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1. Generically free representations I: Large representations

   https://doi.org/doi.org/10.2140/ant.2020.14.1577  

   Garibaldi, S.; Guralnick, R. (January 2020, Algebra and number theory)

   This paper concerns a faithful representation V of a simple linear algebraic group G. Under mild assumptions, we show that if V is large enough, then the Lie algebra of G acts generically freely on V. That is, the stabilizer in Lie.G/ of a generic vector in V is zero. The bound on dim V grows like the square of the rank and holds with only mild hypotheses on the characteristic of the underlying field. The proof relies on results on generation of Lie algebras by conjugates of an element that may be of independent interest. We use the bound in subsequent works to determine which irreducible faithful representations are generically free, with no hypothesis on the characteristic of the field. This in turn has applications to the question of which representations have a stabilizer in general position. 

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2. GENERICALLY FREE REPRESENTATIONS II: IRREDUCIBLE REPRESENTATIONS

   https://doi.org/10.1007/s00031-020-09591-3  

   Garibaldi, S.; Guralnick, R. (January 2020, Transformation groups)

   We determine which faithful irreducible representations V of a simple linear algebraic group G are generically free for Lie(G), i.e., which V have an open subset consisting of vectors whose stabilizer in Lie(G) is zero. This relies on bounds on dim V obtained in prior work (part I), which reduce the problem to a finite number of possibilities for G and highest weights for V , but still infinitely many characteristics. The remaining cases are handled individually, some by computer calculation. These results were previously known for fields of characteristic zero, although new phenomena appear in prime characteristic; we provide a shorter proof that gives the result with very mild hypotheses on the characteristic. (The few characteristics not treated here are settled in part III.) These results are related to questions about invariants and the existence of a stabilizer in general position. 

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3. Hodge Representations

   https://doi.org/10.1017/exp.2020.55  

   Han, Xiayimei; Robles, Colleen (January 2020, Experimental Results)

   Clingher, Adrian (Ed.)

   Abstract Green–Griffiths–Kerr introduced Hodge representations to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford–Tate subdomains. We summarize how, given a fixed period domain $$ \mathcal{D} $$ , to enumerate the Hodge representations and corresponding Mumford–Tate subdomains $$ D \subset \mathcal{D} $$ . The procedure is illustrated in two examples: (i) weight two Hodge structures with $$ {p}_g={h}^{2,0}=2 $$ ; and (ii) weight three CY-type Hodge structures. 

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4. Representation via Representations: Domain Generalization via Adversarially Learned Invariant Representations

   Deng, Zhun; Ding, Frances; Dwork, Cynthia; Hong, Rachel; Parmigiani, Giovanni; Patil, Prasad; Sur, Pragya (June 2020, ArXivorg)

   We investigate the power of censoring techniques, first developed for learning {\em fair representations}, to address domain generalization. We examine {\em adversarial} censoring techniques for learning invariant representations from multiple "studies" (or domains), where each study is drawn according to a distribution on domains. The mapping is used at test time to classify instances from a new domain. In many contexts, such as medical forecasting, domain generalization from studies in populous areas (where data are plentiful), to geographically remote populations (for which no training data exist) provides fairness of a different flavor, not anticipated in previous work on algorithmic fairness. We study an adversarial loss function for k domains and precisely characterize its limiting behavior as k grows, formalizing and proving the intuition, backed by experiments, that observing data from a larger number of domains helps. The limiting results are accompanied by non-asymptotic learning-theoretic bounds. Furthermore, we obtain sufficient conditions for good worst-case prediction performance of our algorithm on previously unseen domains. Finally, we decompose our mappings into two components and provide a complete characterization of invariance in terms of this decomposition. To our knowledge, our results provide the first formal guarantees of these kinds for adversarial invariant domain generalization. 

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5. Regular Supercuspidal Representations

   Tasho Kaletha (July 2019, Journal of the American Mathematical Society)

   We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups. 

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