C-172 Maintenance technician reported incorrect alternator belt installed.

Narrative

If I remember correctly; the parts box was labeled with same part number as IPC (Illustrated Parts Catalog) called for. XXXX had part number same as IPC and box. XXXX had same serial number as alternator. I installed the alternator on day 0 ; aircraft flew for roughly a week; then was in hangar for roughly a week for annual inspection done by someone else. A couple days after return to service from annual; aircraft was reported as down for torn alternator belt. I do not know if aircraft lost generation during flight; but I'm told flight crew did inspect alternator belt after flight and found damage to belt. I do not know if that is a normal part of post-flight inspection. If not; I assume they looked at it because they experienced loss of electrical power.Error was discovered today. The aircraft is currently grounded and I removed the alternator with the wrong pulley before leaving the aircraft in the hangar for the night. It currently has no alternator; spinner; or nose cowl and is parked in hangar so it will not be flying accidentally.Unfortunately; I have more electrical experience than mechanical. This was somewhere between my 3rd-5th install of a belted appliance in my life. I installed an alternator with a V belt pulley. Belt that was in use was a ribbed belt; along with a ribbed ring gear behind prop aft bulkhead. Ribbed belt was narrow enough to fit inside the V pulley. As the belt sat centered on the gears at each end; I incorrectly assumed this was normal. I had guessed that the 'smooth' alternator pulley was to accommodate the pulley slipping on the belt during the belt tensioning test part of alternator torquing. Thinking about it further; I realize that the other style pulley (ribbed) can also slip during tensioning because of the direction the ribs are oriented.The parts box has since left with the old alternator core. The XXXX was retained with logbook. XXXX and logbook match; part#s and serial#s. However; XXXX part number does not match installed alternator part number. At this time; I do not have documentation with me showing it to be an approved alternate part number; though it might be. XXXX serial number matches installed alternator serial number.