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We prove several positive results regarding representation of homotopy classes of spheres and algebraic groups by regular mappings. Most importantly we show that every mapping from a sphere to an orthogonal or a unitary group is homotopic to a regular one. Furthermore we prove that algebraic homotopy classes of spheres form a subgroup of the homotopy group, and that a similar result holds also for cohomotopy groups of arbitrary varieties. [ABSTRACT FROM AUTHOR]

We begin with the observation that the p -adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts ϕ that are "Lie invariant mod p " in the sense that the "normalized" action of ϕ on 1-forms preserves mod p the space of invariant 1-forms. Our main result is that, after removing the 2-torsion sections, the above situation can be "infinitesimally deformed" in the sense that the above mod p result has a mod p 2 analogue. We end by showing that, in contrast with the case of elliptic curves, the following holds: if G is a linear algebraic group over a number field and if G is not a torus then for all but finitely many primes p the p -adic completion of G does not possess a Frobenius lift that is Lie invariant mod p. [ABSTRACT FROM AUTHOR]

The power word problem for a group G asks whether an expression u 1 x 1 ⋯ u n x n , where the u i are words over a finite set of generators of G and the x i binary encoded integers, is equal to the identity of G . It is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over G ). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group G is uNC 1 -many-one reducible to the power word problem for a finite-index subgroup of G . For our main result, we consider graph products of groups that do not have elements of order two. We show that the power word problem in a fixed such graph product is AC 0 -Turing-reducible to the word problem for the free group F 2 and the power word problems of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups C without order two elements, the uniform power word problem in a graph product can be solved in AC 0 [ C = L UPowWP (C) ] , where UPowWP (C) denotes the uniform power word problem for groups from the class C . As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is NP -complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. In the present work we correct this mistake. While we strongly conjecture that the result as stated in Stober and Weiß (2022a) is true, our proof relies on this additional assumption. [ABSTRACT FROM AUTHOR]

Let G${\bf G}$ be a linear algebraic group, not necessarily connected or reductive, over the field of real numbers R${\mathbb {R}}$. We describe a method, implemented on computer, to find the first Galois cohomology set H1(R,G)${\rm H}^1({\mathbb {R}},{\bf G})$. The output is a list of 1‐cocycles in G${\bf G}$. Moreover, we describe an implemented algorithm that, given a 1‐cocycle z∈Z1(R,G)$z\in {\rm Z}^1({\mathbb {R}}, {\bf G})$, finds the cocycle in the computed list to which z$z$ is equivalent, together with an element of G(C)${\bf G}({\mathbb {C}})$ realizing the equivalence. [ABSTRACT FROM AUTHOR]

Let |$ G $| be a complex connected reductive algebraic group and let |$ G_{{\mathbb {R}}} $| be a real form of |$ G $|⁠. We construct a sequence of functors |$ L_{n}\mathcal {R}$| from admissible (resp. finite-length) representations of |$ G $| to admissible (resp. finite-length) representations of |$ G_{{\mathbb {R}}} $|⁠. We establish many basic properties of these functors, including their behavior with respect to infinitesimal character, associated variety, and restriction to a maximal compact subgroup. We deduce that each |$ L_{n}\mathcal {R}$| takes unipotent representations of |$ G $| to unipotent representations of |$ G_{{\mathbb {R}}} $|⁠. Taking the alternating sum of |$ L_{n}\mathcal {R}$|⁠ , we get a well-defined homomorphism on the level of characters. We compute this homomorphism in the case when |$ G_{{\mathbb {R}}} $| is split. [ABSTRACT FROM AUTHOR]

Let |$G$| be a connected reductive group over |$\overline{{\mathbb{F}}}_{q}$| and let |$\rho ^{\vee }:G^{\vee }\rightarrow \mathrm{GL}_{n}$| be an algebraic representation of the dual group |$G^{\vee }$|⁠. Assuming that |$G$| and |$\rho ^{\vee }$| are defined over |${\mathbb{F}}_{q}$|⁠ , Braverman and Kazhdan defined an operator on the space |${{\mathcal{C}}}(G({\mathbb{F}}_{q}))$| of complex valued functions on |$G({\mathbb{F}}_{q})$|⁠. In this paper we are interested in the case where |$G$| is either |$\mathrm{SL}_{2}$| or |$\mathrm{PGL}_{2}$| and |$\rho ^{\vee }$| is the symmetric square representation of |$G^{\vee }$|⁠. We construct a natural |$G\times G$| -equivariant embedding |$G\hookrightarrow{{\mathcal{G}}}={{\mathcal{G}}}_{\rho }$| and an involutive operator (Fourier transform) |${{\mathcal{F}}}^{{{\mathcal{G}}}}$| on the space of functions |${{\mathcal{C}}}({{\mathcal{G}}}({\mathbb{F}}_{q}))$| that extends Braverman-Kazhdan's operator. [ABSTRACT FROM AUTHOR]

