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1. Normal Reflection Subgroups

   Arreche, Carlos; Williams, Nathan (January 2020, Séminaire lotharingien de combinatoire)

   We study normal reflection subgroups of complex reflection groups. Our point of view leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalized exponents. Our refinement gives a uniform proof and generalization of a recent theorem of the second author. 

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2. Products of normal subsets

   https://doi.org/10.1090/tran/8960  

   Larsen, Michael; Shalev, Aner; Tiep, Pham (February 2024, Transactions of the American Mathematical Society)

   In this paper we consider which families of finite simple groups $G$have the property that for each $\mathrm{ϵ<\#comment/>}>0$there exists $N>0$such that, if $|G|\ge <\#comment/>N$and $S,T$are normal subsets of $G$with at least $\mathrm{ϵ<\#comment/>}|G|$elements each, then every non-trivial element of $G$is the product of an element of $S$and an element of $T$. We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form ${\mathrm{P}\mathrm{S}\mathrm{L}}_n\left(q\right)$where $q$is fixed and $n\to <\#comment/>\mathrm{\infty <\#comment/>}$. However, in the case $S=T$and $G$alternating this holds with an explicit bound on $N$in terms of $\mathrm{ϵ<\#comment/>}$. Related problems and applications are also discussed. In particular we show that, if $w_1,w_2$are non-trivial words, $G$is a finite simple group of Lie type of bounded rank, and for $g\in <\#comment/>G$, $P_{w_1\left(G\right),w_2\left(G\right)}\left(g\right)$denotes the probability that $g_1{g}_2=g$where $g_i\in <\#comment/>w_i\left(G\right)$are chosen uniformly and independently, then, as $|G|\to <\#comment/>\mathrm{\infty <\#comment/>}$, the distribution $P_{w_1\left(G\right),w_2\left(G\right)}$tends to the uniform distribution on $G$with respect to the $L^{\mathrm{\infty <\#comment/>}}$norm. 

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3. Normal weak eigenstate thermalization

   https://doi.org/10.1103/PhysRevB.110.104202  

   Łydżba, Patrycja; Świętek, Rafał; Mierzejewski, Marcin; Rigol, Marcos; Vidmar, Lev (September 2024, Physical Review B)
4. Normal Form for Renormalization Groups

   https://doi.org/10.1103/PhysRevX.9.021014  

   Raju, Archishman; Clement, Colin B.; Hayden, Lorien X.; Kent-Dobias, Jaron P.; Liarte, Danilo B.; Rocklin, D. Zeb; Sethna, James P. (April 2019, Physical Review X)
5. Mixed Volumes of Normal Complexes

   https://doi.org/10.1007/s00454-024-00662-w  

   Nowak, Lauren; O’Melveny, Patrick; Ross, Dustin (June 2024, Discrete & Computational Geometry)