Breakdown of Je regrette que tu aies refusé la prime.
je
I
tu
you (informal singular)
avoir
to have
que
that (introducing a clause)
Questions & Answers about Je regrette que tu aies refusé la prime.
Why do we use aies refusé instead of the passé composé (as refusé)?
Because regretter que is an expression of emotion, it forces the following verb into the subjunctive mood. Since the action of refusing happened in the past, we use the past subjunctive. This is formed with the present subjunctive of the auxiliary verb (avoir -> tu aies) plus the past participle (refusé).
Is aies an exception to regular subjunctive conjugation rules?
Yes, avoir (along with être) is highly irregular in the present subjunctive. For tu, the form is aies. You simply have to memorize it, as it serves as the auxiliary building block for the past subjunctive of most French verbs!
What would this look like if the sentence was "I regret that I refused the bonus"?
When the subject of both verbs is the same (you regret your own action), you do not use que + subjunctive. Instead, you use de + the past infinitive: Je regrette d'avoir refusé la prime.