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1. On the codimension-two cohomology of SLn(ℤ).

   https://doi.org/10.1016/j.aim.2024.109795  

   Brück, Benjamin; Miller, Jeremy; Patzt, Peter; Sroka, Robin J; Wilson, Jennifer CH (August 2024, Advances in Mathematics)

   Borel–Serre proved that SL_n(Z) is a virtual duality group of dimension (n choose 2) and the Steinberg module St_n(Q) is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to SL_n(Q). We determine the “relations among the relations” of this Steinberg module. That is, we construct an explicit partial resolution of length two of the SL_n(Z)-module St_n(Q). We use this partial resolution to show the codimension-2 rational cohomology group of SLn(Z) vanishes for n ≥ 3. This resolves a case of a conjecture of Church–Farb–Putman. We also produce lower bounds for the codimension-1 cohomology of certain congruence subgroups of SLn(Z). 

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2. Effective Density for Inhomogeneous Quadratic Forms I: Generic Forms and Fixed Shifts

   https://doi.org/10.1093/imrn/rnaa206  

   Ghosh, Anish; Kelmer, Dubi; Yu, Shucheng (August 2020, International Mathematics Research Notices)

   Abstract We establish effective versions of Oppenheim’s conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result, which implies the optimal density for values of generic inhomogeneous forms. We also obtain a similar density result for fixed irrational shifts satisfying an explicit Diophantine condition. The main technical tool is a formula for the 2nd moment of Siegel transforms on certain congruence quotients of $$SL_n(\mathbb{R}),$$ which we believe to be of independent interest. In a sequel, we use different techniques to treat the companion problem concerning generic shifts and fixed quadratic forms. 

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3. PLETHORA OF CLUSTER STRUCTURES ON GL_n

   Gekhtman, M.; Shapiro, M.; Vainshtein, A. (January 2022, Memoirs of the American Mathematical Society)

   We continue the study of multiple cluster structures in the rings of regular functions on $$GL_n$$, $$SL_n$$ and $$\operatorname{Mat}_n$$ that are compatible with Poisson-Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin-Drinfeld classification of Poisson--Lie structures on a semisimple complex group $$\mathcal G$$ corresponds to a cluster structure in $$\mathcal O(\mathcal G)$$. Here we prove this conjecture for a large subset of Belavin-Drinfeld (BD) data of $$A_n$$ type, which includes all the previously known examples. Namely, we subdivide all possible $$A_n$$ type BD data into oriented and non-oriented kinds. In the oriented case, we single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson-Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on $$SL_n$$ compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $$SL_n$$ equipped with two different Poisson-Lie brackets. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address this situation in future publications. 

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4. Critical points of master functions and mKdV hierarchy of type $$A^{(2)}_{2n}$$, volume on Representations of Lie algebras, quantum groups and related topics, Contemp. Math., 713, Amer. Math. Soc., Providence, RI, 2018

   Alexander Varchenko, Tyler Woodruff (January 2018, Contemporary mathematics - American Mathematical Society)

   It is considered the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra A(2)2n. The population is naturally partitioned into an infinite collection of complex cells C^m, where m are some positive integers. For each cell it is defined an injective rational map C^m→M(A^{(2)}_{2n}) of the cell to the space M(A^{(2)}_{2n}) of Miura opers of type A^{(2)}_{2n}. It is shown that the image of the map is invariant with respect to all mKdV flows on M(A^{(2)}_{2n}) and the image is point-wise fixed by all mKdV flows ∂/∂t_r with index r greater than 4m. 

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5. Embeddings Among Quantum Affine sl_n

   https://doi.org/10.1007/s10114-023-2073-2  

   Li, Yi Qiang (June 2023, Acta Mathematica Sinica, English Series)

   We establish an explicit embedding of a quantum affine sl_n into a quantum affine sl_{n+1} . This embedding serves as a common generalization of two natural, but seemingly unrelated embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine sl_n and gl_n. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation theory of general linear groups over a finite field. The same embedding is also used in a work of Maksimau on the categorical representations of affine general linear algebras. We further provide a more natural compatibility statement of the em- bedding on the idempotent version with that on the quantum affine Schur algebra level. A gl_n-variant of the embedding is also established. 

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