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What do Ou and Wood mean by

Harmonic

and

Horizontally Weakly Conformal?

Let PHI be a quadratic map from Rm to Rn.

Write PHI as

PHI(X) = ( X* A1 X,  X* A2 X,  ...,   X* An X )

where X is a vector in Rm, X* is the transpose of X, and
the Ai (i=1,...,n) are symmetric mxm real matrices.

Then:

PHI is HARMONIC if and only if all the Ai are traceless,
i.e.:
tr(Ai) = 0  (i=1,...,n)

and

PHI is HORIZONTALLY WEAKLY CONFORMAL if and only if

Ai Aj  +  Aj Ai  =  0  (i,j=1,...,n; i=/=j)
and
Ai Ai  =  Aj Aj    (i,j=1,...,n)

Let PHI be an orthogonal multiplication on Rn:

PHI:  Rn x Rn  ---}  Rn

Then PHI is a Harmonic Morphism if and only if n = 1,2,4,8.

Let PHI be a Hopf construction map:

PHI:  Rn x Rn  ---}  R(n+1)

Then PHI is a Harmonic Morphism if and only if n = 1,2,4,8.

Let PHI be a Clifford System map:

PHI:  Rm x Rm = R(2m)  ---}  R(n+1)

represented by 2mx2m symmetric matrices Ai (i=0,1,...,n)

such that
Ai Aj  +  Aj Ai  =  2dij ID  (i,j=1,...,n)

where dij is the Kronecker delta and ID is the 2mx2m identity matrix.

Then PHI is a Harmonic Morphism,
and the Clifford System is irreducible for the following m and n:

m              n
1              1
2              2
4              3
4              4
8              5
8              6
8              7
8              8
...            ...
16m            n+8

Sigmundur Gudmundsson and Stefano Montaldo have
a WWW site, THE  ATLAS  OF  HARMONIC  MORPHISMS,
and Sigmundur Gudmundsson is also editor
of THE HARMONIC MORPHISMS BIBLIOGRAPHY.

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