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I am doing this exercise from wasabi-jpn.com. The exercise includes this sentence:
The sweater (I) bought last week was expensive.
Apparently, the correct answer is
先週買ったセーターは高かったです。
I realize that the particle は cannot be used after 先週 because it's a relative clause, but why there is no particle at all?
Firstly, using the particle が would make no sense at all. が marks the subject of a verb, so 先週がセーターを買いました would mean that "last week bought a jumper". Unless you have some bizarre world where you have anthropomorphised the weeks then this sentence is pure gibberish.
は and が are not interchangeable (see this). は can mark a subject, but it does not do so in this case. The subject in this sentence is the hidden/implied "I".
Adding は to 先週 denotes that either 'last week' is the topic of the sentence, or that you are contrasting last week with some other time see this post.
Finally, you are right that は is not normally allowed in a relative clause, but there is an exception. は can be retained if it is used as a contrast marker rather than as a topic (see this post).
In conclusion, no particle is needed in either the full sentence above or in the sentence which is just the relative clause.
