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1. Products and powers of principal symmetric ideals

   https://doi.org/10.1142/S0219498825503207  

   Dannetun, Eric; Fang, Bruce; Formenti, Riccardo; Gao, Bo Y; Geraci, Juliann; Kogel, Ross; Li, Yuelin; Mandal, Shreya; Rupasinghe, Vinuge; Seceleanu, Alexandra; et al (July 2024, Journal of Algebra and Its Applications)

   Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case. 

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2. 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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3. Blowup algebras of determinantal ideals in prime characteristic

   https://doi.org/10.1112/jlms.12969  

   De_Stefani, Alessandro; Montaño, Jonathan; Núñez‐Betancourt, Luis (August 2024, Journal of the London Mathematical Society)

   Abstract We study when blowup algebras are ‐split or strongly ‐regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew‐symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results, we develop the notion of ‐split filtrations and symbolic ‐split ideals. 

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4. 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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5. Symmetric group fixed quotients of polynomial rings

   https://doi.org/10.1016/j.jpaa.2023.107537  

   Pevzner, Alexandra (April 2024, Journal of Pure and Applied Algebra)

   Given a representation of a finite group G over some commutative base ring k, the cofixed space is the largest quotient of the representation on which the group acts trivially. If G acts by k-algebra automorphisms, then the cofixed space is a module over the ring of G-invariants. When the order of G is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that G is the symmetric group on n letters acting on a polynomial ring by permuting its variables. When k has characteristic 0, the cofixed space is isomorphic to an ideal of the ring of symmetric polynomials in n variables. Localizing k at a prime integer p while letting n vary reveals striking behavior in these ideals. As n grows, the ideals stay stable in a sense, then jump in complexity each time n reaches a multiple of p. 

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