### M Theory Lesson 281

Laurent Manivel, of the CNRS, has a paper that discusses how the hyperdeterminant arises as the restriction of a quartic form for the Weyl group $W(E_7)$. Here $E_7$ means the root system of that name.

It turns out that we should be interested in a product of seven copies of $A_2$ with the automorphism group of the Fano plane. The latter group is the nice group $PSL(2, F_7)$. The hyperdeterminant is a quartic form for an eight dimensional space $A_{ijk}$ that appears in a $56$ dimensional representation of $E_7$ made of seven copies of the form $A_{ijk}$ for different $i$, $j$ and $k$. The entanglement of qudits really does have a lot of wonderful geometry associated to it.

It turns out that we should be interested in a product of seven copies of $A_2$ with the automorphism group of the Fano plane. The latter group is the nice group $PSL(2, F_7)$. The hyperdeterminant is a quartic form for an eight dimensional space $A_{ijk}$ that appears in a $56$ dimensional representation of $E_7$ made of seven copies of the form $A_{ijk}$ for different $i$, $j$ and $k$. The entanglement of qudits really does have a lot of wonderful geometry associated to it.

## 4 Comments:

This also corresponds to what Duff and Ferrara wrote about the E7 suzy black hole duality when string theory is reduced to 4 dimensions.

There should also be more interest in the similar situation with E8 where the 2x2x2x2 hyperdeterminant (degree 24) should be obtained in a similar way from invariants of degree 8 and 12 on the 248 dimensional rep. Manivel shows how this reduces to 14 2x2x2x2 hypermatrices and 8 SL(2) matrices (248 = 14*16 + 8*3) but as far as I know nobody has done the algebra to show exactly how the hyperdeterminant would fall out in this case.

It is interesting because you should get E8 when you reduce M-theory to 3 dimensions (or so I have read). In Witten's 3D theory the black hole entropy is related to the J-function which is great because the 2x2x2x2 hyperdeterminant appears in the J-invariant. Question is, how does the monster get in?

Hi Phil. I found a paper that discusses the degree 24 case, which was only fully constructed in 2007, as it has 2894276 terms!

The full 2x2x2x2 hyperdeterminant was worked out in this paper in 2006. They also looked at its Newton Polytope which is 11 dimensional :)

But as I said, nobody has worked out the corresponding E8 invariant which would use a configuration of 14 2x2x2x2 hypermatrices based on the E8 code and a Chevalley algebra for the remaining 24 dimensions. It would be big but not beyond the possibility of explicit construction.

If that is not bad enough, there may be a similar invariant which uses the Golay code. There would be 759 copies of 2x2x2x2x2x2x2x2 hypermatrices configured like the weight 8 codes and 2576 singlets for the weight 12 codes. It could be related to the Griess algebra whose automorphism group is the monster, but its existance is pure speculation on my part. The 2x2x2x2x2x2x2x2 hyperdeterminant is a polynomial of degree 60032 in 256 variables so imagine how many terms that would have.

Ah, thanks!

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