According to the main view the word 'Ameen' آمين is used to mean 'O God, answer'. So, in this case there is nothing wrong with the expression itself, even though this word does not come from the Quran, nor is it a part of any verse.
This is something agreed upon by both Sunnis and Shi'ah.
However, the issue is whether it is permissible to say it in a daily obligatory prayer, after reciting the Fatihah.
Shi'a scholars say that if Ameen is said with the intention of it being a part of Salat, then the prayer will be invalid. It is a foreign word and has nothing related to the Surah, or correct dhikrs mentioned as parts of the Salat.
We are obliged to adhere to how the Prophet (s.a.w.) has taught us to pray. In the narrations there is no mentioning that the word 'Ameen' should be said after the Fatihah.
We cannot add something that does not exist in the Sunnah of the Prophet (s.a.w.), and therefore doing such a thing would nullify our Salat.
It is a consensus of Shi'ah scholars that saying 'Ameen' in Salat, after Fatihah is an innovation and therefore would invalidate the prayer.
Ameen is a word foreign to the prayer, and not a 'supplication'. This is contrary to the expression 'alhamdulillah rabb al-'alameen' which would be permissible to say after reciting the Fatihah, due to it being a dua and also it being mentioned to do so in authentic traditions.
For further information on the word Ameen, see:
And Allah knows best.