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It is proved that the discriminant of $n\times n$ real symmetric matrices canbe written as a sum of squares, where the number of summands equals thedimension of the space of $n$-variable spherical harmonics of degree $n$. Therepresentation theory of the orthogonal group is applied to express thediscriminant of three by three real symmetric matrices as a sum of fivesquares, and to show that it can not be written as the sum of less than fivesquares. It is proved that the discriminant of four by four real symmetricmatrices can be written as a sum of seven squares. These improve results ofKummer from 1843 and Borchardt from 1846.

Discriminant of symmetric matrices as a sum of squares and the orthogonal group