Let |$G$| be a real reductive algebraic group, and let |$H\subset G$| be an algebraic subgroup. It is known that the action of |$G$| on the space of functions on |$G/H$| is "tame" if this space is spherical. In particular, the multiplicities of the space |${\mathcal {S}}(G/H)$| of Schwartz functions on |$G/H$| are finite in this case. In this paper, we formulate and analyze a generalization of sphericity that implies finite multiplicities in |${\mathcal {S}}(G/H)$| for small enough irreducible representations of |$G$|⁠. [ABSTRACT FROM AUTHOR]

Let \mathsf G be a connected reductive linear algebraic group defined over \mathbb R, and let C: \mathsf G\rightarrow \mathsf G be a fundamental Chevalley involution. We show that for every g\in \mathsf G(\mathbb R), C(g) is conjugate to g^{-1} in the group \mathsf G(\mathbb R). Similar result on the Lie algebras is also obtained. [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) $ \mathcal {V} $ V , and let $ {\Bbbk [\mathcal {V}]^{G}} $ k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set $ S \subseteq {\Bbbk [\mathcal {V}]^{G}} $ S ⊆ k [ V ] G is a set of polynomials with the property that for all $ v,w \in \mathcal {V} $ v , w ∈ V , if there exists $ f \in {\Bbbk [\mathcal {V}]^{G}} $ f ∈ k [ V ] G separating $v$ and $w$, then there exists $ f \in S $ f ∈ S separating $v$ and $w$. In this article, we consider the action of $ G = \operatorname {GL}_2(\mathbb {C}) $ G = GL 2 ⁡ (C) on the $ \mathbb {C} $ C -vector space $ {\mathcal {M}}_2^n $ M 2 n of n-tuples of $ 2 \times 2 $ 2 × 2 matrices by simultaneous conjugation. Minimal generating sets $ S_n $ S n of $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G are well known and $ |S_n| = \frac 16(n^3+11n) $ | S n | = 1 6 (n 3 + 11 n). In recent work, Kaygorodov et al. [Kaygorodov I, Lopatin A, Popov Y. Separating invariants for $ 2 \times 2 $ 2 × 2 matrices. Linear Algebra Appl. 2018;559:114-124.] showed that for all $ n \geq ~1 $ n ≥ 1 , $ S_n $ S n is a minimal separating set by inclusion, i.e. that no proper subset of $ S_n $ S n is a separating set. This does not necessarily mean that $ S_n $ S n has minimum cardinality among all separating sets for $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G . Our main result shows that any separating set for $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G has cardinality $ \geq ~5n-5 $ ≥ 5 n − 5. In particular, there is no separating set of size $ \dim (\mathbb {C}[{\mathcal {M}}_2^n]^G) = 4n-3 $ dim ⁡ (C [ M 2 n ] G) = 4 n − 3 for $ n \geq ~3 $ n ≥ 3. Further, $ S_3 $ S 3 has indeed minimum cardinality as a separating set, but for $ n \geq ~4 $ n ≥ 4 there may exist a smaller separating set than $ S_n $ S n . We show that a smaller separating set does in fact exist for all $ n \geq ~5 $ n ≥ 5. We also prove similar results for the left–right action of $ \operatorname {SL}_2(\mathbb {C}) \times \operatorname {SL}_2(\mathbb {C}) $ SL 2 ⁡ (C) × SL 2 ⁡ (C) on $ {\mathcal {M}}_2^n $ M 2 n . [ABSTRACT FROM AUTHOR]

