What's the significance of Caesar's robe?

Towards the end of Act II, Scene 2, the eagerly ambitious Julius Caesar is willingly persuaded to believe that Calpurnia's dream about his death is not worth taking seriously. He says:
How foolish do your fears seem now, Calpurnia!
I am ashamed I did yield to them.
Give me my robe, for I will go.
It takes a little time, however, for his robe to be brought to him by a servant. Caesar has some chitchat with the men who have come to escort him to the Senate House. During this time the audience would get a good look at Caesar's robe while he is putting it on. No doubt the robe used by Shakespeare in the play's production was conspicuously different from the robes all the other men were wearing. Then in Act III, Scene 2, when Antony shows the dead Caesar's shredded and blood-stained robe to the assembled plebeians during his funeral oration, a second robe which is otherwise identical to the one Caesar put on in Act II, Scene 2 will create the impression that this is the same robe Caesar was wearing when he was attacked by all the conspirators. Shakespeare must have felt that it was more effective to show the torn and bloody robe than to try to show Caesar's mutilated body. The audience would never actually see the body. Antony says it is inside a coffin.
My heart is in the coffin there with Caesar,
And I must pause till it come back to me.
Antony first shows the plebeians the robe supposedly covering the dead body and pretends to know which rents in the garment were caused by which of the conspirators. He has the men in tears. Then he says:
Kind souls—what—weep you when you but behold
Our Caesar's vesture wounded? Look you here.
Here is himself, marred, as you see, with traitors.
Antony holds up the robe to its full length and shows the audience how thoroughly it has been ruined by the conspirators.