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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 III: EXTREMELY BAD CHARACTERISTIC

   https://doi.org/10.1007/s00031-020-09590-4  

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

   In parts I and II, we determined 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, with some assumptions on the characteristic of the field. This paper settles the remaining cases, which are of a different nature because Lie(G) has a more complicated structure and there need not exist general dimension bounds of the sort that exist in good characteristic. 

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3. Representations and Tensor Product Growth

   https://doi.org/10.1093/imrn/rnac172  

   Larsen, Michael; Shalev, Aner; Tiep, Pham Huu (July 2022, International Mathematics Research Notices)

   Abstract The deep theory of approximate subgroups establishes three-step product growth for subsets of finite simple groups $$G$$ of Lie type of bounded rank. In this paper, we obtain two-step growth results for representations of such groups $$G$$ (including those of unbounded rank), where products of subsets are replaced by tensor products of representations. Let $$G$$ be a finite simple group of Lie type and $$\chi $$ a character of $$G$$. Let $$|\chi |$$ denote the sum of the squares of the degrees of all (distinct) irreducible characters of $$G$$ that are constituents of $$\chi $$. We show that for all $$\delta>0$$, there exists $$\epsilon>0$$, independent of $$G$$, such that if $$\chi $$ is an irreducible character of $$G$$ satisfying $$|\chi | \le |G|^{1-\delta }$$, then $$|\chi ^2| \ge |\chi |^{1+\epsilon }$$. We also obtain results for reducible characters and establish faster growth in the case where $$|\chi | \le |G|^{\delta }$$. In another direction, we explore covering phenomena, namely situations where every irreducible character of $$G$$ occurs as a constituent of certain products of characters. For example, we prove that if $$|\chi _1| \cdots |\chi _m|$$ is a high enough power of $|G|$, then every irreducible character of $$G$$ appears in $$\chi _1\cdots \chi _m$$. Finally, we obtain growth results for compact semisimple Lie groups. 

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4. Temperature dependent charge transport in electrostatically doped poly[benzimidazobenzophenanthroline] thin films

   https://doi.org/10.1002/app.53470  

   Cruz‐Arzon, Alejandro J.; Serrano‐Garcia, William; Pinto, Nicholas J.; Gupta, Nikita; Johnson, Alan T. Charlie (November 2022, Journal of Applied Polymer Science)

   Abstract Charge transport in electrostatically doped poly[benzimidazobenzophenanthroline]‐BBL thin films in a field‐effect transistor geometry were investigated in the temperature range 150 K < T < 370 K. At low temperatures activation and hopping transport mechanisms dominated, while phonon scattering dominated at high temperatures. The activation energies (E_A) were found to lie in the range 140 meV < E_A < 400 meV implying the existence of deep traps within the polymer bandgap of 1.8 eV. Two quasi‐linear dependencies ofE_Aon the gate voltage (V_g) were observed withE_Adecreasing asV_gincreased. An unexpected “metallic‐like” transport characteristic appeared forT > 335 K which depended onV_g. Enhanced electron delocalization combined with increased carrier density could be responsible for this “metallic‐like” behavior. Our results show that the existence of deep traps with multiple energy distributions, combined with increased carrier density led to the unusual temperature dependence of charge transport observed in BBL. 

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5. Quantum entropy and central limit theorem

   https://doi.org/10.1073/pnas.2304589120  

   Bu, Kaifeng; Gu, Weichen; Jaffe, Arthur (June 2023, Proceedings of the National Academy of Sciences)

   We introduce a framework to study discrete-variable (DV) quantum systems based on qudits. It relies on notions of a mean state (MS), a minimal stabilizer-projection state (MSPS), and a new convolution. Some interesting consequences are: The MS is the closest MSPS to a given state with respect to the relative entropy; the MS is extremal with respect to the von Neumann entropy, demonstrating a “maximal entropy principle in DV systems.” We obtain a series of inequalities for quantum entropies and for Fisher information based on convolution, giving a “second law of thermodynamics for quantum convolutions.” We show that the convolution of two stabilizer states is a stabilizer state. We establish a central limit theorem, based on iterating the convolution of a zero-mean quantum state, and show this converges to its MS. The rate of convergence is characterized by the “magic gap,” which we define in terms of the support of the characteristic function of the state. We elaborate on two examples: the DV beam splitter and the DV amplifier. 

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