Let H be a complex linear algebraic group with internally graded unipotent radical acting on a complex projective variety X . Given an ample linearisation of the action and an associated Fubini–Study Kähler form which is invariant for a maximal compact subgroup Q of H , we define a notion of moment map for the action of H , and under suitable conditions (that the linearisation is well-adapted and semistability coincides with stability) we describe the (non-reductive) GIT quotient X / / H introduced in (Bérczi et al. in J. Topol. 11(3):826–855, 2018) in terms of this moment map. Using this description we derive formulas for the Betti numbers of X / / H and express the rational cohomology ring of X / / H in terms of the rational cohomology ring of the GIT quotient X / / T H , where T H is a maximal torus in H . We relate intersection pairings on X / / H to intersection pairings on X / / T H , obtaining a residue formula for these pairings on X / / H analogous to the residue formula of (Jeffrey and Kirwan in Topology 34(2):291–327, 1995). As an application, we announce a proof of the Green–Griffiths–Lang and Kobayashi conjectures for projective hypersurfaces with polynomial degree. [ABSTRACT FROM AUTHOR]

Let G be a smooth connected linear algebraic group over a field k, and let X be a G-torsor. Totaro asked: if X admits a zero-cycle of degree d \geq 1, then does X have a closed étalé point of degree dividing d? We give an affirmative answer for absolutely simple classical adjoint groups of types A_1 and A_{2n} over fields of characteristic \neq 2. [ABSTRACT FROM AUTHOR]

We determine the essential dimension of an arbitrary semisimple group of type B of the form G = (Spin (2 n 1 + 1) × ⋯ × Spin (2 n m + 1)) / μ over a field of characteristic 0, for all n 1 , ... , n m ≥ 7 , and a central subgroup μ of Spin (2 n 1 + 1) × ⋯ × Spin (2 n m + 1) not containing the center of Spin (2 n i + 1) as a direct factor for every i = 1 , ... , m. [ABSTRACT FROM AUTHOR]

A function that is analytic on a domain of C n is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. We define and study the Bernstein–Sato polynomial of a holonomic function on a smooth algebraic variety. We analyze the structure of certain sheaves of holonomic functions, such as the algebraic functions along a hypersurface, determining their direct sum decompositions into indecomposables, that further respect decompositions of Bernstein–Sato polynomials. When the space is endowed with the action of a linear algebraic group G, we study the class of G-finite analytic functions, i.e. functions that under the action of the Lie algebra of G generate a finite dimensional rational G-module. These are automatically algebraic functions on a variety with a dense orbit. When G is reductive, we give several representation-theoretic techniques toward the determination of Bernstein–Sato polynomials of G-finite functions. We classify the G-finite functions on all but one of the irreducible reduced prehomogeneous vector spaces, and compute the Bernstein–Sato polynomials for distinguished G-finite functions. The results can be used to construct explicitly equivariant D -modules. [ABSTRACT FROM AUTHOR]

Let 퐺 be a linear algebraic group defined over a finite field F q . We present several connections between the isogenies of 퐺 and the finite groups of rational points (G ⁢ (F q n)) n ≥ 1 . We show that an isogeny ϕ : G ′ → G over F q gives rise to a subgroup of fixed index in G ⁢ (F q n) for infinitely many 푛. Conversely, we show that if 퐺 is reductive, the existence of a subgroup H n of fixed index 푘 for infinitely many 푛 implies the existence of an isogeny of order 푘. In particular, we show that the infinite sequence H n is covered by a finite number of isogenies. This result applies to classical groups GL m , SL m , SO m , SU m , Sp 2 ⁢ m and can be extended to non-reductive groups if 푘 is prime to the characteristic. As a special case, we see that if 퐺 is simply connected, the minimal indices of proper subgroups of G ⁢ (F q n) diverge to infinity. Similar results are investigated regarding the sequence (G ⁢ (F p)) p by varying the characteristic 푝. [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V , and let k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set S ⊆ k [ V ] G is a set of polynomials with the property that for all v , w ∈ V , if there exists f ∈ k [ V ] G separating v and w , then there exists f ∈ S separating v and w. In this article we consider the action of G = SL 2 (C) × SL 2 (C) on the C -vector space M 2 , 2 n of n -tuples of 2 × 2 matrices by multiplication on the left and the right. Minimal generating sets S n of C [ M 2 , 2 n ] G are known, and | S n | = 1 24 (n 4 − 6 n 3 + 23 n 2 + 6 n). In recent work, Domokos [8] showed that for all n ≥ 1 , S n is a minimal separating set by inclusion, i.e. that no proper subset of S n is a separating set. This does not necessarily mean that S n has minimum cardinality among all separating sets for C [ M 2 , 2 n ] G. Our main result shows that any separating set for C [ M 2 , 2 n ] G has cardinality ≥ 5 n − 9. In particular, there is no separating set of size dim ⁡ (C [ M 2 n ] G) = 4 n − 6 for n ≥ 4. Further, S 4 has indeed minimum cardinality as a separating set, but for n ≥ 5 there may exist a smaller separating set than S n. We also consider the action of G = SL l (C) on M l , n by left multiplication. In that case the algebra of invariants has a minimum generating set of size ( n l ) (the l × l minors of a generic matrix) and dimension l n − l 2 + 1. We show that a separating set for C [ M l , n ] G must have size at least (2 l − 2) n − 2 (l 2 − l). In particular, C [ M l , n ] G does not contain a separating set of size dim ⁡ (C [ M l , n ] G) for l ≥ 3 and n ≥ l + 2. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem. [ABSTRACT FROM AUTHOR]

