Strongly real classes in finite unitary groups of odd characteristic

Abstract

Abstract. We classify all strongly real conjugacy classes of the finite unitary group U( n ,𝔽 q ) when q is odd. In particular, we show that g ∈ U( n ,𝔽 q ) is strongly real if and only if g is an element of some embedded orthogonal group O ± ( n ,𝔽 q ). Equivalently, g is strongly real in U( n ,𝔽 q ) if and only if g is real and every elementary divisor of g of the form ( t ± 1) 2 m has even multiplicity. We apply this to obtain partial results on strongly real classes in the finite symplectic group Sp(2 n ,𝔽 q ), q odd, and a generating function for the number of strongly real classes in U( n ,𝔽 q ), q odd, and we also give partial results on strongly real classes in U( n ,𝔽 q ) when q is even.

Locations

• Journal of Group Theory - View - PDF
• arXiv (Cornell University) - View - PDF