In a computation of characters of certain representations of finite cyclic groups which appear as equivariant indices of Dirac operators (using the Atiyah-Bott fixed point formula, cf. [1, Theorem 8.35]), the following elementary problem emerged.

# Question 1

Let $d > 2$ be an integer and set $n = \lceil\frac{d}{2}\rceil -1$. For $k \in \{1, \dotsc, n\}$ with $\gcd(k,d)=1$, we set $$ v_k = \begin{pmatrix} \csc^2(\frac{k \pi}{d})& \csc^2(\frac{2k \pi}{d}) &\cdots&\csc^2(\frac{n k\pi}{d}) \end{pmatrix} \in \mathbb{R}^n, $$ where $\csc x = \frac{1}{\sin x}$ denotes the cosecant. If $\gcd(k,d) \neq 1$, we let $v_k = (0\ \cdots\ 1\ \cdots 0)$ be the standard basis vector with entry $1$ at the $k$-th position.

*Question*: Do the vectors $v_1, \dots, v_n$ generate the vector space $\mathbb{R}^n$?

As this appears somewhat intimidating at first glance, let us simplify it to the case where $d$ is an odd prime number. Then it reads as follows:

# Question 2

Let $p$ be an odd prime number.

*Question:* Is the following $\frac{p-1}{2} \times \frac{p-1}{2}$-matrix invertible?
$$M_p =
\begin{pmatrix}
\csc^2(\frac{\pi}{p}) & \csc^2(\frac{2\pi}{p}) &\cdots & \csc^2(\frac{(p-1)\pi}{2p}) \\
\csc^2(\frac{2\pi}{p}) & \csc^2(\frac{4\pi}{p}) &\cdots & \csc^2(\frac{2(p-1)\pi}{2p}) \\
\vdots & \vdots & \ddots & \vdots\\
\csc^2(\frac{(p-1)\pi}{2p}) & \csc^2(\frac{2(p-1)\pi}{2p}) &\cdots & \csc^2(\frac{(p-1)^2\pi}{4p})
\end{pmatrix}
$$

# Discussion

The questions appears to be of a number-theoretic nature. Potentially relevant formulas involving the cosecant appear for instance in Cauchy's elementary solution to the classical Basel problem.

Moreover, numerical experiments suggest that

- Question 1 has an affirmative answer at least for $d \leq 200$.
- The determinant of the matrix in Question 2 is an integer times $\frac{1}{\sqrt{p}}$ if $p \equiv 1 \mod 4$, and an integer otherwise. This suggests a relation to quadratic reciprocity.

Spotting a concrete formula for the determinants from numerical computations is difficult, however, because the numbers grow very rapidly. But in [2, Lemma 3.1], the formula $$ \prod_{j=1}^{\lfloor\frac{d}{2}\rfloor} \sin(\frac{j \pi}{d}) = \frac{\sqrt{d}}{2^{\frac{d-1}{2}}} $$ is provided which is reminiscent of our numerical observations. A more complicated formula involving quadratic reciprocity is provided in [2, Lemma 3.2].

All of this makes it plausible that it should be possible to derive a concrete formula for the determinants of the relevant matrices appearing in Question 1 and 2, but it remained elusive to me so far.

## References

[1] *Atiyah, Michael F.; Bott, Raoul*, **A Lefschetz fixed point formula for elliptic complexes. II: Applications**, Ann. Math. (2) 88, 451-491 (1968). ZBL0167.21703.

[2] *Miatello, Roberto J.; Podestá, Ricardo A.*, **Eta invariants and class numbers**, Pure Appl. Math. Q. 5, No. 2, 729-753 (2009). ZBL1183.58021.