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1. Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules

   https://doi.org/10.1090/cams/19  

   Edie-Michell, Cain (May 2023, Communications of the American Mathematical Society)

   This paper is the first of a pair that aims to classify a large number of the type I I II quantum subgroups of the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . In this work we classify the braided auto-equivalences of the categories of local modules for all known type I I quantum subgroups of C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). These exceptional cases are the orbifolds C ( s l 2 , 16 ) Rep ⁡ ( Z 2 ) 0 \mathcal {C}(\mathfrak {sl}_{2}, 16)^0_{\operatorname {Rep}(\mathbb {Z}_{2})} , C ( s l 3 , 9 ) Rep ⁡ ( Z 3 ) 0 \mathcal {C}(\mathfrak {sl}_{3}, 9)^0_{\operatorname {Rep}(\mathbb {Z}_{3})} , C ( s l 4 , 8 ) Rep ⁡ ( Z 4 ) 0 \mathcal {C}(\mathfrak {sl}_{4}, 8)^0_{\operatorname {Rep}(\mathbb {Z}_{4})} , and C ( s l 5 , 5 ) Rep ⁡ ( Z 5 ) 0 \mathcal {C}(\mathfrak {sl}_{5}, 5)^0_{\operatorname {Rep}(\mathbb {Z}_{5})} . We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . Our methods here are general, and the techniques developed will generalise to give skein theory for any orbifold of a braided tensor category. We also give a formulation of orthogonal level-rank duality in the type D D - D D case, which is used to construct one of the exceptionals. We uncover an unexpected connection between quadratic categories and exceptional braided auto-equivalences of the orbifolds. We use this connection to construct two of the four exceptionals. In the sequel to this paper we will use the classified braided auto-equivalences to construct the corresponding type I I II quantum subgroups of the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . This will essentially finish the type I I II classification for s l n \mathfrak {sl}_n modulo type I I classification. When paired with Gannon’s type I I classification for r ≤ 6 r\leq 6 , our results will complete the type I I II classification for these same ranks. This paper includes an appendix by Terry Gannon, which provides useful results on the dimensions of objects in the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . 

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2. Invariant random subgroups of semidirect products

   https://doi.org/10.1017/etds.2018.46  

   BIRINGER, IAN; BOWEN, LEWIS; TAMUZ, OMER (January 2018, Ergodic Theory and Dynamical Systems)

   We study invariant random subgroups (IRSs) of semidirect products $$G=A\rtimes \unicode[STIX]{x1D6E4}$$ . In particular, we characterize all IRSs of parabolic subgroups of $$\text{SL}_{d}(\mathbb{R})$$ , and show that all ergodic IRSs of $$\mathbb{R}^{d}\rtimes \text{SL}_{d}(\mathbb{R})$$ are either of the form $$\mathbb{R}^{d}\rtimes K$$ for some IRS of $$\text{SL}_{d}(\mathbb{R})$$ , or are induced from IRSs of $$\unicode[STIX]{x1D6EC}\rtimes \text{SL}(\unicode[STIX]{x1D6EC})$$ , where $$\unicode[STIX]{x1D6EC}<\mathbb{R}^{d}$$ is a lattice. 

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3. Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

   https://doi.org/10.1093/imrn/rnaa281  

   Liaw, Constanze; Treil, Sergei; Volberg, Alexander (November 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $$d$$ perturbations $$A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$$, $${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$$, with $${\operatorname{Ran}}{\textbf{B}}$$ being cyclic for $$A$$, parametrized by $$d\times d$$ Hermitian matrices $${\boldsymbol{\alpha }}$$, the singular parts of the spectral measures of $$A$$ and $$A_{\boldsymbol{\alpha }}$$ are mutually singular for all $${\boldsymbol{\alpha }}$$ except for a small exceptional set $$E$$. It was shown earlier by the 1st two authors, see [4], that $$E$$ is a subset of measure zero of the space $$\textbf{H}(d)$$ of $$d\times d$$ Hermitian matrices. In this paper, we show that the set $$E$$ has small Hausdorff dimension, $$\dim E \le \dim \textbf{H}(d)-1 = d^2-1$$. 

