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1. Cluster algebra structures on Poisson nilpotent algebras

   https://doi.org/10.1090/memo/1445  

   Goodearl, K; Yakimov, M (October 2023, Memoirs of the American Mathematical Society)

   Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements. 

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2. Upper Triangular Linear Relations on Mmultiplicities and the Stanley-Stembridge Conjecture

   https://doi.org/10.37236/10489  

   Harada, Megumi; Precup, Martha (July 2022, The Electronic Journal of Combinatorics)

   In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian-Wachs conjecture, which links the Stanley-Stembridge conjecture in combinatorics to the geometry of Hessenberg varieties through Tymoczko's permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. In previous work, the authors exploited this connection to prove a graded version of the Stanley-Stembridge conjecture in a special case. In this manuscript, we derive a new set of linear relations satisfied by the multiplicities of certain permutation representations in Tymoczko's representation. We also show that these relations are upper-triangular in an appropriate sense, and in particular, they uniquely determine the multiplicities. As an application of these results, we prove an inductive formula for the multiplicity coefficients corresponding to partitions with a maximal number of parts. 

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3. The essential dimension of congruence covers

   https://doi.org/10.1112/S0010437X21007594  

   Farb, Benson; Kisin, Mark; Wolfson, Jesse (November 2021, Compositio Mathematica)

   Abstract Consider the algebraic function $$\Phi _{g,n}$$ that assigns to a general $$g$$ -dimensional abelian variety an $$n$$ -torsion point. A question first posed by Klein asks: What is the minimal $$d$$ such that, after a rational change of variables, the function $$\Phi _{g,n}$$ can be written as an algebraic function of $$d$$ variables? Using techniques from the deformation theory of $$p$$ -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $$p$$ -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $$p$$ -dimension of congruence covers of the moduli space of genus $$g$$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $$M$$ is proper . As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety. 

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4. Quiver varieties and symmetric pairs

   https://doi.org/10.1090/ert/522  

   Li, Yiqiang (January 2019, Representation theory)

   We study fixed-point loci of Nakajima varieties under symplectomorphisms and their antisymplectic cousins, which are compositions of a diagram isomorphism, a reflection functor, and a transpose defined by certain bilinear forms. These subvarieties provide a natural home for geometric representation theory of symmetric pairs. In particular, the cohomology of a Steinberg-type variety of the symplectic fixed-point subvarieties is conjecturally related to the universal enveloping algebra of the subalgebra in a symmetric pair. The latter symplectic subvarieties are further used to geometrically construct an action of a twisted Yangian on a torus equivariant cohomology of Nakajima varieties. In the type A case, these subvarieties provide a quiver model for partial Springer resolutions of nilpotent Slodowy slices of classical groups and associated symmetric spaces, which leads to a rectangular symmetry and a refinement of Kraft-Procesi row/column removal reductions. 

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5. Bott vanishing for Fano threefolds

   https://doi.org/10.1007/s00209-024-03468-x  

   Totaro, Burt (May 2024, Mathematische Zeitschrift)

   Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that the higher cohomology of every bundle of differential forms tensored with an ample line bundle is zero. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano threefolds that satisfy Bott vanishing. There are many more than expected. 

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