Classical method of constructing a complete set of irreducible representations of semidirect product of a compact group with a finite group

• Autorzy: Hirai T.
• Języki publikacji: EN

Abstrakty

EN

Let G = U x S be a group of semidirect product of U compact and S finite. For an irreducible representation (= IR) ρ of U, let S([ρ]) be the stationary subgroup in S of the equivalence class [ρ] ∈ Û. Intertwining operators J_ρ(s) (s ∈ S([ρ])) between ρ and ^s−1ρ gives in general a spin (= projective) representation of S([ρ]), which is lifted up to a linear representation J′_ρ of a covering group S([ρ])′ of S([ρ]). Put π⁰:= ρ ·J′_ρ, and take a spin representation π¹ of S([ρ]) corresponding to the factor set inverse to that of J_ρ, and put Π(π⁰, π¹) = Ind^G_{U x S([ρ])} (π⁰ ▪ π¹). We give a simple proof that Π(π⁰, π¹) is irreducible and that any IR of G is equivalent to some of Π(π⁰, π¹).

Słowa kluczowe

EN

semidirect product group; construction of irreducible representations; projective representation; finite group; compact group

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2013
• Tom: Vol. 33, Fasc. 2
• Strony: 353--362
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Hirai T.

• 22-8 Nakazaichi-Cho, Iwakura, Sakyo-Ku, Kyoto 606-0027, Japan

Bibliografia

• [1] A. H. Clifford, Representations induced in an invariant subgroup, Ann. Math. 38 (1937), pp. 533-550.
• [2] T. Hirai, E. Hirai, and A. Hora, Towards projective representations and spin characters of finite and infinite complex reflection groups, in: Proceedings of the Fourth German-Japanese Symposium: Infinite Dimensional Harmonic Analysis IV, World Scientific, 2009, pp. 112-128.
• [3] T. Hirai, E. Hirai, and A. Hora, Projective representations and spin characters of complex reflection groups G(m, p, n) and G(m, p, ∞). I, MSJ Mem., Vol. 29, Math. Soc. Japan, 2013, pp. 49-122.
• [4] T. Hirai, A. Hora, and E. Hirai, Introductory expositions on projective representations of groups, MSJ Mem., Vol. 29, Math. Soc. Japan, 2013, pp. 1-47.
• [5] T. Hirai, A. Hora, and E. Hirai, Projective representations and spin characters of complex reflection groups G(m, p, n) and G(m, p, ∞). II: Case of generalized symmetric groups, MSJ Mem., Vol. 29, Math. Soc. Japan, 2013, pp. 123-272.
• [6] G. W. Mackey, Theory of Group Representations, Mimeographed Lecture Notes, University of Chicago, 1955.
• [7] G. W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), pp. 265-311.
• [8] J. Schur, Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 127 (1904), pp. 20-50.
• [9] J. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), pp. 155-255.
• [10] K. Yamazaki, On projective representations and ring extensions of finite groups, J. Fac. Sci. Univ. Tokyo, Sect. I, 10 (1964), pp. 147-195.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).