-7

I have cross-posted the question

and

It seems feasible that the new random matrices features in Version 11 might facilitate addressing this question. Any specific thoughts in this matter? One of the new commands is CircularQuaternionMatrixDistribution. Can one create an octonionic counterpart?

The two formulas $P_1(\alpha)$ and $P_2(\alpha)$ were developed based solely on analyses of matrices with real and complex (and not quaternionic and octonionic) entries. To be more specific, the ascending moments of determinants (emphasis added) of the 4 x 4 matrices and of their “partial transposes” were computed, and formulas found for them. (These were, then, used in the Mathematica density approximation procedure of Provost [http://www.mathematica-journal.com/issue/v9i4/contents/DensityApproximants/DensityApproximants.pdf], to eventually arrive at the expressions for $P_1(\alpha)$ and $P_2(\alpha)$ .)

The two formulas (Charles Dunkl observed) could be absorbed into one, by regarding the parameter in the complex case to be twice that in the real case (hence the apparent [Dyson-index-like] connection to random matrix theory).

Now, although the calculation of determinants is straightforward with matrices the entries of which are restricted to real and complex values, it becomes more subtle with the quaternions, and, a fortiori, it would seem with the octonions. E. H. Moore (Bull. Amer. Math. Soc. 28 [1922], 161-162) gave a definition in the quaternionic case—and Wikipedia has a brief article, I see, about the “Dieudonne determinant” (“which is a generalization of the determinant of a matrix over division rings and local rings”). So, I think the originally stated problem I posed hinges on to what extent the formula Dunkl developed can be “extrapolated” to the octonionic domain. (I note, however, that Fei and Joynt in the cited paper appear to have by-passed the use of determinants, in their quaternionic analysis).

In preparing this “answer”, I found a (rather remarkable) series of (unpublished) June 2012 emails from Dunkl in which he does a highly in-depth [using Maple] analysis of the use of the Moore determinant in the quaternionic case, apparently succeeding in confirming its appropriateness. Here is part of his treatment (this, of course, deal with the quaternionic scenario, and the original octonionic question remains). Dunkl writes more of interest in this series of detailed emails (but I don’t see how to really present his remarks here).

The Maple code Dunkl employed for the application of the Moore determinant to the quaternionic case was:

qm := proc (z1, z2) local zq1, zq2, w1, w2, w3, w4; global qco, iq, jq, kq;
zq1 := qco(z1); zq2 := qco(z2); w1 := zq1[1]*zq2[1]-zq1[2]*zq2[2]-zq1[3]*zq2[3
]-zq1[4]*zq2[4]; w2 := zq1[1]*zq2[2]+zq1[2]*zq2[1]+zq1[3]*zq2[4]-zq1[4]*zq2[3]
; w3 := zq1[1]*zq2[3]+zq1[3]*zq2[1]+zq1[4]*zq2[2]-zq1[2]*zq2[4]; w4 := zq1[1]*
zq2[4]+zq1[4]*zq2[1]+zq1[2]*zq2[3]-zq1[3]*zq2[2]; w1+w2*iq+w3*jq+w4*kq end
proc;
qconj := proc (f) options operator, arrow; subs({kq = -kq, iq = -iq, jq = -jq}
,f) end proc;
qco := proc (f) local fq; global iq, jq, kq, lq; fq := collect(f,[iq, jq, kq])
; lq[2] := coeff(fq,iq); lq[3] := coeff(fq,jq); lq[4] := coeff(fq,kq); lq[1]
:= subs(iq = 0,jq = 0,kq = 0,f); [lq[1], lq[2], lq[3], lq[4]] end proc;
qdet4x := [[[1, 1], [2, 2], [3, 3], [4, 4], 1], [[4, 4], [3, 3], [1, 2], [2, 1
], -1], [[4, 4], [2, 3], [3, 2], [1, 1], -1], [[4, 4], [2, 2], [1, 3], [3, 1],
-1], [[3, 3], [2, 4], [4, 2], [1, 1], -1], [[3, 4], [4, 3], [2, 2], [1, 1], -1
], [[3, 4], [4, 3], [1, 2], [2, 1], 1], [[2, 3], [3, 2], [1, 4], [4, 1], 1], [
[2, 4], [4, 2], [1, 3], [3, 1], 1], [[4, 4], [1, 2], [2, 3], [3, 1], -1], [[4,
4], [1, 3], [3, 2], [2, 1], -1], [[3, 3], [1, 4], [4, 2], [2, 1], -1], [[2, 2]
, [1, 3], [3, 4], [4, 1], -1], [[2, 2], [1, 4], [4, 3], [3, 1], -1], [[2, 3],
[3, 4], [4, 2], [1, 1], -1], [[2, 4], [4, 3], [3, 2], [1, 1], -1], [[1, 2], [2
, 3], [3, 4], [4, 1], 1], [[1, 2], [2, 4], [4, 3], [3, 1], 1], [[1, 3], [3, 2]
, [2, 4], [4, 1], 1], [[1, 3], [3, 4], [4, 2], [2, 1], 1], [[1, 4], [4, 2], [2
, 3], [3, 1], 1], [[1, 4], [4, 3], [3, 2], [2, 1], 1], [[3, 3], [2, 2], [1, 4]
, [4, 1], -1], [[3, 3], [1, 2], [2, 4], [4, 1], -1]];
qmdet4 := proc (mx) local dt, i, tm, ppq; global qm, qdet4x; dt := 0; for i to
24 do tm := op(i,qdet4x); ppq := qm(mx[tm[1][1],tm[1][2]],qm(mx[tm[2][1],tm[2]
[2]],qm(mx[tm[3][1],tm[3][2]],mx[tm[4][1],tm[4][2]]))); dt := dt+tm[5]*
simplify(ppq) end do; simplify(dt) end proc;