We show that for Chevalley groups G(R) of rank at least 2 over an integral domain R each root subgroup is (essentially) the double centralizer of a corresponding root element. In many cases, this implies that R and G(R) are bi-interpretable, yielding a new approach to biinterpretability for algebraic groups over a wide range of rings and fields. For such groups it then follows that the group G(R) is (finitely) axiomatizable in the appropriate class of groups provided R is (finitely) axiomatizable in the corresponding class of rings. [ABSTRACT FROM AUTHOR]

Henstock and Macbeath asked in 1953 whether the Brunn–Minkowski inequality can be generalized to nonabelian locally compact groups; questions along the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. The proof follows an induction on dimension strategy; new ingredients include an understanding of the role played by maximal compact subgroups of Lie groups, a necessary modified form of the inequality which is also applicable to nonunimodular locally compact groups, and a proportionated averaging trick. [ABSTRACT FROM AUTHOR]

Let |$G$| be a connected linear algebraic group over a number field |$K$|⁠ , let |$\Gamma $| be a finitely generated Zariski dense subgroup of |$G(K)$|⁠ , and let |$Z\subseteq G(K)$| be a thin set, in the sense of Serre. We prove that, if |$G/\textrm {R}_{u}(G)$| is either trivial or semisimple and |$Z$| satisfies certain necessary conditions, then a long random walk on a Cayley graph of |$\Gamma $| hits elements of |$Z$| with negligible probability. We deduce corollaries to Galois covers, characteristic polynomials, and fixed points in group actions. We also prove analogous results in the case where |$K$| is a global function field. [ABSTRACT FROM AUTHOR]

We study the prime-to- p Hecke action on the projective limit of the sets of connected components of Shimura varieties with fixed parahoric or Bruhat–Tits level at p. In particular, we construct infinitely many Shimura varieties for CM unitary groups in odd variables for which the considered actions are not transitive. We prove this result by giving negative examples on the question of Bruhat–Colliot-Thélène–Sansuc–Tits and its variant, which are related to the weak approximation on tori over Q. [ABSTRACT FROM AUTHOR]

The article focuses on complements, counterexamples, and conjectures in the context of a topological form of the Gabriel-Rosenberg theorem for varieties characterized by constructible étale sheaves. Topics include the determination of the Zariski topological space of a scheme based on the category of constructible étale sheaves, analogues of Gabriel's reconstruction theorem, and examples illustrating inability to recover a field from Zariski topology of a projective space over a finite field.

The article focuses on the problem of reconstructing a variety from its topological space, which is equivalent to reconstructing linear equivalence on divisors. Topics include the introduction of linear similarity of divisors, the recognition of irreducible and ample Q-Cartier divisors, and the use of the notion of linkage to address various aspects of algebraic geometry.

The article focuses on the set-theoretic complete intersection property, which examines the intersection of an irreducible curve with a divisor on a normal, projective variety. The topics covered include the behavior of abelian varieties over different types of fields (finite fields, number fields, and geometric fields), providing insights into the qualitative properties of these fields based on the topology of the variety.

