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1. Higher resonance schemes and Koszul modules of simplicial complexes

   https://doi.org/10.1007/s10801-024-01313-2  

   Aprodu, Marian ; Farkas, Gavril ; Raicu, Claudiu ; Sammartano, Alessio ; Suciu, Alexander I. ( March 2024 , Journal of Algebraic Combinatorics)

   Each connected graded, graded-commutative algebraAof finite type over a field$$\Bbbk $$$k$of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the(higher) Koszul modulesofA. In this note, we investigate the geometry of the support loci of these modules, called theresonance schemesof the algebra. When$$A=\Bbbk \langle \Delta \rangle $$$A=k⟨\Delta ⟩$is the exterior Stanley–Reisner algebra associated to a finite simplicial complex$$\Delta $$$\Delta$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

    

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2. A minimaj-preserving crystal on ordered multiset partitions

   Georgia Benkart, Laura Colmenarejo ( April 2018 , Advances in applied mathematics)

   We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization $R_{n,k}$ due to Haglund, Rhoades and Shimozono of the coinvariant algebra $R_n$. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions. 

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3. Homological mirror symmetry for higher-dimensional pairs of pants

   https://doi.org/10.1112/S0010437X20007150  

   Lekili, Yankı ; Polishchuk, Alexander ( July 2020 , Compositio Mathematica)

   Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$ , for $k\geqslant n$ , with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ( $n$ -dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$ . By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$ -dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$ . We also prove similar equivalences for finite abelian covers of the $n$ -dimensional pair of pants. 

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4. Lefschetz Theory for Exterior Algebras and Fermionic Diagonal Coinvariants

   https://doi.org/10.1093/imrn/rnaa203  

   Kim, Jongwon ; Rhoades, Brendon ( August 2020 , International Mathematics Research Notices)

   null (Ed.)

   Abstract Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W:= \wedge (V \oplus V^*)/ \langle \wedge (V \oplus V^*)^{W}_+ \rangle $ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = {{\mathfrak{S}}}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped ${{\mathfrak{S}}}_n$-modules. We relate the Hilbert series of $DR_W$ to the (type A) Catalan and Narayana numbers and describe a standard monomial basis of $DR_W$ using a variant of Motzkin paths. Our methods are type-uniform and involve a Lefschetz-like theory, which applies to the exterior algebra $\wedge (V \oplus V^*)$. 

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5. Presenting the cohomology of a Schubert variety: Proof of the minimality conjecture

   https://doi.org/10.1112/jlms.12832  

   Dizier, Avery St. ; Yong, Alexander ( November 2023 , Journal of the London Mathematical Society)

   A minimal presentation of the cohomology ring of the flag manifold was given in A. Borel (1953). This presentation was extended by E. Akyildiz–A. Lascoux–P. Pragacz (1992) to a nonminimal one for all Schubert varieties. Work of V. Gasharov–V. Reiner (2002) gave a short, that is, polynomial‐size, presentation for a subclass of Schubert varieties that includes the smooth ones. In V. Reiner–A. Woo–A. Yong (2011), a general shortening was found; it implies an exponential upper bound of on the number of generators required. That work states a minimality conjecture whose significance would be an exponential lower bound of on the number of generators needed in worst case, giving the first obstructions to short presentations. We prove the minimality conjecture. Our proof uses the Hopf algebra structure of the ring of symmetric functions.

    

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