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Mathematics  2015 

A proof of the shuffle conjecture

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Abstract:

We present a proof of the compositional shuffle conjecture \cite{haglund2012compositional}, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra \cite{haglund2005diagcoinv}. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space $V_\cdot$ whose degree zero part is the ring of symmetric functions $Sym[X]$ over $\mathbb{Q}(q,t)$. We then extend these operators to two larger algebras $\mathbb{A}$ and $\mathbb{A}^*$ acting on this space, and interpret the right generalization of the $\nabla$ operator as an intertwiner between the two actions, which is antilinear with respect to the conjugation $(q,t)\mapsto (q^{-1},t^{-1})$.

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