Algebr Represent Theor (2014) 17:199–212 DOI 10.1007/s10468-012-9394-7 Semi-invariant Matrices over Finite Groups Yuval Ginosar · Ofir Schnabel Received: 11 July 2012 / Accepted: 12 November 2012 / Published online: 24 November 2012 © Springer Science+Business Media Dordrecht 2012 Abstract The semi-center of an artinian semisimple module-algebra over a finite group G can be described using the projective representations of G. In particular, the semi-center of the endomorphism ring of an irreducible projective representation over an algebraically closed field has a structure of a twisted group algebra. The following group-theoretic result is deduced: the center of a group of central type embeds into the group of its linear characters. Keywords Semi-invariants · Projective representation · Twisted group algebra Mathematics Subject Classifications (2010) 16S35 · 16W22 · 20C25 1 Introduction Let A be a K-algebra and let G be a group. A group homomorphism from G to the K-algebra automorphisms of A furnishes A with a G-module-algebra structure. Throughout this note we assume that G is a finite group whose order is prime to char( K). This non-modularity condition is needed for a complete decomposition of A into irreducible G-representations. An interesting goal is to understand this decomposition. The properties of the G-trivial constituent of A is the concern of the classical invariant theory. More generally, the 1-dimensional G-modules in A, namely its semi-invariants, are investigated mainly with regard to enveloping algebras of finite dimensional Lie algebras, see, e.g. [2, 4, 7, 9, 12, 13]. Presented by: Alain Verschoren Y. Ginosar (B ) · O. Schnabel Department of Mathematics, University of Haifa, Haifa 31905, Israel e-mail: ginosar@math.haifa.ac.il O. Schnabel e-mail: os2519@yahoo.com