More Like this

1. Coordinate rings and birational charts

   https://doi.org/https://doi.org/10.1090/ert/592  

   Fomin, Sergey; Lusztig, George (January 2022, Representation theory)

   Let $$G$$ be a semisimple simply connected complex algebraic group. Let $$U$$ be the unipotent radical of a Borel subgroup in $$G$$. We describe the coordinate rings of $$U$$ (resp., $G/U$, $$G$$) in terms of two (resp., four, eight) birational charts introduced by Lusztig [Total positivity in reductive groups, Birkhäuser Boston, Boston, MA, 1994; Bull. Inst. Math. Sin. (N.S.) 14 (2019), pp. 403–459] in connection with the study of total positivity. 

   more » « less
2. Contact graphs, boundaries, and a central limit theorem for $$\mathrm{CAT}(0)$$ cubical complexes

   https://doi.org/10.4171/GGD/775  

   Fernós, Talia; Lécureux, Jean; Mathéus, Frédéric (April 2024, Groups, Geometry, and Dynamics)

   LetXbe a nonelementary\mathrm{CAT}(0)cubical complex. We prove that ifXis essential and irreducible, then the contact graph ofX(introduced by Hagen (2014)) is unbounded and its boundary is homeomorphic to the regular boundary ofX(defined by Fernós (2018) and Kar–Sageev (2016)). Using this, we reformulate the Caprace–Sageev’s rank-rigidity theorem in terms of the action on the contact graph. LetGbe a group with a nonelementary action onX, and let (Z_{n})be a random walk corresponding to a generating probability measure onGwith finite second moment. Using this identification of the boundary of the contact graph, we prove a central limit theorem for (Z_{n}), namely that\frac{d(Z_{n} o,o)-nA}{\sqrt{n}}converges in law to a non-degenerate Gaussian distribution (A\hspace{-0.7pt}=\hspace{-0.7pt}\lim_{n\to\infty}\hspace{-0.7pt}\frac{d(Z_{n}o,o)}{n}is the drift of the random walk, ando\in Xis an arbitrary basepoint). 

   more » « less
3. Sums of proper divisors follow the Erdős–Kac law

   https://doi.org/10.1090/proc/16167  

   Pollack, Paul; Troupe, Lee (March 2023, Proceedings of the American Mathematical Society)

   Let s ( n ) = ∑ d ∣ n ,   d > n d s(n)=\sum _{d\mid n,~d>n} d denote the sum of the proper divisors of n n . The second-named author proved that ω ( s ( n ) ) \omega (s(n)) has normal order log ⁡ log ⁡ n \log \log {n} , the analogue for s s -values of a classical result of Hardy and Ramanujan [ The normal number of prime factors of a number n [Quart. J. Math. 48 (1917), 76–92], AMS Chelsea Publ., Providence, RI, 2000, pp. 262–275]. We establish the corresponding Erdős–Kac theorem: ω ( s ( n ) ) \omega (s(n)) is asymptotically normally distributed with mean and variance log ⁡ log ⁡ n \log \log {n} . The same method applies with s ( n ) s(n) replaced by any of several other unconventional arithmetic functions, such as β ( n ) ≔ ∑ p ∣ n p \beta (n)≔\sum _{p\mid n} p , n − φ ( n ) n-\varphi (n) , and n + τ ( n ) n+\tau (n) ( τ \tau being the divisor function). 

   more » « less
4. The trace of the affine Hecke category

   https://doi.org/10.1112/plms.12523  

   Gorsky, Eugene; Neguț, Andrei (April 2023, Proceedings of the London Mathematical Society)

   Abstract We compare the (horizontal) trace of the affine Hecke category with the elliptic Hall algebra, thus obtaining an “affine” version of the construction of Gorsky et al. (Int. Math. Res. Not. IMRN2022(2022) 11304–11400). Explicitly, we show that the aforementioned trace is generated by the objects as , where denote the Wakimoto objects of Elias and denote Rouquier complexes. We compute certain categorical commutators between the 's and show that they match the categorical commutators between the sheaves on the flag commuting stack that were considered in Neguț (Publ. Math. Inst. Hautes Études Sci. 135 (2022) 337–418). At the level of ‐theory, these commutators yield a certain integral form of the elliptic Hall algebra, which we can thus map to the ‐theory of the trace of the affine Hecke category. 

   more » « less
5. Proper proximality among various families of groups

   https://doi.org/10.4171/GGD/778  

   Ding, Changying; Kunnawalkam_Elayavalli, Srivatsav (June 2024, Groups, Geometry, and Dynamics)

   In this paper, the notion of proper proximality (introduced by Boutonnet, Ioana, and Peterson [Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), 445–482]) is studied and classified in various families of groups. We show that if a group acts non-elementarily by isometries on a tree such that, for any two edges, the intersection of their edge stabilizers is finite, thenGis properly proximal. We show that the wreath productG\wr His properly proximal if and only ifHis non-amenable. We then completely classify proper proximality among graph products of non-trivial groups. Our results generalize the recent work of Duchesne, Tucker-Drob, and Wesolek classifying inner amenability for these families of groups. Our results also recover some rigidity results associated to the group von Neumann algebras by virtue of being properly proximal. A key idea in the proofs of our theorems is a technique to upgrade from relative proper proximality using computations in the double dual of the small at infinity boundary. 

   more » « less