You cannot grant permissions to execute a trigger, since users cannot execute triggers: SQL Anywhere fires them in response to actions on the database. Nevertheless, a trigger does have permissions associated with it as it executes, defining its right to perform certain actions.
Triggers execute using the permissions of the owner of the table on which they are defined, not the permissions of the user who caused the trigger to fire, and not the permissions of the user who created the trigger.
When a trigger refers to a table, it uses the group memberships of the table creator to locate tables with no explicit owner name specified. For example, if a trigger on user_1.Table_A references Table_B and does not specify the owner of Table_B, then either Table_B must have been created by user_1 or user_1 must be a member of a group (directly or indirectly) that is the owner of Table_B. If neither condition is met, a table not found message results when the trigger fires.
Also, user_1 must have permissions to perform the operations specified in the trigger.