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1. LLT POLYNOMIALS IN THE SCHIFFMANN ALGEBRA

   BLASIAK, J; HAIMAN, M; MORSE, J; PUN, A; SEELINGER, GH (April 2024, Journal für die reine und angewandte Mathematik)

   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 in the algebra of symmetric functions embedded in the elliptic Hall algebra of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the nabla operator applied to any LLT polynomial. In particular, we obtain a formula for ∇msλ 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. The classifying element for quotients of Fermat curves

   https://doi.org/10.1080/00927872.2024.2346299  

   Duque-Rosero, Juanita; Pries, Rachel (May 2024, Communications in Algebra)

   Suppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and ∞. Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of Q. 

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3. 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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4. Calibrated representations of the double Dyck path algebra

   https://doi.org/10.1007/s00208-024-02937-2  

   González, Nicolle; Gorsky, Eugene; Simental, José (August 2024, Mathematische Annalen)

   Abstract The double Dyck path algebra$$\mathbb {A}_{q,t}$$ $A_{q,t}$and its polynomial representation first arose as a key figure in the proof of the celebrated Shuffle Theorem of Carlsson and Mellit. A geometric formulation for an equivalent algebra$$\mathbb {B}_{q,t}$$ $B_{q,t}$was then given by the second author and Carlsson and Mellit using the K-theory of parabolic flag Hilbert schemes. In this article, we initiate the systematic study of the representation theory of the double Dyck path algebra$$\mathbb {B}_{q,t}$$ $B_{q,t}$. We define a natural extension of this algebra and study its calibrated representations. We show that the polynomial representation is calibrated, and place it into a large family of calibrated representations constructed from posets satisfying certain conditions. We also define tensor products and duals of these representations, thus proving (under suitable conditions) the category of calibrated representations is generically monoidal. As an application, we prove that tensor powers of the polynomial representation can be constructed from the equivariant K-theory of parabolic Gieseker moduli spaces. 

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5. Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem

   Giordano Cotti, Alexander Varchenko (January 2021, Proceedings of symposia in pure mathematics)

   Novikov, Krichever (Ed.)

   In [TV19a] the equivariant quantum differential equation (qDE) for a projective space was considered and a compatible system of difference qKZ equations was introduced; the space of solutions to the joint system of the qDE and qKZ equations was identified with the space of the equivariant K-theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant K-theory algebra. This paper is a continuation of [TV19a]. We describe the relation between solutions to the joint system of the qDE and qKZ equations and the topological-enumerative solution to the qDE only, defined as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant K-theory algebra, the equivariant Gamma class of the projective space, and the equivariant first Chern class of the tangent bundle of the projective space. We consider a Stokes basis, the associated exceptional basis in the equivariant K-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincaré pairing wrt to the basis, which is the left dual to the associated exceptional basis. We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space. These statements are equivariant analogs of results of G. Cotti, B. Dubrovin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani. 

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