Let k be a number field and X a smooth, geometrically integral quasi-projective variety over k. For any linear algebraic group G over k and any G-torsor g: Z \to X, we observe that if the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for all twists of Z by elements in H^1_{\text {\'{e}t}}(k,G), then the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for X. As an application, we show that any homogeneous space of the form G/H with G a connected linear algebraic group over k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some compactness assumptions when k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G/H with G semisimple simply connected and H finite, using the theory of torsors and descent. [ABSTRACT FROM AUTHOR]

We produce a short and elementary algorithm to compute an upper bound for the canonical dimension of a spit semisimple linear algebraic group. Using this algorithm we confirm previously known bounds by Karpenko [ 6 – 9 ] and Devyatov [ 5 ] as well as we produce new bounds (e.g. for groups of types |$F_4$|⁠ , adjoint |$E_6$|⁠ , for some semisimple groups). [ABSTRACT FROM AUTHOR]

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl's book: for the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings, and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with SG⊆S being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring SG is a pure subring of S, equivalently, SG is a direct summand of S as an SG-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion SG⊆S is pure. It turns out that if SG⊆S is pure, then either the invariant ring SG is regular, or the group G is linearly reductive. [ABSTRACT FROM AUTHOR]

It is well known that all Borel subgroups of a linear algebraic group are conjugate. Berest, Eshmatov, and Eshmatov have shown that this result also holds for the automorphism group Aut ⁢ (픸 2) of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir ⁢ (ℙ 2) up to conjugation, proving in particular that they are not necessarily conjugate. In principle, this fact answers a question of Popov. More precisely, we prove that Bir ⁢ (ℙ 2) admits Borel subgroups of any rank r ∈ { 0 , 1 , 2 } and that all Borel subgroups of rank r ∈ { 1 , 2 } are conjugate. In rank 0, there is a one-to-one correspondence between conjugacy classes of Borel subgroups of rank 0 and hyperelliptic curves of genus ℊ ≥ 1 . Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus ℊ , and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus ℊ . This moduli space is of dimension 2 ⁢ ℊ - 1 . [ABSTRACT FROM AUTHOR]

In this paper we extend to algebraic extensions of local and global fields and their completions some classical results due to Borel-Serre, Tits, Conrad, Douai, Kneser and Sansuc concerning the finiteness, the surjectivity of maps between Galois cohomology groups and the obstruction to weak approximation and some related results of connected linear algebraic groups. [ABSTRACT FROM AUTHOR]

We extend previous results by Cumplido, Martin and Vaskou on parabolic subgroups of large‐type Artin groups to a broader family of two‐dimensional Artin groups. In particular, we prove that an arbitrary intersection of parabolic subgroups of a (2,2)‐free two‐dimensional Artin group is itself a parabolic subgroup. An Artin group is (2,2)‐free if its defining graph does not have two consecutive edges labeled by 2. As a consequence of this result, we solve the conjugacy stability problem for this family by applying an algorithm introduced by Cumplido. All of this is accomplished by considering systolic‐by‐function complexes, which generalize systolic complexes. Systolic‐by‐function complexes have a more flexible structure than systolic complexes since we allow the edges to have different lengths. At the same time, their geometry is rigid enough to satisfy an analogue of the Cartan–Hadamard theorem and other geometric properties similar to those of systolic complexes. [ABSTRACT FROM AUTHOR]

For any smooth connected linear algebraic group G over an algebraically closed field k, we describe the Picard group of the universal moduli stack of principal G-bundles over pointed smooth k-projective curves. [ABSTRACT FROM AUTHOR]

Given a Chevalley group G.q and a parabolic subgroup P G.q we prove that for any set A there is a certain growth of A relatively to P, namely, either AP or PA is much larger than A. Also, we study a question about the intersection of An with parabolic subgroups P for large n. We apply our method to obtain some results on a modular form of Zaremba's conjecture from the theory of continued fractions, and make the first step towards Hensley's conjecture about some Cantor sets with Hausdorff dimension greater than 1=2. [ABSTRACT FROM AUTHOR]

