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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. 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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3. 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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4. 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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5. Conformal Field Theories with Sporadic Group Symmetry

   https://doi.org/10.1007/s00220-021-04207-7  

   Bae, Jin-Beom; Harvey, Jeffrey A.; Lee, Kimyeong; Lee, Sungjay; Rayhaun, Brandon C. (October 2021, Communications in Mathematical Physics)

   null (Ed.)

   The monster sporadic group is the automorphism group of a central charge $c=24$ vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its $c=24$ stress tensor $T(z)$, this theory contains many other conformal vectors of smaller central charge; for example, it admits $48$ commuting $$c=\frac12$$ conformal vectors whose sum is $T(z)$. Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for the existence of a number of CFTs with sporadic symmetry groups and employ a variety of techniques, including Hecke operators and modular linear differential equations, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups $${}^2\tsl{E}_6(2)$$ and $$\tsl{F}_4(2)$$ of Lie type. Many of these examples are naturally associated to McKay's $$\widehat{E_8}$$ correspondence, and we use the structure of Norton's monstralizer pairs more generally to organize our presentation. 

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