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1. LLT polynomials in the Schiffmann algebra

   https://doi.org/10.1515/crelle-2024-0012  

   Blasiak, Jonah; Haiman, Mark; Morse, Jennifer; Pun, Anna; Seelinger, George H (April 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies $\mathrm{\Lambda }\left(X^{m,n}\right)\subset \mathcal{E}$\Lambda(X^{m{,}n})\subset\mathcal{E}of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann.As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial.In particular, we obtain a formula for ${\nabla }^m{s}_{\lambda }$\nabla^{m}s_{\lambda}which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one. 

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2. A Proof of the Extended Delta Conjecture

   https://doi.org/10.1017/fmp.2023.3  

   Blasiak, Jonah; Haiman, Mark; Morse, Jennifer; Pun, Anna; Seelinger, George H. (January 2023, 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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3. A Shuffle Theorem for Paths Under Any Line

   https://doi.org/10.1017/fmp.2023.4  

   Blasiak, Jonah; Haiman, Mark; Morse, Jennifer; Pun, Anna; Seelinger, George H. (January 2023, 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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4. A raising operator formula for Macdonald polynomials

   https://doi.org/10.1017/fms.2025.8  

   Blasiak, J; Haiman, M; Morse, J; Pun, A; Seelinger, G H (January 2025, Forum of Mathematics, Sigma)

   Abstract We give an explicit raising operator formula for the modified Macdonald polynomials$$\tilde {H}_{\mu }(X;q,t)$$, which follows from our recent formula for$$\nabla $$on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions$$\tilde {H}^{1,n}(X;q,t)$$that we call$$1,n$$-Macdonald polynomials, which reduce to a scalar multiple of$$\tilde {H}_{\mu }(X;q,t)$$when$$n=1$$. We conjecture that the coefficients of$$1,n$$-Macdonald polynomials in terms of Schur functions belong to$${\mathbb N}[q,t]$$, generalizing Macdonald positivity. 

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5. Braided Frobenius algebras from certain Hopf algebras

   https://doi.org/10.1142/S0219498823500123  

   Saito, Masahico; Zappala, Emanuele (September 2021, Journal of Algebra and Its Applications)

   A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation [Formula: see text], that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations. 

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