"not" comes after auxiliary verb. For example: "I can go." "I can not go."
Then, What is the negative question form of "Shall we~?"? "Shall not we go?" In this sentence "not" came after auxiliary verb "Shall". Then, "Shall not we go?" Is this correct sentence?
But in the real conversation, I often heard this: "Shall we not go?"
Which one is correct?
The examples are a little confusing for me, other users may not find it to be the case.
Are the sentences meant to be negative or negative questions?
For example:
(Affirmative) We shall go.
(Negative) We shall not go.
(Question) Shall we go?
Note the inversion of the subject pronoun "we" with the auxiliary "shall".
(Negative question)
The former is the contraction form, it's slightly old-fashioned and mostly used in speaking. The latter is grammatical and acceptable in formal writing.
to be
(Aff) We are late.
(Neg) We are not late.
(Que) Are we late?
(Neg.Q) Are we not late?
(Cont'n) Aren't we late?
To think about this, you need to consider the meaning.
The form "Shall we ..." is used to make a suggestion of an activity. In the affirmative, this makes sense. You suggest a action. "Shall we play tennis"
What would the negative mean. Well, not making a suggestion is the same as doing nothing. That isn't a sensible thing to describe.
It is possible to suggest not doing something. In that case the "not" is applied to the verb in the infintive clause. "Shall we not play tennis". But this is odd. You see how can you tell the difference between "not playing tennis" and "not playing football"!? It would nearly always be better to positively suggest the alternative "We always play tennis on Sunday. This weekend, shall we go for a walk together instead?" It is better to say "shall we go for a walk" instead of "Shall we not play tennis."
Finally there is the possibility of the negative question. Negative questions carry a strong implication (compare the neutral "can you play tennis" with the marked "can't you play tennis" - the latter means that the speaker would be surprised that you can't) But this implication doesn't work well with "Shall we" type suggestions.
In conclusion, English speakers would tend not to use a negative form of "shall we".
The meanings are different.
Shall not we go?
You believe/want to go, but someone asks you not to go, so you ask him with this question.
Shall we not go?
Someone asks you to go, but you believe it's better not to go. So you use this question.
