In Act 2 Scene 1, what does Brutus refusing to swear an oath reveal about his character traits?
To start with, it is Cassius, in this scene, who suggests that all the conspirators take an oath that they will stand together. The fact that Shakespeare uses this speech as a way for Brutus to disagree with Cassius foreshadows all the conflict that will emerge between these two characters later in the play.
Aside from opposing Cassius, Brutus is taking a stand based upon the driving force of his character -- honor. He wants the conspirators to stand together based upon their word as Romans ("what other oath than honesty to honest engaged"), rather than swearing to it. Shakespeare is able to show, in this moment when the conspirators don't swear an oath, that they are more strongly swayed by Brutus than by Cassius, so his honor is a strength for him among his comrades.
However, honor is key to Brutus' strength and his weakness. The Rome that Shakespeare portrays, doesn't offer much political success to Brutus and his honor. He is driven out of Rome on the heels of Antony's emotional plea to the citizens to avenge the death of Caesar. It is interesting to note that Antony repeatedly calls Brutus an "honorable" man in this speech, and it's unclear whether it intends it as a compliment or not.