This paper gives an explicit argument to show strong boundedness for Sp 2 n (R) for R a ring of S-algebraic integers or a semi-local ring thereby giving a quantitative version of the abstract result in the paper [15]. The results presented further generalize older results regarding strong boundedness by Kedra, Libman and Martin [6] and Morris [9] from SL n to Sp 2 n. The results also completely solve the question of the asymptotic of strong boundedness for Sp 2 n (R) for R semi-local case with an argument that immediately generalizes to all other split Chevalley groups. [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group over an algebraically closed field k and Aut alg (G) the group of all algebraic group automorphisms of G. For every φ ∈ Aut alg (G) let R (φ) denote the set of all orbits of the φ -twisted conjugacy action of G on itself (given by (g , x) ↦ g x φ (g − 1) , for all g , x ∈ G). We say that G has the algebraic R ∞ -property if R (φ) is infinite for every φ ∈ Aut alg (G). In [1] we have shown that this property is satisfied by every connected non-solvable algebraic group. From a theorem due to Steinberg it follows that if a connected algebraic group G has the algebraic R ∞ -property, then G φ (the fixed-point subgroup of G under φ) is infinite for all φ ∈ Aut alg (G). In this article we show that the condition is also sufficient. We also show that a Borel subgroup of any semisimple algebraic group has the algebraic R ∞ -property and identify certain classes of solvable algebraic groups for which the property fails. [ABSTRACT FROM AUTHOR]

The category of linear algebraic groups admits non-surjective epimorphisms. For simple algebraic groups of rank 2 defined over algebraically closed fields, we show that the minimal dimension of a closed epimorphic subgroup is 3. [ABSTRACT FROM AUTHOR]

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric. [ABSTRACT FROM AUTHOR]

Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between Severi-Brauer varieties and central simple algebras. For more general varieties, one might use endomorphism algebras of line bundles, of indecomposable vector bundles, or of exceptional objects in their derived categories. Using Galois cohomology, we describe a new invariant of reductive algebraic groups that captures precisely when this strategy will fail. Our main result characterizes this invariant in terms of coflasque resolutions of linear algebraic groups introduced by Colliot-Thélène. We determine whether or not this invariant is trivial for many fields. For number fields, we show it agrees with the Tate-Shafarevich group of the linear algebraic group, up to behavior at real places. [ABSTRACT FROM AUTHOR]

Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of degree at most the degree of the group as an algebraic variety. However, the degree of a linear algebraic group G ⊂ GLn(C) can be arbitrarily large even for n = 1. One of the key ingredients of Hrushovski's algorithm for computing the Galois group of a linear differential equation was an idea to "approximate" every algebraic subgroup of GLn(C) by a "similar" group so that the degree of the latter is bounded uniformly in n. Making this uniform bound computationally feasible is crucial for making the algorithm practical. In this paper, we derive a single-exponential degree bound for such an approximation (we call it a toric envelope), which is qualitatively optimal. As an application, we improve the quintuply exponential bound due to Feng for the first step of Hrushovski's algorithm to a single-exponential bound. For the cases n = 2, 3 often arising in practice, we further refine our general bound. [ABSTRACT FROM AUTHOR]

We set HT ht . We use H, M, and HT ht in the proof of Proposition 4.1. Using the argument in [[32], Claim 3.6] word-for-word, we have that HT ht is not k-defined. Also, H is G-cr since M is HT ht -ir by [[19], Table 10, ID 3]. [Extracted from the article]

Let X be a real algebraic variety (resp. nonsingular real algebraic variety) and let Y be a homogeneous space for some linear real algebraic group. We prove that a continuous (resp. C ∞) map f : X → Y can be approximated by regular maps in the C 0 (resp. C ∞) topology if and only if it is homotopic to a regular map. Taking Y = S p , the unit p -dimensional sphere, we obtain solutions of several problems that have been open since the 1980's and which concern approximation of maps with values in the unit spheres. This has several consequences for approximation of maps between unit spheres. For example, we prove that for every positive integer n every C ∞ map from S n into S n can be approximated by regular maps in the C ∞ topology. Up to now such a result has only been known for five special values of n , namely, n = 1 , 2 , 3 , 4 or 7. [ABSTRACT FROM AUTHOR]

We formulate and analyze several finiteness conjectures for linear algebraic groups over higher-dimensional fields. In fact, we prove all of these conjectures for algebraic tori as well as in some other situations. This work relies in an essential way on several finiteness results for unramified cohomology. [ABSTRACT FROM AUTHOR]

