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  1. ; ; ; ; (, Forum of Mathematics, Pi)
    Abstract We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $$\operatorname {\mathrm {GL}}_{l}$$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials. 
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  2. ; ; ; ; (, Forum of Mathematics, Pi)
    Abstract We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $$\Delta _{h_l}\Delta ' _{e_k} e_{n}$$ , where $$\Delta ' _{e_k}$$ and $$\Delta _{h_l}$$ are Macdonald eigenoperators and $$e_n$$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $$\operatorname {\mathrm {GL}}_m$$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions. 
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