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2022 ◽  
Vol 26 (1) ◽  
pp. 1-16
Author(s):  
Sergey Fomin ◽  
George Lusztig

Let G G be a semisimple simply connected complex algebraic group. Let U U be the unipotent radical of a Borel subgroup in  G G . We describe the coordinate rings of U U (resp., G / U G/U , G 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.


2021 ◽  
Vol 157 (12) ◽  
pp. 2657-2698
Author(s):  
Runlin Zhang

In the present article, we study the following problem. Let $\boldsymbol {G}$ be a linear algebraic group over $\mathbb {Q}$ , let $\Gamma$ be an arithmetic lattice, and let $\boldsymbol {H}$ be an observable $\mathbb {Q}$ -subgroup. There is a $H$ -invariant measure $\mu _H$ supported on the closed submanifold $H\Gamma /\Gamma$ . Given a sequence $(g_n)$ in $G$ , we study the limiting behavior of $(g_n)_*\mu _H$ under the weak- $*$ topology. In the non-divergent case, we give a rather complete classification. We further supplement this by giving a criterion of non-divergence and prove non-divergence for arbitrary sequence $(g_n)$ for certain large $\boldsymbol {H}$ . We also discuss some examples and applications of our result. This work can be viewed as a natural extension of the work of Eskin–Mozes–Shah and Shapira–Zheng.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Falk Bannuscher ◽  
Alastair Litterick ◽  
Tomohiro Uchiyama

Abstract Let 𝑘 be a non-perfect separably closed field. Let 𝐺 be a connected reductive algebraic group defined over 𝑘. We study rationality problems for Serre’s notion of complete reducibility of subgroups of 𝐺. In particular, we present the first example of a connected non-abelian 𝑘-subgroup 𝐻 of 𝐺 that is 𝐺-completely reducible but not 𝐺-completely reducible over 𝑘, and the first example of a connected non-abelian 𝑘-subgroup H ′ H^{\prime} of 𝐺 that is 𝐺-completely reducible over 𝑘 but not 𝐺-completely reducible. This is new: all previously known such examples are for finite (or non-connected) 𝐻 and H ′ H^{\prime} only.


2021 ◽  
Author(s):  
Eli Amzallag ◽  
Andrei Minchenko ◽  
Gleb Pogudin

2021 ◽  
Vol 103 (3) ◽  
pp. 36-43
Author(s):  
A.A. Ibrayeva ◽  

In this paper we calculate cohomology of a classical Lie algebra of type A2 over an algebraically field k of characteristic p = 3 with coefficients in simple modules. To describe their structure we will consider them as modules over an algebraic group SL3(k). In the case of characteristic p = 3, there are only two peculiar simple modules: a simple that module isomorphic to the quotient module of the adjoint module by the center, and a one-dimensional trivial module. The results on the cohomology of simple nontrivial module are used for calculating the cohomology of the adjoint module. We also calculate cohomology of the simple quotient algebra Lie of A2 by the center.


2021 ◽  
Vol 25 (27) ◽  
pp. 780-806
Author(s):  
Pierre Baumann ◽  
Arnaud Demarais

Let G G be a connected reductive algebraic group over C \mathbb C . Through the geometric Satake equivalence, the fundamental classes of the Mirković–Vilonen cycles define a basis in each tensor product V ( λ 1 ) ⊗ ⋯ ⊗ V ( λ r ) V(\lambda _1)\otimes \cdots \otimes V(\lambda _r) of irreducible representations of G G . We compute this basis in the case G = S L 2 ( C ) G=\mathrm {SL}_2(\mathbb C) and conclude that in this case it coincides with the dual canonical basis at q = 1 q=1 .


Author(s):  
Hédi Nabli

In this paper, we introduce a new topic on geometric patterns issued from Truchet tile, that we agree to call combinatorial arabesque. The originally Truchet tile, by reference to the French scientist of 17th century Sébastien Truchet, is a square split along the diagonal into two triangles of contrasting colors. We define an equivalence relation on the set of all square tiling of same size, leading naturally to investigate the equivalence classes and their cardinality. Thanks to this class notion, it will be possible to measure the beauty degree of a Truchet square tiling by means of an appropriate algebraic group. Also, we define many specific arabesques such as entirely symmetric, magic and hyper-maximal arabesques. Mathematical characterizations of such arabesques are established facilitating thereby their enumeration and their algorithmic generating. Finally the notion of irreducibility is introduced on arabesques.


Author(s):  
Moshe Kamensky ◽  
Sergei Starchenko ◽  
Jinhe Ye

Abstract We consider G, a linear algebraic group defined over $\Bbbk $ , an algebraically closed field (ACF). By considering $\Bbbk $ as an embedded residue field of an algebraically closed valued field K, we can associate to it a compact G-space $S^\mu _G(\Bbbk )$ consisting of $\mu $ -types on G. We show that for each $p_\mu \in S^\mu _G(\Bbbk )$ , $\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$ is a solvable infinite algebraic group when $p_\mu $ is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of $\mathrm {Stab}\left (p_\mu \right )$ in terms of the dimension of p.


Author(s):  
János Kollár ◽  
Aaron Landesman

AbstractWe prove a generalization of the Fulton–Hansen connectedness theorem, where $${\mathbb {P}}^n$$ P n is replaced by a normal variety on which an algebraic group acts with a dense orbit.


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