Recently in JHEP09 (2021) 053, Wang et al. discussed the polynomial ring formed by flavor invariants in the leptonic sector with massive Majorana neutrinos. They have explicitly constructed the finite generating sets of the polynomial rings for both two-generation scenario and three-generation scenario. However, Wang et al.'s claim of the finiteness of the generating sets of the polynomial rings and their calculation by the approach of Hilbert series with generalized Molien-Weyl formula are both based on their assertion that the unitary group U(n, ℂ) is reductive, which is unfortunately incorrect. The property of being reductive is only applicable to linear algebraic groups. And it is well-known that the unitary group U(n, ℂ) is not even a linear algebraic group. In this paper, we point out the above issue and provide a solution to fill in the accompanying logical gaps in JHEP09 (2021) 053. Some important results from the theory of linear algebraic group, the invariant theory of square matrices and group theory are needed in the analysis. We also clarify some somewhat misleading or vague statements in JHEP09 (2021) 053 about the scope of flavor invariants. Note that, although built from incorrect assertion, Wang et al.'s calculation results in JHEP09 (2021) 053 are nonetheless correct, which is ultimately because the ring of invariants of U(n, ℂ) is isomorphic to that of GL(n, ℂ) which is itself reductive. [ABSTRACT FROM AUTHOR]

Let X denote a quasi-projective variety over a field on which a connected linear algebraic group G acts with finitely many orbits. Then, the G-orbits define a stratification of X. We establish several key properties of the category of equivariant perverse sheaves on X, which have locally constant cohomology sheaves on each of the orbits. Under the above assumptions, we show that this category comes close to being a highest weight category in the sense of Cline, Parshall and Scott and defines a quasi-hereditary algebra. We observe that the above hypotheses are satisfied by all toric varieties and by all spherical varieties associated to connected reductive groups over any algebraically closed field. Next we show that the odd dimensional intersection cohomology sheaves vanish on all spherical varieties defined over algebraically closed fields of positive characteristics, extending similar results for spherical varieties defined over the field of complex numbers by Michel Brion and the author in prior work. Assuming that the linear algebraic group G and the action of G on X are defined over a finite field F q , and where the odd dimensional intersection cohomology sheaves on the orbit closures vanish, we also establish several basic properties of the mixed category of mixed equivariant perverse sheaves so that the associated terms in the weight filtration are finite sums of the shifted equivariant intersection cohomology complexes on the orbit closures. [ABSTRACT FROM AUTHOR]

Consider a connected reductive algebraic group G and a symmetric subgroup K. Let X = K / B K × G / P be a double flag variety of finite type, where B K is a Borel subgroup of K , and P a parabolic subgroup of G. A general argument shows that the orbit space C X / K inherits a natural action of the Hecke algebra H = H (K , B K) of double cosets via convolutions. However, it is a quite different problem to find out the explicit structure of the Hecke module. In this paper, for the double flag variety of type AIII, we determine the explicit action of H on C X / K in a combinatorial way using graphs. As a by-product, we also get the description of the representation of the Weyl group on C X / K as a direct sum of induced representations. [ABSTRACT FROM AUTHOR]

In this article, we introduce a partial Poisson transform on the affine symmetric space SU(n, n) / SL(n, C) x R+* and prove that this transform is a continuous SU(n, n)-homomorphism. We also give the form for the Fourier transform of the Poisson kernel. [ABSTRACT FROM AUTHOR]

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups, provided the maximal normal torsion-free definable subgroup is nilpotent. As a consequence, we show definable algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is Lie isomorphic to a linear algebraic group. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We show the boundedness of finite subgroups in any anisotropic reductive group over a perfect field that contains all roots of 1. Also, we provide explicit bounds for orders of finite subgroups of automorphism groups of Severi–Brauer varieties and quadrics over such fields. [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group over a field k of characteristic 0. We show that any two connected semisimple k-subgroups of G that are conjugate over an algebraic closure of k are actually conjugate over a finite field extension of k of degree bounded independently of the subgroups. Moreover, if k is a real number field , we show that any two connected semisimple k-subgroups of G that are conjugate over the field of real numbers R are actually conjugate over a finite real extension of k of degree bounded independently of the subgroups. [ABSTRACT FROM AUTHOR]

We show that the Brauer-Manin obstructions to the Hasse principle and weak approximation for homogeneous spaces under connected reductive groups over global function fields with connected reductive stabilizers are the only ones, extending some of Borovoi's results (and thus also proving a partial case of a conjecture of Colliot-Thelene) in this regard. Along the way, we extend some perfect pairings and an important local-global exact sequence (an analog of a Cassels-Tate's exact sequence) proved by Sansuc for connected linear algebraic groups defined over number fields, to the case of connected reductive groups over global function fields and beyond. [ABSTRACT FROM AUTHOR]

We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups. [ABSTRACT FROM AUTHOR]