Small interactions in the groups $\mathrm{Sp}_4(q)$ for even $q$

V. A. Belonogov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 4, pp. 19-37

Résumé

The investigation of nonclassical interactions (orthogonality relations) in finite groups is continued. We obtain a description of all small interactions (small $D$-blocks and small $\Phi$-blocks) in the symplectic groups $\mathrm{Sp}_4(q)$ for even $q$. In particular, two known conjectures on these interactions are confirmed for these groups. Note that the existence of a small $D$-block in a group is equivalent to the existence of a pair of semiproportional rows in its character table and the existence of a small $\Phi$-block in a group is equivalent to the existence of a pair of semiproportional columns in its character table.

Détail

Keywords: finite symplectic groups, character table, small interactions, semiproportional characters.

Bibliographie

[1] Belonogov V. A., “$D$-bloki kharakterov konechnoi gruppy”, Issledovaniya po teorii grupp, UrO AN SSSR, Sverdlovsk, 1984, 3–31 ; Belonogov V. A., “$D$-blocks of characters of finite group”, Amer. Math. Soc. Transl. (2), 143 (1989), 103–128 | MR

[2] Belonogov V. A., Predstavleniya i kharaktery v teorii konechnykh grupp, UrO AN SSSR, Sverdlovsk, 1990, 380 pp. | MR

[3] Belonogov V. A., “Vzaimodeistviya i $D$-bloki v konechnykh gruppakh”, Podgruppovaya struktura grupp, UrO AN SSSR, Sverdlovsk, 1988, 4–44 | MR

[4] Belonogov V. A., “O malykh vzaimodeistviyakh v konechnykh gruppakh”, Trudy Instituta matematiki i mekhaniki UrO RAN, 2, 1992, 3–18 | MR | Zbl

[5] Belonogov V. A., “Malye vzaimodeistviya v gruppakh $\mathrm{GL}_3(q)$, $\mathrm{GU}_3(q)$, $\mathrm{PGL}_3(q)$ i $\mathrm{PGU}_3(q)$”, Trudy Instituta matematiki i mekhaniki UrO RAN, 4, 1996, 17–47 | MR | Zbl

[6] Belonogov V. A., “Malye vzaimodeistviya v gruppakh $\mathrm{SL}_3(q)$, $\mathrm{SU}_3(q)$, $\mathrm{PSL}_3(q)$ i $\mathrm{PSU}_3(q)$”, Trudy Instituta matematiki i mekhaniki UrO RAN, 5, 1998, 3–27 | Zbl

[7] Belonogov V. A., “O neprivodimykh kharakterakh grupp $S_n$ i $A_n$”, Sib. mat. zhurn., 45:5 (2004), 977–994 | MR | Zbl

[8] Belonogov V. A., “O neprivodimykh kharakterakh gruppy $S_n$, poluproportsionalnykh na $A_n$ ili na $S_n\setminus A_n$. VII”, Trudy Instituta matematiki i mekhaniki UrO RAN, 17, no. 1, 2011, 3–16

[9] J. H. Conway, R. T. Curtis, S. P. Norten, R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985, 252 pp. | MR | Zbl

[10] Belonogov V. A., “K gipoteze o poluproportsionalnykh kharakterakh”, Sib. mat. zhurn., 46:2 (2005), 299–314 | MR | Zbl

[11] Belonogov V. A., “Novyi metod vychisleniya $p$-blokov”, Trudy Instituta matematiki i mekhaniki UrO RAN, 1, 1992, 3–12 | MR | Zbl

[12] Enomoto H., “The Characters of the finite symplectic group $\mathrm{Sp}(4,q)$, $q=2^f$”, Osaka J. Math., 9 (1972), 75–94 | MR | Zbl

[13] Lam T. Y., Leung K. H., “On vanishing sums of roots of unit”, J. Algebra, 224 (2000), 91–109 | DOI | MR | Zbl