M Theory Lesson 38

Carl Brannen (who has a new blog called Mass) likes to talk about 1-circulant matrices, which are of the form

A B C
C A B
B C A

but in M theory we also find 2-circulants

A B C
B C A
C A B

Recall that 1-circulants obey a relation PM = MP for a permutation matrix P, and similarly 2-circulants obey a relation PM = MPP. Lam [1] studied circulants with rational entries, and in particular circulants M such that M.M is of the form

(d + s) s s
s (d + s) s
s s (d + s)

For 3x3 matrices, the classification of solutions involves cubed roots of unity associated to the cyclotomic polynomial defining the field

$\frac{\mathbb{Q} [x]}{(x^3 - 1)}$

Carl Brannen's 1-circulant matrices often take the form

$A$ $B$ $B^{-1}$
$B^{-1}$ $A$ $B$
$B$ $B^{-1}$ $A$

and we observe that a sum of two such matrices, with B and its inverse interchanged, will result in a matrix of the form studied by Lam et al. Lam's results generalise to all primes p.

[1] C. W. H. Lam, Lin. Alg. and Appl. 12 (1975) 139-150