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1. Evaluations of annular Khovanov–Rozansky homology

   https://doi.org/10.1007/s00209-022-03163-9  

   Gorsky, Eugene; Wedrich, Paul (January 2023, Mathematische Zeitschrift)

   Abstract We describe the universal target of annular Khovanov–Rozansky link homology functors as the homotopy category of a free symmetric monoidal linear category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov–Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types. 

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2. A category $$\mathcal{O}$$ for oriented matroids

   https://doi.org/10.4171/JCA/71  

   Kowalenko, Ethan; Mautner, Carl (June 2023, Journal of Combinatorial Algebra)

   We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite-dimensional algebra, whose representation theory is analogous to blocks of Bernstein—Gelfand—Gelfand category\mathcal{O}. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden—Licata—Proudfoot—Webster. Applying our construction to non-linear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, as in the linear setting. In the non-Euclidean case, we obtain algebras that are not quasi-hereditary and not known to be Koszul, but still have a natural class of standard modules and satisfy numerical analogues of quasi-heredity and Koszulity on the level of graded Grothendieck groups. 

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3. The lowest discriminant ideal of a Cayley–Hamilton Hopf algebra

   https://doi.org/10.1090/tran/9354  

   Mi, Zhongkai; Wu, Quanshui; Yakimov, Milen (May 2025, Transactions of the American Mathematical Society)

   Discriminant ideals of noncommutative algebras $A$, which are module finite over a central subalgebra $C$, are key invariants that carry important information about $A$, such as the sum of the squares of the dimensions of its irreducible modules with a given central character. There has been substantial research on the computation of discriminants, but very little is known about the computation of discriminant ideals. In this paper we carry out a detailed investigation of the lowest discriminant ideals of Cayley–Hamilton Hopf algebras in the sense of De Concini, Reshetikhin, Rosso and Procesi, whose identity fiber algebras are basic. The lowest discriminant ideals are the most complicated ones, because they capture the most degenerate behaviour of the fibers in the exact opposite spectrum of the picture from the Azumaya locus. We provide a description of the zero sets of the lowest discriminant ideals of Cayley–Hamilton Hopf algebras in terms of maximally stable modules of Hopf algebras, irreducible modules that are stable under tensoring with the maximal possible number of irreducible modules with trivial central character. In important situations, this is shown to be governed by the actions of the winding automorphism groups. The results are illustrated with applications to the group algebras of central extensions of abelian groups, big quantum Borel subalgebras at roots of unity and quantum coordinate rings at roots of unity. 

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4. Affine cluster monomials are generalized minors

   https://doi.org/10.1112/S0010437X19007292  

   Rupel, Dylan; Stella, Salvatore; Williams, Harold (June 2019, Compositio Mathematica)

   We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with $$\mathbf{g}$$ -vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type $$A_{1}^{(1)}$$ , we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras. 

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5. Khovanov Laplacian and Khovanov Dirac for knots and links

   https://doi.org/10.1088/2632-072X/adde9f  

   Jones, Benjamin; Wei, Guo-Wei (June 2025, Journal of Physics: Complexity)

   Abstract Khovanov homology has been the subject of much study in knot theory and low dimensional topology since 2000. This work introduces a Khovanov Laplacian and a Khovanov Dirac to study knot and link diagrams. The harmonic spectrum of the Khovanov Laplacian or the Khovanov Dirac retains the topological invariants of Khovanov homology, while their non-harmonic spectra reveal additional information that is distinct from Khovanov homology. 

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