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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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