Which river might be used to transport materials mined near Irkutsk to manufacturing centers?
Are you referring to the city of Irkutsk or the district (Oblast)?
The rivers that flow through the city go to Lake Baikal, where there are no manufacturing centers.
Possibly useful navigable rivers from the Irkutsk Oblast towards the west are:
Chadobets, Yennisey, Kan and Chuna