Linear Algebraic Group Research Articles

The Picard group of the universal moduli stack of principal bundles on pointed smooth curves

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

Algebraic Groups

Linear algebraic groups is an active research area in contemporary mathematics. It has rich connections to algebraic geometry, representation theory, algebraic combinatorics, number theory, algebraic topology, and differential equations. The foundations of this theory were laid by A. Borel, C. Chevalley, J.-P. Serre, T. A. Springer and J. Tits in the second half of the 20th century. The Oberwolfach workshops on algebraic groups, led by Springer and Tits, played an important role in this effort as a forum for researchers, meeting at approximately 3 year intervals since the 1960s. The present workshop continued this tradition, covering a range of topics, with an emphasis on recent developments in the subject.

Toric principal bundles, piecewise linear maps and Tits buildings

We define the notion of a piecewise linear map from a fan $\Sigma$ to $\tilde{\mathfrak{B}}(G)$, the cone over the Tits building of a linear algebraic group $G$. Let $X_\Sigma$ be a toric variety with fan $\Sigma$. We show that when $G$ is reductive the set of integral piecewise linear maps from $\Sigma$ to $\tilde{\mathfrak{B}}(G)$ classifies the isomorphism classes of (framed) toric principal $G$-bundles on $X_\Sigma$. This in particular recovers Klyachko's classification of toric vector bundles, and gives new classification results for the orthogonal and symplectic toric principal bundles.

Local-Global Principles for Constant Reductive Groups over Semi-Global Fields

We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; that is, over one-variable function fields over complete discretely valued fields. We provide conditions on the group and the semi-global field under which the local-global principle holds, and we compute the obstruction to the local-global principle in certain classes of examples. Using our description of the obstruction, we give the first example of a semisimple simply connected group over a semi-global field where the local-global principle fails. Our methods include patching and R-equivalence.

Hilbert’s Irreducibility Theorem via Random Walks

Abstract 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.

Local-global principle for classical groups over function fields of $p$-adic curves

Let K be a local field with residue field \kappa and F the function field of a curve over K . Let G be a connected linear algebraic group over F of classical type. Suppose \operatorname{char}(\kappa) is a good prime for G . Then we prove that projective homogeneous spaces under G over F satisfy a local-global principle for rational points with respect to discrete valuations of F . If G is a semisimple simply connected group over F , then we also prove that principal homogeneous spaces under G over F satisfy a local-global principle for rational points with respect to discrete valuations of F .

On birational Torelli theorems

Let G be a simple simply-connected connected linear algebraic group over \({\mathbb {C}}\). We prove a 2-birational Torelli theorem for the moduli space of semistable principal G-bundles over a smooth curve of genus \(\ge 3\), which says that if two such moduli spaces are 2-birational, then the curves are isomorphic. We also prove a 3-birational Torelli theorem for the moduli space of stable symplectic parabolic bundles over a smooth curve of genus \(\ge 4\).

Twisted conjugacy in linear algebraic groups II

Let G be a linear algebraic group over an algebraically closed field k and Autalg(G) the group of all algebraic group automorphisms of G. For every φ∈Autalg(G) let R(φ) denote the set of all orbits of the φ-twisted conjugacy action of G on itself (given by (g,x)↦gxφ(g−1), for all g,x∈G). We say that G has the algebraic R∞-property if R(φ) is infinite for every φ∈Autalg(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 Gφ (the fixed-point subgroup of G under φ) is infinite for all φ∈Autalg(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.

On approximation of maps into real algebraic homogeneous spaces

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 C0 (resp. C∞) topology if and only if it is homotopic to a regular map. Taking Y=Sp, 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 Sn into Sn 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.

A Compactification of the Universal Moduli Space of Principal G-Bundles

Let G be a semisimple linear algebraic group, rho :Ghookrightarrow text {SL}(V) a finite-dimensional faithful representation, gge 2 a natural number, and delta a positive rational number. We prove the existence of a compactification of the universal moduli space of semistable principal G-bundles over overline{text {M}}_{g}, provided that delta is sufficiently large, having the following property: the fibers over singular curves are the moduli spaces of delta -semistable singular principal G-bundles.

Comment on “flavor invariants and renormalization-group equations in the leptonic sector with massive Majorana neutrinos”

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.

Algebraic groups as difference Galois groups of linear differential equations

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field C ( x ) with derivation d d x and endomorphism f ( x ) ↦ f ( x + 1 ) . Our main result is that every linear algebraic group , considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over C ( x ) .

Galois cohomology of real quasi-connected reductive groups

By a quasi-connected reductive group (a term of Labesse) over an arbitrary field we mean an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group. We compute the first Galois cohomology set \(H^1({\mathbb {R}},G)\) of a quasi-connected reductive group G over the field \({\mathbb {R}}\) of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-torus of G.

Overgroups of regular unipotent elements in reductive groups

Abstract We study reductive subgroups H of a reductive linear algebraic group G – possibly nonconnected – such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre. This generalises results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.

Translates of homogeneous measures associated with observable subgroups on some homogeneous spaces

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.

Relative geometric invariant theory

Relative geometric invariant theory is an invariant theory for equivariant projective morphisms between algebraic varieties endowed with an action of a reductive linear algebraic group. We will give brief accounts of the basic results of relative geometric invariant theory and present alternative proofs for recent results obtained by Halle, Hulek, and Zhang on the variation of quotients in relative geometric invariant theory.

The Canonical Dimension of a Semisimple Group and the Unimodular Degree of a Root System

AbstractWe 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).

Equivariant perverse sheaves and quasi-hereditary algebras

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 Fq, 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.

Pathological behavior of arithmetic invariants of unipotent groups

We show that all of the nice behavior for Tamagawa numbers, Tate-Shafarevich sets, and other arithmetic invariants of pseudo-reductive groups over global function fields proved in \cite{rospred} fails in general for non-commutative unipotent groups. We also give some positive results which show that Tamagawa numbers do exhibit some reasonable behavior for arbitrary connected linear algebraic groups over global function fields.

Real Liouvillian Extensions of Partial Differential Fields

In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally p-adic differential fields with a p-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally p-adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities of further development of algebraic methods in real dynamical systems.