Sir Gawain is a young knight who has learned a valuable lesson in honesty, integrity and honor. The green sash represents his experience with the Green Knight but also reminds him that he is still human and is weak. At that time, knights were considered elite and practically indestructible. Not only that, but they were held to a very high standard of manners and chivalry. Knights were also supposed to be brave and never shrink in a fight or challenge. Sir Gawain does this by hiding the sash from the Green Knight's knowledge and by flinching during the repayment due him on New Year's Day. So, he carries the sash as a reminder to keep himself humble as well as beholden to high standards of character.