?? linear structure type carried by finite set ... "assign to set s direct sum over [space of semi-simple comm k-alg structures on s] of ..." ... ??? ....

?? l-function .... ???? ...

?? extent to which we've been through this before ..... ???? .....

??? upward vs downward .... factor ... irrep ... vs ... ?? bigger thing ... ?? .... ?? underlying vs free .... induced ..... ???? .....

?? euler factor structure type here ... ???? .....