Also, here is the list of the 24 (4!) factors of the Moore determinant, in order, with the sign:

# this is a list of the 24 factors, in order, with the sign

qdet4x;

[[[1, 1], [2, 2], [3, 3], [4, 4], 1],

```
[[4, 4], [3, 3], [1, 2], [2, 1], -1],
[[4, 4], [2, 3], [3, 2], [1, 1], -1],
[[4, 4], [2, 2], [1, 3], [3, 1], -1],
[[3, 3], [2, 4], [4, 2], [1, 1], -1],
[[3, 4], [4, 3], [2, 2], [1, 1], -1],
[[3, 4], [4, 3], [1, 2], [2, 1], 1],
[[2, 3], [3, 2], [1, 4], [4, 1], 1],
[[2, 4], [4, 2], [1, 3], [3, 1], 1],
[[4, 4], [1, 2], [2, 3], [3, 1], -1],
[[4, 4], [1, 3], [3, 2], [2, 1], -1],
[[3, 3], [1, 4], [4, 2], [2, 1], -1],
[[2, 2], [1, 3], [3, 4], [4, 1], -1],
[[2, 2], [1, 4], [4, 3], [3, 1], -1],
[[2, 3], [3, 4], [4, 2], [1, 1], -1],
[[2, 4], [4, 3], [3, 2], [1, 1], -1],
[[1, 2], [2, 3], [3, 4], [4, 1], 1],
[[1, 2], [2, 4], [4, 3], [3, 1], 1],
[[1, 3], [3, 2], [2, 4], [4, 1], 1],
[[1, 3], [3, 4], [4, 2], [2, 1], 1],
[[1, 4], [4, 2], [2, 3], [3, 1], 1],
[[1, 4], [4, 3], [3, 2], [2, 1], 1],
[[3, 3], [2, 2], [1, 4], [4, 1], -1],
[[3, 3], [1, 2], [2, 4], [4, 1], -1]]
```

# this adds the 24 terms to get the Moore determinant

print(qmdet4);

proc(mx) local dt, i, tm, ppq; global qm, qdet4x; dt := 0; for i to 24 do tm := op(i, qdet4x); ppq := qm(mx[tm[1][1], tm[1][2]], qm( mx[tm[2][1], tm[2][2]], qm( mx[tm[3][1], tm[3][2]], mx[tm[4][1], tm[4][2]]))) ; dt := dt + tm[5]*simplify(ppq) end do; simplify(dt) end proc

# here is a PT of a matrix with 4 off-diagonal entries nonzero,

# a14,a23,a41,a32

mtqp;

```
[ 2
[h[1] , 0 , 0 ,
]
h[2] g2[0] + h[2] g2[1] iq + h[2] g2[2] jq + h[2] g2[3] kq]
[ 2
[0 , h[2] ,
h[1] g1[0] + h[1] g1[1] iq + h[1] g1[2] jq + h[1] g1[3] kq ,
]
0]
[
[0 ,
h[1] g1[0] - h[1] g1[1] iq - h[1] g1[2] jq - h[1] g1[3] kq ,
2 2 2 2 2 ]
g2[0] + g2[1] + g2[2] + g2[3] + h[3] , 0]
[
[h[2] g2[0] - h[2] g2[1] iq - h[2] g2[2] jq - h[2] g2[3] kq ,
2 2 2 2 2]
0 , 0 , g1[0] + g1[1] + g1[2] + g1[3] + h[4] ]
```

# the Cholesky factor is [[h1,0,0,g1],[0,h2,g2,0],[0,0,h3,0],[0,0,0,h4]

# where g1,g2 are quaternions

# this is the PT det (Moore formula)

det4;

```
2 2 2 2 2 2 2 2
```

-(-h[2] g2[0] - h[2] g2[1] - h[2] g2[3] - h[2] g2[2]

```
2 2 2 2 2 2 2 2
+ h[1] g1[0] + h[1] g1[1] + h[1] g1[2] + h[1] g1[3]
2 2 2 2 2 2 2 2
+ h[4] h[1] ) (h[1] g1[3] + h[1] g1[0] + h[1] g1[1]
2 2 2 2 2 2 2 2
+ h[1] g1[2] - h[2] h[3] - h[2] g2[0] - h[2] g2[1]
2 2 2 2
- h[2] g2[2] - h[2] g2[3] )
```

1Maybe you could describe which random matrix functions you mean, and how they might apply? – bill s – 2016-11-14T03:19:58.917

Thanks, bill s. Peter Forrester posted (https://arxiv.org/abs/1610.08081) a paper, "Octonions in Random Matrix Theory", outlining certain difficulties in dealing with a theory of Hermitian random matrices with octonion entries. The case N=3 is problematical, while the conjecture in question deals with the even more challenging case N=4. However, there seems to must be some way of addressing the question that quite naturally arises from application of the formulas I give. So, I want/need 4 x 4 Hermitian random matrices with octonion entries (and unit trace).

– Paul B. Slater – 2016-11-14T05:07:02.7231What is the purpose of posting this and your self-answer on three different sites? Please format your code snippets correctly, and make sure that the code is correct and complete. – Yves Klett – 2016-11-26T22:16:41.693

Thanks, Yves Klett. The problem seemed to have mathematical/physical/Mathematica aspects. Sorry, for possibly over-reaching here. – Paul B. Slater – 2016-11-29T12:33:11.257

I have expanded my answer into "Octonionic separability probability conjectures" (https://arxiv.org/abs/1612.02798)

– Paul B. Slater – 2016-12-12T03:19:39.583