A proof of the shuffle conjecture. (English) Zbl 1387.05265

Carlsson, Erik; Mellit, Anton

J. Am. Math. Soc. 31, No. 3, 661-697 (2018).

Summary: We present a proof of the compositional shuffle conjecture by J. Haglund et al. [Can. J. Math. 64, No. 4, 822–844 (2012; Zbl 1246.05163)], which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra by J. Haglund et al. [Duke Math. J. 126, No. 2, 195–232 (2005; Zbl 1069.05077)]. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space \( V_\ast\) whose degree zero part is the ring of symmetric functions \( \operatorname {Sym}[X]\) over \( \mathbb{Q}(q,t)\). We then extend these operators to an action of an algebra \( \tilde {\mathbb{A}}\) acting on this space, and interpret the right generalization of the \( \nabla \) using an involution of the algebra which is antilinear with respect to the conjugation \( (q,t)\mapsto (q^{-1},t^{-1})\).

MSC:

05E10	Combinatorial aspects of representation theory
05E05	Symmetric functions and generalizations
05A30	\(q\)-calculus and related topics
33D52	Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)

Keywords:

compositional shuffle conjecture; diagonal coinvariant algebra

Citations:

Zbl 1246.05163; Zbl 1069.05077

References:

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