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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 GLl 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. [ABSTRACT FROM AUTHOR]

We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for ΔhlΔ′eken, where Δ′ek and Δhl are Macdonald eigenoperators and en is an elementary symmetric function. We actually prove a stronger identity of infinite series of GLm 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. [ABSTRACT FROM AUTHOR]