History
Robin Arryn is the sickly, immature son of Jon and Lysa Arryn, who died after being poisoned and pushed out a window respectively. Nominally the Lord of the Vale and the commander of its armies, Robin is unduly influenced by people like Petyr “Littlefinger” Baelish, who is known to pry him with gifts.
In season 6, Littlefinger convinced Robin to send the Knights of the Vale to the aid of Sansa Stark. The Knights played a huge part in defeating Ramsay Bolton’s forces outside Winterfell, and are now ensconced at the castle.
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Status
We actually don’t know where Robin is at the moment. Did Littlefinger leave him in the Vale while he led the Knights of the Vale into the North? Or did Littlefinger take Robin to Winterfell, where he’s even now getting settled in a spare bedroom? Wherever Robin is, he has the final say on what the Knights of the Vale do, which is an unsettling thought.
Wherever he is, there are things Robin doesn’t know that could change his loyalties. He is currently unaware, for example, that Littlefinger killed his mother Lysa in season 4. Sansa Stark, however, does know. If she wants to undermine Littlefinger’s authority, she could try to sway Robin Arryn with this information.