During the tense negotiations, we suspected our competitors had a major concession up their sleeve.

had 是动词 have 的过去式，表示"拥有"或"持有"。 a 是不定冠词，表示"一个"或"某种"。 major 是形容词，意思是"主要的"或"重大的"。 concession 是名词，来自动词 concede（让步），指"让步"或"妥协"。 up their sleeve 是一个习语，字面意思是"在他们的袖子里"，比喻"隐藏的"或"留作后手的"。 整个短语 had a major concession up their sleeve 表示"暗藏着一个重大让步作为后手"，指谈判中保留着重要筹码以备关键时刻使用。