No abstract

No abstract

Abstract. We address the problem of finding necessary and sufficient conditions for an arbitrary group, not necessarily finite, to admit a faithful irreducible representation over an arbitrary field.

We study hermitian forms and unitary groups defined over a local ring, not necessarily commutative, equipped with an involution. When the ring is finite we obtain formulae for the order of the unitary groups as well as their point stabilizers, and use these to compute the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring.

Abstract. For any matrix X let X denote its transpose. It is known that if A is an n-by-n matrix over a field F , then A and A are congruent over F , i.e., XAX = A for some X ∈ GLn(F ). Moreover, X can be chosen so that X 2 = In, where In is the identity matrix. An algorithm is constructed to compute such an X for a given matrix A. Consequently, a new and completely elementary proof of that result is obtained.As a by-product another interesting result is also established. Let G be a semisimple complexLie group with Lie algebra g. Let g = g 0 ⊕ g 1 be a Z 2 -gradation such that g 1 contains a Cartan subalgebra of g. Then L.V. Antonyan has shown that every G-orbit in g meets g 1 . It is shown that, in the case of the symplectic group, this assertion remains valid over an arbitrary field F of characteristic different from 2. An analog of that result is proved when the characteristic is 2.