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1. Tiered Trees and Theta Operators

   https://doi.org/10.1093/imrn/rnac258  

   D'Adderio, Michele; Iraci, Alessandro; Le Borgne, Yvan; Romero, Marino; Vanden Wyngaerd, Anna (September 2022, International Mathematics Research Notices)

   In [10], the authors introduced tiered trees to define combinatorial objects counting absolutely indecomposable representations of certain quivers and torus orbits on certain homogeneous varieties. In this paper, we use Theta operators, introduced in [6], to give a symmetric function formula that enumerates these trees. We then formulate a general conjecture that extends this result, a special case of which might give some insight about how to formulate a unified Delta conjecture [20]. 

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2. Globally irreducible Weyl modules for quantum groups

   Garibaldi, Skip; Nakano, Daniel K.; Guralnick, Robert M. (January 2018, Springer Proceedings in Mathematics and Statistics (Geometric and Topological Aspects of the Representation Theory of Finite Groups), Springer-Verlag)

   The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $$E_{8}$$ or its highest weight is minuscule. In this paper, we prove an analogous criteria for irreducibility of Weyl modules over the quantum group $$U_{\zeta}({{\mathfrak g}})$$ where $${\mathfrak g}$$ is a complex simple Lie algebra and $$\zeta$$ ranges over roots of unity. 

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3. Counterexamples to the Low-Degree Conjecture

   Holmgren, Justin; Wein, Alexander S (April 2021, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021))

   Lee, James R (Ed.)

   A conjecture of Hopkins (2018) posits that for certain high-dimensional hypothesis testing problems, no polynomial-time algorithm can outperform so-called "simple statistics", which are low-degree polynomials in the data. This conjecture formalizes the beliefs surrounding a line of recent work that seeks to understand statistical-versus-computational tradeoffs via the low-degree likelihood ratio. In this work, we refute the conjecture of Hopkins. However, our counterexample crucially exploits the specifics of the noise operator used in the conjecture, and we point out a simple way to modify the conjecture to rule out our counterexample. We also give an example illustrating that (even after the above modification), the symmetry assumption in the conjecture is necessary. These results do not undermine the low-degree framework for computational lower bounds, but rather aim to better understand what class of problems it is applicable to. 

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4. A counterexample to the HK-conjecture that is principal

   https://doi.org/10.1017/etds.2022.25  

   DEELEY, ROBIN J. (June 2023, Ergodic Theory and Dynamical Systems)

   Abstract Scarparo has constructed counterexamples to Matui’s HK-conjecture. These counterexamples and other known counterexamples are essentially principal but not principal. In the present paper, a counterexample to the HK-conjecture that is principal is given. Like Scarparo’s original counterexample, our counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free. The examples discussed here do satisfy the rational version of the HK-conjecture. 

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5. On the Finkelberg–Ginzburg Mirabolic Monodromy Conjecture

   https://doi.org/10.1093/imrn/rnae245  

   Toledano_Laredo, Valerio; Walters, Robin (November 2024, International Mathematics Research Notices)

   Abstract We compute the monodromy of the mirabolic $$\mathcal{D}$$-module for all values of the parameters $$(\vartheta ,c)$$ in rank 1 and outside an explicit codimension 2 set of values in ranks 2 and higher. This shows in particular that the Finkelberg–Ginzburg conjecture, which is known to hold for generic values of $$(\vartheta ,c)$$, fails at special values even in rank 1. Our main tools are Opdam’s shift operators and intertwiners for the extended affine Weyl group, which allow for the resolution of resonances outside the codimension two set. 

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