### Eleven

This paper by Kostant (a great mathematician who is currently busy giving seminars on Lisi's E8) mentions the $11 \times 11$ circulant matrix (associated to a $12 \times 12$ Hadamard matrix) with first row

$1, −1 ,−1, 1, −1, −1, −1, 1, 1, 1, −1$

The paper looks at $PSL(2,11)$ for the finite field $\mathbb{F}_{11}$, which is a little larger than the 5 or 7 that we are used to, but also interesting. One can make a planar geometry for this group, like the Fano plane, but with 11 lines.

$1, −1 ,−1, 1, −1, −1, −1, 1, 1, 1, −1$

The paper looks at $PSL(2,11)$ for the finite field $\mathbb{F}_{11}$, which is a little larger than the 5 or 7 that we are used to, but also interesting. One can make a planar geometry for this group, like the Fano plane, but with 11 lines.

## 7 Comments:

That's quite intriguing. I knocked up another java app and found that you can get circulant matrices whose elements are 1 and -1 for sizes N = 3,7,11,15,19,23,31.

I expect that people have looked a lot further than this because they have searched for Hadamard matrices up to much larger sizes, but is there any general theory about such circulants?

Kostant's paper says:

"... for the three exceptional cases ... one has ...

PSl(2; 5) = A4 Z5

PSl(2; 7) = S4 Z7

PSl(2; 11) = A5 Z11

... precisely the symmetry groups of the Platonic solids. ...".

However,

there is also the Platonic connection related to Fano-type diagrams:

5 = 4 vertices of tetrahedron + 1 center

7 = 6 vertices of octahedron + 1 center

13 = 12 vertices of icosahedron + 1 center

so,

it seems to me that something is breaking down for the icosahedron

which seems to correspond both to

PSL(2,11) by PSL(2,11) = A5 Z11

and to

PSL(2,13) by 12 icosahedron vertices + 1 center

I see that Kostant sort of avoids the break-down by working geometrically with 11 = 12 - 1

instead of 13 = 12 + 1

but I would like to know:

Why does the icosahedron seem to be related to

both PSL(2,11) and PSL(2,13) ?

Is it significant in any way ?

Tony Smith

PS - The order of PSL(2,13) is 1092 = 84 x 13

and

2x84 = 168 = order of PSL(2,7).

Hi Phil and Tony. Good questions! I like thinking about 11 = 12-1 as an analogue of 5 = 6-1, and 7 as a prime in between that comes along for the ride. I find modular geometry far more fascinating than Platonic solids by themselves.

Kea, you say that you "... like thinking about 11 = 12-1 as an analogue of 5 = 6-1, and 7 as a prime in between that comes along for the ride. ...".

If that line of thought involves Hadamard matrices,

then

how about 7 as 8-1 where 8x8 Hadamard matrices describe the Octonions?

Tony Smith

PS - Dorit Aharonov in quant-ph/0301040 said

"... Hadamard is all that one needs to add to classical computations in order to achieve the full quantum computation power; It perhaps explains the important role that the Hadamard gate plays in quantum algorithms, and can be interpreted as saying that Fourier transform is really all there is to quantum computation on top of classical, since the Hadamard gate is the Fourier transform over the group Z2

...

The Toffoli gate T can perform exactly all classical reversible computation

...

Toffoli and Hadamard are universal for quantum computation. This is perhaps the simplest universal set of gates that one can hope for ... The fact that {T,H} is universal has philosophical interpretations

...

the set {T,H} ... generates a dense subgroup in the group of orthogonal matrices ...".

Note that orthogonal matrices are related to the Bn and Dn Lie algebras and therefore related to Clifford algebras.

Tony, I'm glad you can appreciate a connection with Clifford algebra, but I am focusing on computational/algorithmic maths, which is why I am indeed interested in Hadamards! By the same token, the octonions are secondary to whatever higher categorical invariant uses such matrices.

I'm probably being a bit slow on the uptake here but now I realise from the paper that the circulant matrices are generated using quadratic residues. So they exist whenever N is a prime which is 3 (mod 4). Then the case N = 23 is beased on the binary Golay code, but I suppose it does not fit in with the rest of the paper so it does not get a mention.

It is curious that N=15 also works, so some non-primes are OK too.

philg: Circulant matrices with elements -1 and 1 that give rise to matrices with orthogonal rows are equivalent to certain difference sets. Quite a bit of theory has been developed about these, but a lot remains unknown. The La Jolla Difference Set Repository contains a wealth of useful information. The size 15 example you found exists because 15 is one less than a power of two. (You will find it listed in the La Jolla database as the difference set with parameters v=15, k=7, lambda=3. The difference set itself can be thought of as indicating the positions of the +1 elements.) The general construction for 2^n-1 is not related to the quadratic residue construction.

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