Samsung has not released an official HDMI adapter for the Galaxy Note 10.1 but we have come to know that the one meant for the Galaxy Tab works just fine with it. The only thing not to like about it is the fact you need to have the charger plugged in for it to work.
If you do not get the audio output from it, you must set the sound to ‘stereo’ instead of surround in the HDMI settings on the Note 10.1. Also, it works best if you own a Samsung LED/LCD HDTV. You can buy the adapter from here (just 39 dollars, actual price is $59).