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1. Cluster Algebras and Its Applications

   https://doi.org/10.4171/OWR/2024/2  

   Baur, Karin; Marsh, Bethany; Schiffler, Ralf; Schroll, Sibylle (September 2024, Oberwolfach Reports)

   This workshop focused on recent developments in cluster algebras and their applications as well as interactions with other areas of mathematics. In addition to new advances in the theory of cluster algebras themselves, it included applications to knot theory and geometry as well as interactions with representation theory and categorification, Grassmannians, combinatorics, geometric surfaces models and Lie theory. 

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2. On rationality of C-graded vertex algebras and applications to Weyl vertex algebras under conformal flow

   https://doi.org/10.1063/5.0117895  

   Barron, Katrina; Batistelli, Karina; Orosz_Hunziker, Florencia; Pedić_Tomić, Veronika; Yamskulna, Gaywalee (September 2022, Journal of Mathematical Physics)

   Using the Zhu algebra for a certain category of C-graded vertex algebras V, we prove that if V is finitely Ω-generated and satisfies suitable grading conditions, then V is rational, i.e., it has semi-simple representation theory, with a one-dimensional level zero Zhu algebra. Here, Ω denotes the vectors in V that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal element ωμ parameterized by μ∈C and prove that for certain non-integer values of μ, these vertex algebras, which are non-integer graded, are rational, with a one-dimensional level zero Zhu algebra. In addition, we generalize this result to appropriate C-graded Weyl vertex algebras of arbitrary ranks. 

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3. Equivariant Log Concavity and Representation Stability

   https://doi.org/10.1093/imrn/rnab352  

   Matherne, Jacob P; Miyata, Dane; Proudfoot, Nicholas; Ramos, Eric (December 2021, International Mathematics Research Notices)

   Abstract We expand upon the notion of equivariant log concavity and make equivariant log concavity conjectures for Orlik–Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik–Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $$\mathfrak{s}\mathfrak{l}_n$$, we exploit the theory of representation stability to give computer-assisted proofs of these conjectures in low degree. 

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4. Homomorphisms between standard modules over finite-type KLR algebras

   https://doi.org/10.1112/S0010437X16008204  

   Kleshchev, Alexander S.; Steinberg, David J. (March 2017, Compositio Mathematica)

   Khovanov–Lauda–Rouquier (KLR) algebras of finite Lie type come with families of standard modules, which under the Khovanov–Lauda–Rouquier categorification correspond to PBW bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of a standard module are injective. We present applications to the extensions between standard modules and modular representation theory of KLR algebras. 

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5. Vertex algebras of CohFT-type

   Chiara Damiolini, Angela Gibney (January 2022, Cambridge University Press, Cambridge,)

   Paolo Aluffi, David Anderson (Ed.)

   Representations of certain vertex algebras, here called of CohFT-type, can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable curves [DGT2]. We show that such bundles define semisimple cohomological field theories. As an application, we give an expression for their total Chern character in terms of the fusion rules, following the approach and computation in [MOPPZ] for bundles given by integrable modules over affine Lie algebras. It follows that the Chern classes are tautological. Examples and open problems are discussed. 

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