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1. Three-Dimensional Mirror Symmetry and Elliptic Stable Envelopes

   https://doi.org/10.1093/imrn/rnaa389  

   Rimányi, Richárd; Smirnov, Andrey; Zhou, Zijun; Varchenko, Alexander (February 2021, International Mathematics Research Notices)

   Abstract We consider a pair of quiver varieties $$(X;X^{\prime})$$ related by 3D mirror symmetry, where $$X =T^*{Gr}(k,n)$$ is the cotangent bundle of the Grassmannian of $$k$$-planes of $$n$$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on $$X \times X^{\prime} $$ (the mother function) whose restrictions to $$X$$ and $$X^{\prime} $$ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $$X$$ and $$X^{\prime}$$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side. 

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2. Two-Parameter Quantum Groups and $$R$$-Matrices: Classical Types

   https://doi.org/10.3842/SIGMA.2025.064  

   Martin, Ian; Tsymbaliuk, Alexander (July 2025, Symmetry, Integrability and Geometry: Methods and Applications)

   We construct finite $$R$$-matrices for the first fundamental representation $$V$$ of two-parameter quantum groups $$U_{r,s}(\mathfrak{g})$$ for classical $$\mathfrak{g}$$, both through the decomposition of $$V\otimes V$$ into irreducibles $$U_{r,s}(\mathfrak{g})$$-submodules as well as by evaluating the universal $$R$$-matrix. The latter is crucially based on the construction of dual PBW-type bases of $$U^{\pm}_{r,s}(\mathfrak{g})$$ consisting of the ordered products of quantum root vectors defined via $(r,s)$-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine $$R$$-matrices, both through the Yang-Baxterization technique of [Internat. J. Modern Phys. A 6 (1991), 3735-3779] and as the unique intertwiner between the tensor product of $V(u)$ and $V(v)$, viewed as modules over two-parameter quantum affine algebras $$U_{r,s}(\widehat{\mathfrak{g}})$$ for classical $$\mathfrak{g}$$. The latter generalizes the formulas of [Comm. Math. Phys. 102 (1986), 537-547] for one-parametric quantum affine algebras. 

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3. Boolean Product Polynomials, Schur Positivity, and Chern Plethysm

   https://doi.org/10.1093/imrn/rnz261  

   Billey, Sara C.; Rhoades, Brendon; Tewari, Vasu (November 2019, International Mathematics Research Notices)

   Abstract Let $$k \leq n$$ be positive integers, and let $$X_n = (x_1, \dots , x_n)$$ be a list of $$n$$ variables. The Boolean product polynomial$$B_{n,k}(X_n)$$ is the product of the linear forms $$\sum _{i \in S} x_i$$, where $$S$$ ranges over all $$k$$-element subsets of $$\{1, 2, \dots , n\}$$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $$B_{n,k}(X_n)$$ for certain $$k$$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $$B_{n,n-1}(X_n)$$ to a bigraded action of the symmetric group $${\mathfrak{S}}_n$$ on a divergence free quotient of superspace. 

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4. Classical freeness of orthosymplectic affine vertex superalgebras

   https://doi.org/10.1090/proc/16548  

   Creutzig, Thomas; Linshaw, Andrew; Song, Bailin (October 2024, Proceedings of the American Mathematical Society)

   The question of when a vertex algebra is a quantization of the arc space of its associated scheme has recently received a lot of attention in both the mathematics and physics literature. This property was first studied by Tomoyuki Arakawa and Anne Moreau (see their paper in the references), and was given the name \lq\lq classical freeness by Jethro van Ekeren and Reimundo Heluani [Comm. Math. Phys. 386 (2021), no. 1, pp. 495-550] in their work on chiral homology. Later, it was extended to vertex superalgebras by Hao Li [Eur. J. Math. 7 (2021), pp. 1689–1728]. In this note, we prove the classical freeness of the simple affine vertex superalgebra $L_n\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{m|2r}\right)$for all positive integers $m,n,r$satisfying $-<\#comment/>\frac{m}{2}+r+n+1>0$. In particular, it holds for the rational vertex superalgebras $L_n\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2r}\right)$for all positive integers $r,n$. 

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5. A partial converse ghost lemma for the derived category of a commutative Noetherian ring

   https://doi.org/10.1090/proc/16294  

   Liu, Jian; Pollitz, Josh (January 2023, Proceedings of the American Mathematical Society)

   In this article a condition is given to detect the containment among thick subcategories of the bounded derived category of a commutative noetherian ring. More precisely, for a commutative noetherian ring $R$and complexes of $R$-modules with finitely generated homology $M$and $N$, we show $N$is in the thick subcategory generated by $M$if and only if the ghost index of $N_{\mathfrak{p}}$with respect to $M_{\mathfrak{p}}$is finite for each prime $\mathfrak{p}$of $R$. To do so, we establish a “converse coghost lemma” for the bounded derived category of a non-negatively graded DG algebra with noetherian homology. 

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