;\mathbb{Q})\), and any \(r \in \mathbb{Q}^\times\). With the same notation as above, it is not hard to see that \(\alpha_{rA} A = \alpha_A A\). Thus we the identity

\begin{equation*} \mathrm{Pdet}(rA) = \det(\alpha_{rA} rA) = \det(\alpha_A A) = \mathrm{Pdet}(A) \end{equation*}

as required. ∎

Continuing in this vein, we have:

Proposition 2. The restriction of \(\mathrm{Pdet}\) to \(\mathrm{PGL}^+(n;\mathbb{Q})\) induces a well-defined function

\begin{equation*} \mathrm{PSL}(n;\mathbb{Z}) \backslash \mathrm{PGL}^+(n;\mathbb{Q}) \longrightarrow \mathbb{Z}_{>0} \end{equation*}

Proof: Positivity is immediate from the fact we restrict our domain to \(\mathrm{GL}^+(n;\mathbb{Q}\).

In light of the previous proposition it suffices to show, for all \(A \in \mathrm{SL}(n;\mathbb{Z})\) and \(X \in \mathrm{Mat}(n;\mathbb{Q})\), we have

\begin{equation*} \mathrm{Pdet}(AX) = \mathrm{Pdet}(X) = \mathrm{Pdet}(XA) \end{equation*}

To this end, notice

\begin{equation*} \mathrm{Pdet}(AX) = \alpha_{AX}^n \det(AX) = \alpha_{AX}^n \det(A)\det(X) = \alpha^n_{AX}\det(X) \end{equation*}

and

\begin{equation*} \mathrm{Pdet}(X) = \alpha_X^n\det(X) \end{equation*}

Thus all we need to do is prove the identity \(\alpha_{AX} = \alpha_{X}\) for all \(A \in \mathrm{SL}(n;\mathbb{Z})\). Suppose \(\alpha \in \mathbb{Q}\) is such that \(\alpha AX \in \mathrm{Mat}(n;\mathbb{Z})\), ie \(AX\) has integer entries. Since \(A^{-1} \in \mathrm{SL}(2;\mathbb{Z})\), the matrix

\begin{equation*} A^{-1}\alpha AX = \alpha X \end{equation*}

also has integer entries. Conversely

\begin{equation*} \alpha X \in \mathrm{Mat}(n;\mathbb{Z}) \quad \Longrightarrow \quad \alpha AX \in \mathrm{Mat}(n;\mathbb{Z}) \end{equation*}

As an immediate consequence, we get the following equality of sets:

\begin{equation*} \left\{ \alpha \in \mathbb{Q}^+ ~|~ \alpha AX \in \mathrm{Mat}(n;\mathbb{Z}\right\} = \left\{\alpha \in \mathbb{Q}^+ ~|~ \alpha X \in \mathrm{Mat}(n;\mathbb{Z})\right\} \end{equation*}

Since these two sets are equal, their smallest elements are also equal. In other words, \(\alpha_{AX} = \alpha_X\) and we are done. ∎

In the note Conway’s Big Picture we will use the rational projective determinant to define a (hyper-)distance function on the set of projective lattices in \(\mathbb{C}\).