The discriminant of symmetric matrices as a sum of squares and the orthogonal group

Details The discriminant of symmetric matrices as a sum of squares and the orthogonal group The discriminant of symmetric matrices as a sum of squares and the orthogonal group

2010-03-01

arxiv.org/abs/1003.0475v1

Mathematics

Representation Theory

26 pages

Scientific paper

It is proved that the discriminant of $n\times n$ real symmetric matrices can be written as a sum of squares, where the number of summands equals the dimension of the space of $n$-variable spherical harmonics of degree $n$. The discriminant of three by three real symmetric matrices is explicitly presented as a sum of five squares, and it is shown that the discriminant of four by four real symmetric matrices can be written as a sum of seven squares. These results improve theorems of Kummer from 1843 and Borchardt from 1846.

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