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We study closures of conjugacy classes in the symmetric matrices of theorthogonal group and we determine which one are normal varieties. In contrastto the result for the symplectic group where all classes have normal closure,there is only a relatively small portion of classes with normal closure. Weperform a combinatorial computation on top of the same methods used byKraft-Procesi and Ohta.

Normality of closure of orthogonal nilpotent symmetric orbits