# ALGEBRAIC COMBINATORICS

Radicals of ${S}_{n}$-invariant positive semidefinite hermitian forms
Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 425-440.

Let $G$ be a finite group, $V$ a complex permutation module for $G$ over a finite $G$-set $𝒳$, and $f:V×V\to ℂ$ a $G$-invariant positive semidefinite hermitian form on $V$. In this paper we show how to compute the radical ${V}^{\perp }$ 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.24
Classification: 20C30,  15A63,  05E25,  11E39
Keywords: Hermitian form, Symmetric group, Majorana representation, Monster group, Association scheme, Specht module.
Franchi, Clara 1; Ivanov, Alexander A. 2; Mainardis, Mario 3

1 Dipartimento di Matematica e Fisica Università Cattolica del Sacro Cuore Via Musei 41 I-25121 Brescia, Italy
2 Department of Mathematics Imperial College 180 Qeen’s Gt., London SW7 2AZ, UK
3 Dipartimento di Scienze Matematiche, Informatiche e Fisiche Università degli Studi di Udine via delle Scienze 206 I-33100 Udine, Italy
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Franchi, Clara; Ivanov, Alexander A.; Mainardis, Mario. Radicals of $S_n$-invariant positive semidefinite hermitian forms. Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 425-440. doi : 10.5802/alco.24. https://alco.centre-mersenne.org/articles/10.5802/alco.24/

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