Schur Matrix

The $p\times p$ Square Matrix formed by setting $s_{ij}=\xi^{ij}$, where $\xi$ is a $p$th Root of Unity. The Schur matrix has a particularly simple Determinant given by

$$\mathop{\rm det} {\hbox{\sf S}}=\epsilon_p p^{p/2},$$

where $p$ is an Odd Prime and

$$\epsilon_p\equiv \cases{ 1 & if $p\equiv 1\ \left({{\rm mod... ...)$\cr i & if $p\equiv 3\ \left({{\rm mod\ } {4}}\right)$.\cr}$$

This determinant has been used to prove the Quadratic Reciprocity Law (Landau 1958, Vardi 1991). The Absolute Values of the Permanents of the Schur matrix of order $2p+1$ are given by 1, 3, 5, 105, 81, 6765, ... (Sloane's A003112, Vardi 1991).

Denote the Schur matrix ${\hbox{\sf S}}_p$ with the first row and first column omitted by ${\hbox{\sf S}}_p'$. Then

$$\mathop{\rm perm}{\hbox{\sf S}}_p = p\mathop{\rm perm}{\hbox{\sf S}}'_p,$$

where perm denoted the Permanent (Vardi 1991).

References

Graham, R. L. and Lehmer, D. H. ``On the Permanent of Schur's Matrix.'' J. Austral. Math. Soc. 21, 487-497, 1976.

Landau, E. Elementary Number Theory. New York: Chelsea, 1958.

Sloane, N. J. A. Sequence A003112/M2509 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 119-122 and 124, 1991.