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1. Free resolutions of orbit closures of Dynkin quivers

   https://doi.org/10.1090/tran/7697  

   Lorincz, Andras; Weyman, Jerzy (May 2019, Transactions of the American Mathematical Society)

   We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type $$ \mathbb{A}$$, we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type $$ \mathbb{A}$$ quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation. 

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2. A Volume = Multiplicity formula for p-families of ideals

   https://doi.org/10.1090/proc/16451  

   Das, Sudipta (October 2023, Proceedings of the American Mathematical Society)

   In this article, we work with certain families of ideals called $p$-families in rings of prime characteristic. This family of ideals is present in the theories of tight closure, Hilbert-Kunz multiplicity, and $F$-signature. For each $p$-family of ideals, we attach an Euclidean object called $p$-body, which is analogous to the Newton Okounkov body associated with a graded family of ideals. Using the combinatorial properties of $p$-bodies and algebraic properties of the Hilbert-Kunz multiplicity, we establish a Volume = Multiplicity formula for $p$-families of ${\mathfrak{m}}_R$-primary ideals in a Noetherian local ring $R$. 

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3. The combinatorial structure of symmetric strongly shifted ideals

   https://doi.org/10.4171/JCA/107  

   Costantini, Alessandra; Seceleanu, Alexandra (January 2025, Journal of Combinatorial Algebra)

   Symmetric strongly shifted ideals constitute a class of monomial ideals which are equipped with an action of the symmetric group and are analogous to the well-studied class of strongly stable monomial ideals. In this paper, we focus on algebraic and combinatorial properties of symmetric strongly shifted ideals. On the algebraic side, we elucidate properties that pertain to behavior under ideal operations, primary decomposition, and the structure of their Rees algebra. On the combinatorial side, we develop a notion of partition Borel generators which leads to connections to discrete polymatroids, convex polytopes, and permutohedral toric varieties. 

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4. Hypersurface support and prime ideal spectra for stable categories.

   Negron, C. (January 2022, Annals of Ktheory)

   We use hypersurface support to classify thick (two-sided) ideals in the stable categories of representations for several families of finite-dimensional integrable Hopf algebras: bosonized quantum complete intersections, quantum Borels in type A, Drinfeld doubles of height 1 Borels in finite characteristic, and rings of functions on finite group schemes over a perfect field. We then identify the prime ideal (Balmer) spectra for these stable categories. In the curious case of functions on a finite group scheme G, the spectrum of the category is identified not with the spectrum of cohomology, but with the quotient of the spectrum of cohomology by the adjoint action of the subgroup of connected components in G. 

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5. Gröbner bases, symmetric matrices, and type C Kazhdan–Lusztig varieties

   https://doi.org/10.1112/jlms.12856  

   Escobar, Laura; Fink, Alex; Rajchgot, Jenna; Woo, Alexander (February 2024, Journal of the London Mathematical Society)

   Abstract We study a class of combinatorially defined polynomial ideals that are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan–Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme‐theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan–Lusztig ideals that arise are exactly those where the opposite cell is 123‐avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley–Reisner ideals of subword complexes), and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams. 

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