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Sesshōmaru resents Inuyasha for being given the Tessaiga and not him. Sesshōmaru also seems to feel some sort of superiority over Inuyasha because Sesshōmaru is a full blooded demon - a daiyōkai at that - and Inuyasha is "merely" a half-demon. Also, Sesshōmaru and Inuyasha really do not get along and do not want to be around each other.

It should be noted that by the end of the series Sesshōmaru's hatred of Inuyasha seems to have subsided, and vice-versa.