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Radicals of $$S_n$$-invariant positive semidefinite Hermitian forms. (English) Zbl 06963900
Summary: Let $$G$$ be a finite group, $$V$$ a complex permutation module for $$G$$ over a finite $$G$$-set $$\mathcal{X}$$, and $$f\colon V\times V \to \mathbb{C}$$ a $$G$$-invariant positive semidefinite hermitian form on $$V$$. In this paper we show how to compute the radical $$V^\bot$$ of $$f$$, by extending to nontransitive actions the classical combinatorial methods from the theory of association schemes. We apply this machinery to obtain a result for standard Majorana representations of the symmetric groups.
##### MSC:
 20C30 Representations of finite symmetric groups 15A63 Quadratic and bilinear forms, inner products 05E25 Group actions on posets, etc. (MSC2000) 11E39 Bilinear and Hermitian forms
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