On the center of a compact group

Details On the center of a compact group On the center of a compact group

2003-12-12

arxiv.org/abs/math/0312257v2

Int. Math. Res. Notices 2004, no. 51, 2751-2756

Mathematics

Group Theory

Some improvements of terminology. Final version, to appear in I.M.R.N. latex2e, 6 pages, uses diagrams.tex

Scientific paper

We prove a conjecture due to Baumgaertel and Lledo according to which for every compact group G one has Z(G)^ \cong C(G), where the `chain group' C(G) is the free abelian group (written multiplicatively) generated by the set G^ of isomorphism classes of irreducible representations of G modulo the relations [Z]=[X]\cdot[Y] whenever Z is contained in X \otimes Y. Thus the center Z(G) depends only on the representation ring of G. Furthermore, we prove that every `t-map' phi: G^ -> A into an abelian group, i.e. every map satisfying phi(Z)=phi(X)phi(Y) whenever X,Y,Z in G^ and Z\prec X\otimes Y, factors through the restriction map G^ -> Z(G)^. All these results also hold for proalgebraic groups over algebraically closed fields of characteristic zero.

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