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In [Indiana Univ. Math. J. 60 (2011), pp. 847-857] the first author introduced second order necessary conditions for a commuting square to admit sequential deformations in the moduli space of non-isomorphic commuting squares. In this paper we investigate these conditions for commuting squares CG constructed from finite groups G. We are especially interested in the case G = Zn, since deformations of CZn correspond to deformations of the Fourier matrix Fn in the moduli space of non-equivalent complex Hadamard matrices. We show that for G = Zn the second order conditions follow automatically from the first order conditions, but this is not necessarily true for other finite abelian groups G. Our result gives a complete description of the second order deformations of the Fourier matrix Fn in the moduli space of non-equivalent complex Hadamard matrices. [ABSTRACT FROM AUTHOR]

Let \mathbf {G} be the General Linear or Special Linear group with entries from the finite quotients of the ring of integers of a non-archimedean local field and \mathbf {U} be the subgroup of \mathbf {G} consisting of upper triangular unipotent matrices. We prove that the induced representation \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta) of \mathbf {G} obtained from a non-degenerate character \theta of \mathbf {U} is multiplicity free for all \ell \geq 2. This is analogous to the multiplicity one theorem regarding Gelfand-Graev representation for the finite Chevalley groups. We prove that for many cases the regular representations of \mathbf {G} are characterized by the property that these are the constituents of the induced representation \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta) for some non-degenerate character \theta of \mathbf {U}. We use this to prove that the restriction of a regular representation of General Linear groups to the Special Linear groups is multiplicity free and also obtain the corresponding branching rules in many cases. [ABSTRACT FROM AUTHOR]