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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]

We make a broad conjecture about the k -Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the k -Schur expansion of (1) Hall-Littlewood polynomials, proving the q = 0 case of the strengthened Macdonald positivity conjecture from [24] ; (2) the product of a Schur function and a k -Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum cohomology of complete flag varieties; (3) k -split polynomials, solving a substantial special case of a problem of Broer and Shimozono-Weyman on parabolic Hall-Littlewood polynomials [37]. In addition, we prove the conjecture that the k -Schur functions defined via k -split polynomials [25] agree with those defined in terms of strong tableaux [21]. [ABSTRACT FROM AUTHOR]