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1. Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields

   https://doi.org/10.1515/crelle-2022-0053  

   Grantcharov, Dimitar; Serganova, Vera (September 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let 𝒲 n {{\mathcal{W}}_{n}} be the Lie algebra of polynomial vector fields.We classify simple weight 𝒲 n {{\mathcal{W}}_{n}} -modules M with finite weight multiplicities. We prove that every such nontrivial module M is either a tensor module or the unique simple submodule in a tensor module associatedwith the de Rham complex on ℂ n {\mathbb{C}^{n}} . 

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2. Beyond Lazy Training for Over-parameterized Tensor Decomposition

   Wang, Xiang; Wu, Chenwei; Lee, Jason D; Ma, Tengyu; Ge, Rong (January 2020, NeurIPS)

   Over-parametrization is an important technique in training neural networks. In both theory and practice, training a larger network allows the optimization algorithm to avoid bad local optimal solutions. In this paper we study a closely related tensor decomposition problem: given an l-th order tensor in (Rd)⊗l of rank r (where r≪d), can variants of gradient descent find a rank m decomposition where m>r? We show that in a lazy training regime (similar to the NTK regime for neural networks) one needs at least m=Ω(dl−1), while a variant of gradient descent can find an approximate tensor when m=O∗(r2.5llogd). Our results show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data. 

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3. Rigidity of Ext and Tor with Coefficients in Residue Fields of a Commutative Noetherian Ring

   https://doi.org/10.1017/s0013091518000081  

   Christensen, Lars Winther; Iyengar, Srikanth B.; Marley, Thomas (May 2019, Proceedings of the Edinburgh Mathematical Society)

   Abstract Let 𝔭 be a prime ideal in a commutative noetherian ring R . It is proved that if an R -module M satisfies $${\rm Tor}_n^R $$ ( k (𝔭), M ) = 0 for some n ⩾ R 𝔭 , where k (𝔭) is the residue field at 𝔭, then $${\rm Tor}_i^R $$ ( k (𝔭), M ) = 0 holds for all i ⩾ n . Similar rigidity results concerning $${\rm Tor}_R^{\ast} $$ ( k (𝔭), M ) are proved, and applications to the theory of homological dimensions are explored. 

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4. Higher order Kirillov--Reshetikhin modules for 𝐔 _q ( A _n ⁽¹⁾ ), imaginary modules and monoidal categorification

   https://doi.org/10.1515/crelle-2023-0068  

   Brito, Matheus; Chari, Vyjayanthi (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We study the family of irreducible modules for quantum affine $𝔰{𝔩}_{n+1}${\mathfrak{sl}_{n+1}}whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to $A_m${A_{m}}with $m\le n${m\leq n}. These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category ${\mathcal{𝒞}}^{-}${\mathscr{C}^{-}}. This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc,A cluster algebra approach toq-characters of Kirillov–Reshetikhin modules,J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez,Geometric conditions for $\mathrm{\square }$\square-irreducibility of certain representations of the general linear group over a non-archimedean local field,Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type $D_4${D_{4}}which do not arise from an embedding of $A_r${A_{r}}with $r\le 3${r\leq 3}in $D_4${D_{4}}. 

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5. Tate resolutions and {MCM} approximations

   https://doi.org/10.1090/conm/773/15531  

   David Eisenbud and Frank-Olaf Schreyer (January 2021, Contemporary mathematics)

   Let M be a finitely generated Cohen-Macaulay module of codimension m over a Gorenstein Ring R=S/I, where S is a regular ring. We show how to form a quasi-isomorphism ϕ from an R-free resolution of M to the dual of an R-free resolution of M∨:=ExtmR(M,R) using the S-free resolutions of R and M. The mapping cone of ϕ is then a Tate resolution of M, allowing us to compute the maximal Cohen-Macaulay approximation of M. "In the case when R is 0-dimensional local, and M is the residue field, the formula for ϕ becomes a formula for the socle of R generalizing a well-known formula for the socle of a zero-dimensional complete intersection. "When I⊂J⊂S are ideals generated by regular sequences, the R-module M=S/J is called a quasi-complete intersection, and ϕ was studied in detail by Kustin and Şega. We relate their construction to the sequence of `EagonNorthcott'-like complexes originally introduced by Buchsbaum and Eisenbud. 

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