\titleofpresentation{Asymptotic representation theory\\
of finite groups:\\ Evolution of a representation theory\\
 for locally finite groups?}

\presenter{Felix Leinen}  

In the early 1990s, {\sc A.\,E.\,Zalesskii} discovered a neat and 
natural way to generalize notions from the representation theory of finite groups to
locally finite groups -- see his survey \cite{Leinen15} in the notes of the Istanbul Conference.  
\par
For a locally finite group $G$, the role of representations is taken by so-called 
{\em inductive systems}, that is, by families of sets $\Phi_F$ of irreducible representations 
of the finite subgroups $F$ of $G$ with the compatability condition that $\Phi_{F_1}$    
be precisely the set of irreducible constituents of the restrictions to 
$F_1$ of the representations in $\Phi_{F_2}$ whenever $F_1\le F_2$. 
Clearly, inductive systems reflect the asymptotic behaviour of the 
representations of the finite subgroups of $G$. And {\sc A.\,E.\,Zalesskii} proved 
that the ideals in the 
complex group algebra of $G$ are in 1-1-correspondence with the inductive systems of 
complex representations of the finite subgroups of $G$. 
\par
Further, for our locally finite group $G$, the role of characters is taken by {\em normalized positive definite 
class functions} \,$\mathbb CG\longrightarrow\mathbb C$, \,which are also studied in the theory
of $C^\star$-algebras. Due to a theorem by {\sc A.\,M.\,Vershik} and {\sc S.\,V.\,Kerov} \cite{Leinen14}, 
convexly indecomposable such functions (the ``irreducible characters'' of $G$) arise as 
direct limits of irreducible normalized characters of the finite subgroups of $G$. 
Every normalized positive definite class function gives rise to an ideal in $\mathbb CG$;
\,the converse fails. 
\par
In principle, the above connections open the possibility to generalize classical representation
theory to locally finite groups. But in practice, it is very hard to 
understand the immersion of the finite subgroups in a given locally finite group $G$ and their 
effect on the branching rules for irreducible characters. 
At least, this information is available when the group $G$ is built up from a well-understood 
family of finite subgroups in a systematic uniform way. This happens for example with many simple 
locally finite groups.
\par
First results about positive definite class functions for simple locally finite groups 
were obtained by {\sc E.\,Thoma} \cite{Leinen13} and {\sc H.-L.\,Skudlarek} \cite{Leinen12} 
more than 30 years ago. Around that time, {\sc K.\,Bonvallet}, {\sc B.\,Hartley}, 
{\sc D.\,S.\,Passman}, and {\sc M.\,K.\,Smith} \cite{Leinen01} observed that
the complex group algebra of {\sc P.\,Hall}'s universal locally finite group {\sl ULF} admits only the 
three unavoidable ideals. 
\par
In the early 1990s, {\sc A.\,E.\,Zalesskii} \cite{Leinen16} and {\sc B.\,Hartley} 
\cite{Leinen05, Leinen04, Leinen03} pursued the topic further. 
This was the starting point for a series of papers 
\cite{Leinen11, Leinen10, Leinen09, Leinen08, Leinen07, Leinen06} by {\sc O.\,Puglisi} 
and {\sc F.\,Leinen} with the aim to settle the situation completely for simple locally finite groups.
Our work relies on {\sc S.\,Delcroix}'s distinction of simple locally finite groups into various classes
\cite{Leinen02}. At present, open questions remain for so-called {\em groups of 1-type} 
(these groups are built up from ``tame'' {\sc Kegel} covers with alternating quotients), 
but also for finitary isometry groups over finite fields.

\begin{ourbiblio}{99}

\bibitem{Leinen01}
	{\sc K.\,Bonvallet\,$-$\,B.\,Hartley\,$-$\,D.\,S.\,Passman\,$-$\,M.\,K.\,Smith},
	{\em Group rings with simple augmentation ideals},
	\ Proc.\,Amer.\,Math.\,Soc.\ {\bf 56} (1976), 79--82.

\bibitem{Leinen02}{\sc S.\,Delcroix},
	{\em Non-finitary locally finite simple groups},
	\ Ph.\,D.\ thesis, University of Gent, Belgium 2000.