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4. Tight Bounds for ℓ ₁ Oblivious Subspace Embeddings

   https://doi.org/10.1145/3477537  

   Wang, Ruosong; Woodruff, David P. (January 2022, ACM Transactions on Algorithms)

   An \ell _p oblivious subspace embedding is a distribution over r \times n matrices \Pi such that for any fixed n \times d matrix A , \[ \Pr _{\Pi }[\textrm {for all }x, \ \Vert Ax\Vert _p \le \Vert \Pi Ax\Vert _p \le \kappa \Vert Ax\Vert _p] \ge 9/10,\] where r is the dimension of the embedding, \kappa is the distortion of the embedding, and for an n -dimensional vector y , \Vert y\Vert _p = (\sum _{i=1}^n |y_i|^p)^{1/p} is the \ell _p -norm. Another important property is the sparsity of \Pi , that is, the maximum number of non-zero entries per column, as this determines the running time of computing \Pi A . While for p = 2 there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparsity, for the important case of 1 \le p \lt 2 , much less was known. In this article, we obtain nearly optimal tradeoffs for \ell _1 oblivious subspace embeddings, as well as new tradeoffs for 1 \lt p \lt 2 . Our main results are as follows: (1) We show for every 1 \le p \lt 2 , any oblivious subspace embedding with dimension r has distortion \[ \kappa = \Omega \left(\frac{1}{\left(\frac{1}{d}\right)^{1 / p} \log ^{2 / p}r + \left(\frac{r}{n}\right)^{1 / p - 1 / 2}}\right).\] When r = {\operatorname{poly}}(d) \ll n in applications, this gives a \kappa = \Omega (d^{1/p}\log ^{-2/p} d) lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for p = 1 is optimal up to {\operatorname{poly}}(\log (d)) factors. (2) We give sparse oblivious subspace embeddings for every 1 \le p \lt 2 . Importantly, for p = 1 , we achieve r = O(d \log d) , \kappa = O(d \log d) and s = O(\log d) non-zero entries per column. The best previous construction with s \le {\operatorname{poly}}(\log d) is due to Woodruff and Zhang (COLT, 2013), giving \kappa = \Omega (d^2 {\operatorname{poly}}(\log d)) or \kappa = \Omega (d^{3/2} \sqrt {\log n} \cdot {\operatorname{poly}}(\log d)) and r \ge d \cdot {\operatorname{poly}}(\log d) ; in contrast our r = O(d \log d) and \kappa = O(d \log d) are optimal up to {\operatorname{poly}}(\log (d)) factors even for dense matrices. We also give (1) \ell _p oblivious subspace embeddings with an expected 1+\varepsilon number of non-zero entries per column for arbitrarily small \varepsilon \gt 0 , and (2) the first oblivious subspace embeddings for 1 \le p \lt 2 with O(1) -distortion and dimension independent of n . Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise \ell _p low-rank approximation. Our results give improved algorithms for these applications. 

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5. Polynomial Parametrization for SL2 over Quadratic Number Rings

   https://doi.org/10.1093/imrn/rnz046  

   Larsen, Michael; Nguyen, Dong Quan (March 2019, International Mathematics Research Notices)

   null (Ed.)

   Abstract If $$R$$ is the ring of integers of a number field, then there exists a polynomial parametrization of the set $$\operatorname{SL}_2(R)$$, that is, an element $$A\in{\textrm{SL}}_2(\mathbb{Z}[x_1,\ldots ,x_n])$$ such that every element of $$\operatorname{SL}_2(R)$$ is obtained by specializing $$A$$ via some homomorphism $$\mathbb{Z}[x_1,\ldots ,x_n]\to R$$. 

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