zbMATH — the first resource for mathematics

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.
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
Full Text: DOI
[1] Bannai, Eiichi; Ito, Tatsuro, Algebraic combinatorics. I: Association schemes, xxiv+425 pp., (1984), The Benjamin/Cummings Publishing Company · Zbl 0555.05019
[2] Castillo-Ramirez, Alonso; Ivanov, Alexander A., Groups of exceptional type, Coxeter groups and related geometries (Bangalore, 2012), 82, The axes of a Majorana representation of \(A_{12}\), 159-188, (2014), Springer · Zbl 1341.20010
[3] Franchi, Clara; Ivanov, Alexander A.; Mainardis, Mario, Standard Majorana representations of the symmetric groups, J. Algebr. Comb., 44, 2, 265-292, (2016) · Zbl 1351.05228
[4] Franchi, Clara; Ivanov, Alexander A.; Mainardis, Mario, The 2A-Majorana representations of the harada-norton group, Ars Math. Contemp., 11, 1, 175-187, (2016) · Zbl 1394.20009
[5] Higman, Donald G., Coherent configurations. I: ordinary representation theory, Geom. Dedicata, 4, 1-32, (1975) · Zbl 0333.05010
[6] Isaacs, I. Martin, Character theory of finite groups, xii+303 pp., (1994), Dover Publications · Zbl 0849.20004
[7] Ivanov, Alexander A., The Monster group and Majorana involutions, 176, xiii+252 pp., (2009), Cambridge University Press · Zbl 1205.20014
[8] Ivanov, Alexander A., On Majorana representations of \(A _6\) and \(A _7\), Commun. Math. Phys., 307, 1, 1-16, (2011) · Zbl 1226.17023
[9] Ivanov, Alexander A.; Pasechnik, Dmitrii V.; Seress, Ákos; Shpectorov, Sergey V., Majorana representations of the symmetric group of degree 4, J. Algebra, 324, 9, 2432-2463, (2010) · Zbl 1257.20011
[10] Ivanov, Alexander A.; Seress, Ákos, Majorana representations of \(A_5\), Math. Z., 272, 1-2, 269-295, (2012) · Zbl 1260.20019
[11] James, Gordon D., The representation theory of the symmetric groups, 682, (1978), Springer · Zbl 0393.20009
[12] Lang, Serge, Algebra, 211, xv+914 pp., (2002), Springer · Zbl 0984.00001
[13] Norton, Simon P., F and other simple groups, (1975)
[14] Norton, Simon P., Moonshine, the monster, and related topics. Joint summer research conference on moonshine, the monster, and related topics (Mount Holyoke College, 1994), 193, The monster algebra: some new formulae, 297-306, (1996), American Mathematical Society · Zbl 0847.11023
[15] Serre, Jean-Pierre, Linear representations of finite groups, 42, x+170 pp., (1977), Springer · Zbl 0355.20006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.