\bibitem{Leinen03}{B.\,Hartley\,$-$\,A.\,E.\,Zalesskii},
	{\em Confined subgroups of simple locally finite groups and ideals of their group rings},
	\ J.\,London Math.\,Soc.\,(2) {\bf 55} (1997) 210--230.

\bibitem{Leinen04}{B.\,Hartley\,$-$\,A.\,E.\,Zalesskii},
	{\em The ideal lattice of the complex group ring of finitary special and
	general linear groups over finite fields},
	\ Math.\,Proc.\,Camb.\,Phil.\,Soc.\ {\bf 116} (1994) 7--25.

\bibitem{Leinen05}{B.\,Hartley\,$-$\,A.\,E.\,Zalesskii},
	{\em On simple periodic linear groups: Dense subgroups,
	permutation representations and induced modules},
	\ Israel J.\,Math.\ {\bf 82} (1993) 299--327.

\bibitem{Leinen06}{\sc F.\,Leinen\,$-$\,O.\,Puglisi},
	{\em Positive definite functions of finitary isometry groups over fields of odd characteristic}, 
	 \ J.\,Pure Appl.\,Algebra {\bf 208} (2007), 1003--1021.

\bibitem{Leinen07}{\sc F.\,Leinen\,$-$\,O.\,Puglisi},
	{\em Some results concerning simple locally finite groups of 1-type}, 
	 \ J.\,Algebra {\bf 287} (2005), 32--51.

\bibitem{Leinen08}{\sc F.\,Leinen\,$-$\,O.\,Puglisi},
	{\em Positive definite functions of diagonal limits of finite alternating 
	groups}, \ J.\,London Math.\,Soc.\ {\bf 70} (2004), 678--690.

\bibitem{Leinen09}{\sc F.\,Leinen\,$-$\,O.\,Puglisi},
	{\em Diagonal limits of finite alternating groups: confined subgroups, 
	ideals, and positive definite functions}, \ 
	Illinois J.\,Math.\ {\bf 47} (2003), 345--360.

\bibitem{Leinen10}{\sc F.\,Leinen\,$-$\,O.\,Puglisi},
	{\em Ideals in group algebras of simple locally finite groups of 1-type}, \
	Pacific J.\,Math.\ {\bf 207} (2002), 433--445.

\bibitem{Leinen11}{\sc F.\,Leinen\,$-$\,O.\,Puglisi},
	{\em Confined subgroups in periodic simple finitary linear groups}, \
	Israel J.\,Math.\ {\bf 128} (2002), 285--324.

\bibitem{Leinen12}{H.-L.\,Skudlarek},
	{\em Die unzerlegbaren Charaktere einiger diskreter Gruppen},
	\ Math.\ Ann.\ {\bf 223} (1976) 213--231.

\bibitem{Leinen13}{\sc E.\,Thoma},
	{\em Die unzerlegbaren, positiv-definiten Klassenfunktionen
	der abz\"ahlbar unendlichen, symmetrischen Gruppe},
	\ Math.\,Z.\ {\bf 85} (1964), 40--61.

\bibitem{Leinen14}{\sc A.\,M.\,Vershik\,$-$\,S.\,V.\,Kerov},
	{\em Locally semisimple algebras. Combinatorial theory and the $K_0$-functor},
	\ Sovrem.\,Probl.\,Mat.\,Nov.\,Dost.\ {\bf 26} (1985), 3--56 (Russian).

\bibitem{Leinen15}{\sc A.\,E.\,Zalesskii},
    Group rings of simple locally finite groups, pp.\ 219--246
	\,in:
	{\sc B.\,Hartley\,$-$\,G.\,M.\,Seitz\,$-$\,A.\,V.\,Borovik\,$-$\,R.\,M.\,Bryant}
 	(eds.),
	{\em Finite and locally finite groups,}
	\,NATO ASI Series {\bf C\,471}, Kluwer Academic Publishers,
	Dordrecht - Boston - London 1995.

\bibitem{Leinen16}{\sc A.\,E.\,Zalesskii},
	{\em Group rings of inductive limits of alternating groups},
	\ Leningrad Math.\,J.\ {\bf 2} (1991), 1287--1303.

\end{ourbiblio}

\presenteraddress{Institute of Mathematics, University of Mainz, D-55099 Mainz, Germany.}

\email{Leinen@uni-mainz.de}

\web{www.mathematik.uni-mainz.de/~leinen}


