I've reviewed the parts list and wiring diagram for the 917.270912. I can not find one for a 917.270912 TIA.
I can find no relay that tuke2 mentions. This model is wired almost identically to my 917.270821.
For the engine to start, the seat switch (normally open contact) must be closed and both safety interlock switches (brake and pto) must be open. Two circuits in the ignition switch must also be closed.
Since his engine starts, these functions must be working.
The seat switch is a dual contact switch, with both normally open and normally closed contacts. One set is used to ensure the seat is occupied when attempting to start the mower. The other is used to make sure the seat is occupied when either the pto is engaged or the clutch/brake is released in order to put the machine in motion and to ensure they both are disengaged when attempting to start.
If either the clutch/brake or pto interlock is closed and the normally closed contact on the seat switch is closed (it should be open when the seat is occupied) the ignition will be grounded and the engine will shut down.
Again, I believe the dual circuit seat switch is probably the culprit, unleass the "TIA" after the model number uses some special wiring scheme. It is most likely not the clutch/brake or pto safety interlock. There is an outside possibility that the ignition switch is bad.