# ALGEBRAIC COMBINATORICS

Radicals of ${S}_{n}$-invariant positive semidefinite hermitian forms
Algebraic Combinatorics, Volume 1 (2018) no. 4, p. 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 : 2018-06-04
Accepted : 2018-06-10
Published online : 2018-09-10
DOI : https://doi.org/10.5802/alco.24
Classification:  20C30,  15A63,  05E25,  11E39
Keywords: Hermitian form, Symmetric group, Majorana representation, Monster group, Association scheme, Specht module.
@article{ALCO_2018__1_4_425_0,
author = {Franchi, Clara and Ivanov, Alexander A. and Mainardis, Mario},
title = {Radicals of $S\_n$-invariant positive semidefinite hermitian forms},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
volume = {1},
number = {4},
year = {2018},
pages = {425-440},
doi = {10.5802/alco.24},
zbl = {06963900},
language = {en},
url = {https://alco.centre-mersenne.org/item/ALCO_2018__1_4_425_0}
}

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/item/ALCO_2018__1_4_425_0/

[1] Bannai, Eiichi; Ito, Tatsuro Algebraic combinatorics. I: Association schemes, The Benjamin/Cummings Publishing Company, Mathematics Lecture Note Series (1984), xxiv+425 pages | Zbl 0555.05019

[2] Castillo-Ramirez, Alonso; Ivanov, Alexander A. The axes of a Majorana representation of ${A}_{12}$, Groups of exceptional type, Coxeter groups and related geometries (Bangalore, 2012), Springer (Springer Proceedings in Mathematics & Statistics) Volume 82 (2014), pp. 159-188 | Article | MR 3207276 | Zbl 1341.20010

[3] Franchi, Clara; Ivanov, Alexander A.; Mainardis, Mario Standard Majorana representations of the symmetric groups, J. Algebr. Comb., Volume 44 (2016) no. 2, pp. 265-292 | Article | MR 3533555 | Zbl 1351.05228

[4] Franchi, Clara; Ivanov, Alexander A.; Mainardis, Mario The 2A-Majorana representations of the Harada-Norton group, Ars Math. Contemp., Volume 11 (2016) no. 1, pp. 175-187 | MR 3546657 | Zbl 06659095

[5] Higman, Donald G. Coherent configurations. I: Ordinary representation theory, Geom. Dedicata, Volume 4 (1975), pp. 1-32 | Article | MR 398868 | Zbl 0333.05010

[6] Isaacs, I. Martin Character theory of finite groups, Dover Publications (1994), xii+303 pages | MR 1280461 | Zbl 0849.20004

[7] Ivanov, Alexander A. The Monster group and Majorana involutions, Cambridge University Press, Cambridge Tracts in Mathematics, Volume 176 (2009), xiii+252 pages | MR 2503090 | Zbl 1205.20014

[8] Ivanov, Alexander A. On Majorana representations of ${A}_{6}$ and ${A}_{7}$, Commun. Math. Phys., Volume 307 (2011) no. 1, pp. 1-16 | Article | 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, Volume 324 (2010) no. 9, pp. 2432-2463 | Article | MR 2684148 | Zbl 1257.20011

[10] Ivanov, Alexander A.; Seress, Ákos Majorana representations of ${A}_{5}$, Math. Z., Volume 272 (2012) no. 1-2, pp. 269-295 | Article | MR 2968225 | Zbl 1260.20019

[11] James, Gordon D. The representation theory of the symmetric groups, Springer Volume 682 (1978) | MR 513828 | Zbl 0393.20009

[12] Lang, Serge Algebra, Springer, Graduate Texts in Mathematics, Volume 211 (2002), xv+914 pages | Zbl 0984.00001

[13] Norton, Simon P. F and other simple groups, University of Cambridge (UK) (1975) (Ph. D. Thesis)

[14] Norton, Simon P. The Monster algebra: Some new formulae, Moonshine, the monster, and related topics. Joint summer research conference on moonshine, the monster, and related topics (Mount Holyoke College, 1994), American Mathematical Society (Contemporary Mathematics) Volume 193 (1996), pp. 297-306 | MR 1372728 | Zbl 0847.11023

[15] Serre, Jean-Pierre Linear representations of finite groups, Springer, Graduate Texts in Mathematics, Volume 42 (1977), x+170 pages | MR 450380 | Zbl 0355.20006