Recent zbMATH articles in MSC 14Lhttps://zbmath.org/atom/cc/14L2022-01-14T13:23:02.489162ZWerkzeug\(F\)-sets and finite automatahttps://zbmath.org/1475.110382022-01-14T13:23:02.489162Z"Bell, Jason"https://zbmath.org/authors/?q=ai:bell.jason-p"Moosa, Rahim"https://zbmath.org/authors/?q=ai:moosa.rahim-nThe notion of a \(k\)-automatic subset of \(\mathbb{N}\) (that is, a subset \(S\subset\mathbb{N}\) with the property that there is a finite automaton which accepts exactly the words arising as base \(k\) expansions of elements of \(S\)) is extended here to an \(F\)-automatic subset of a finitely generated abelian group \(\Gamma\) equipped with an endomorphism \(F\). An \(F\)-subset means a finite union of finite sets of sums of elements of \(\Gamma\), of \(F\)-invariant subgroups in \(\Gamma\), and of \(F\)-cycles in \(\Gamma\) (subsets of the form \(\{\gamma+F^\delta\gamma+F^{2\delta}\gamma+\cdots +F^{\ell\delta}\gamma\}\) with \(\ell,\delta\in\mathbb{N}\) and \(\gamma\in\Gamma\)). \(F\)-automaticity is also defined via finite-state automata and accepted words. The main results (Theorems 4.2, 6.9 and 7.4) prove automaticity of \(F\)-subsets under some mild conditions. The results are a natural generalisations of Derksen's analog of the Skolem-Mahler-Lech theorem [\textit{H. Derksen}, Invent. Math. 168, No. 1, 175--224 (2007; Zbl 1205.11030)] to the Mordell-Lang context studied by \textit{R. Moosa} and \textit{T. Scanlon} [Am. J. Math. 126, No. 3, 473--522 (2004; Zbl 1072.03020)].
Reviewer: Thomas B. Ward (Leeds)Reductions of points on algebraic groups. IIhttps://zbmath.org/1475.111192022-01-14T13:23:02.489162Z"Bruin, Peter"https://zbmath.org/authors/?q=ai:bruin.peter"Perucca, Antonella"https://zbmath.org/authors/?q=ai:perucca.antonellaSummary: Let \(A\) be the product of an abelian variety and a torus over a number field \(K\), and let \(m\geqslant 2\) be a square-free integer. If \(\alpha \in A(K)\) is a point of infinite order, we consider the set of primes \(\mathfrak{p}\) of \(K\) such that the reduction \((\alpha\bmod\mathfrak{p})\) is well defined and has order coprime to \(m\). This set admits a natural density, which we are able to express as a finite sum of products of \(\ell\)-adic integrals, where \(\ell\) varies in the set of prime divisors of \(m\). We deduce that the density is a rational number, whose denominator is bounded (up to powers of \(m)\) in a very strong sense. This extends the results of the paper \textit{Reductions of points on algebraic groups} by \textit{D. Lombardo} and the second author, where the case \(m\) prime is established [Part I, J. Inst. Math. Jussieu 20, No. 5, 1637--1669 (2021; Zbl 1475.11122)].Reductions of points on algebraic groupshttps://zbmath.org/1475.111222022-01-14T13:23:02.489162Z"Lombardo, Davide"https://zbmath.org/authors/?q=ai:lombardo.davide-m"Perucca, Antonella"https://zbmath.org/authors/?q=ai:perucca.antonellaSummary: Let \(A\) be the product of an abelian variety and a torus defined over a number field \(K\). Fix some prime number \(\ell \). If \(\alpha \in A(K)\) is a point of infinite order, we consider the set of primes \(\mathfrak{p}\) of \(K\) such that the reduction \(( \alpha \bmod \mathfrak{p})\) is well-defined and has order coprime to \(\ell \). This set admits a natural density. By refining the method of \textit{R. Jones} and \textit{J. Rouse} [Proc. Lond. Math. Soc. (3) 100, No. 3, 763--794 (2010; Zbl 1244.11057)], we can express the density as an \(\ell \)-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of \(\ell )\) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.Algebraic groups as difference Galois groups of linear differential equationshttps://zbmath.org/1475.120112022-01-14T13:23:02.489162Z"Bachmayr, Annette"https://zbmath.org/authors/?q=ai:bachmayr.annette"Wibmer, Michael"https://zbmath.org/authors/?q=ai:wibmer.michaelLet \(F\) denote a \(\delta\sigma\)-field of characteristic zero, i.e. a field with two commuting operators, a derivation \(\delta\) and an endomorphism \(\sigma\). For such fields it is possible to build the theory of a similar Kolchin's Differential Galois theory. The authors in a number of publications [\textit{A. Bachmayr} et al., Doc. Math. 23, 241--291 (2018; Zbl 1436.12006); Adv. Math. 381, Article ID 107605, 28 p. (2021; Zbl 1461.12003); Trans. Am. Math. Soc. 374, No. 6, 4293--4308 (2021; Zbl 07344666)] make contribution to the development of such theory. This paper discusses the converse problem of the \(\sigma\)-Picard-Vessiot theory. In the usual Picard-Vessiot theory, for the field \(\mathbb{C}(x)\) and the linear algebraic group, the converse problem always has a solution [\textit{C. Tretkoff} and \textit{M. Tretkoff}, Am. J. Math. 101, 1327--1332 (1979; Zbl 0423.12021)]. The authors show that not every difference algebraic group occurs as a \(\sigma\)-Galois group of a \(\sigma\)-Picard-Vessiot extension of \(\mathbb{C}(x)\) . But the main result of the paper says that there are still quite a lot of positive occasions.
\textbf{Theorem.} \textit{Every linear algebraic group over \(\mathbb{C}\), considered as a difference algebraic group, occurs as a \(\sigma\)-Galois group over} \(\mathbb{C}(x)\).
Reviewer: Mykola Grygorenko (Kyïv)Almost-simple affine difference algebraic groupshttps://zbmath.org/1475.120132022-01-14T13:23:02.489162Z"Wibmer, Michael"https://zbmath.org/authors/?q=ai:wibmer.michaelAffine algebraic groups can be described as subgroups of a general linear group defined by polynomials in the matrix entries. Affine difference algebraic groups can be described as subgroups of a general linear group defined by difference polynomials in the matrix entries, i.e., the defining equations involve a formal symbol \(\sigma\) that has to be interpreted as a ring endomorphism. The author shows that isomorphism theorems from abstract group theory have meaningful analogs for these groups and that one can establish a Jordan-Hölder type theorem that allows to decompose any affine difference algebraic group into almost-simple affine difference algebraic groups. The author also characterizes almost-simple affine difference algebraic groups via almost-simple affine algebraic groups.
Reviewer: Vladimir P. Kostov (Nice)Some torsion classes in the Chow ring and cohomology of \(\mathbf{B}PGL_n\)https://zbmath.org/1475.140092022-01-14T13:23:02.489162Z"Gu, Xing"https://zbmath.org/authors/?q=ai:gu.xingSummary: In the integral cohomology ring of the classifying space of the projective linear group \(\mathrm{PGL}_n\) (over \(\mathbb{C} )\), we find a collection of \(p\)-torsion classes \(y_{p , k}\) of degree \(2 ( p^{k + 1} + 1 )\) for any odd prime divisor \(p\) of \(n\), and \(k \geqslant 0\). If, in addition, \( p^2 \nmid n\), there are \(p\)-torsion classes \(\rho_{p , k}\) of degree \(p^{k + 1} + 1\) in the Chow ring of the classifying stack of \(\mathrm{PGL}_n\), such that the cycle class map takes \(\rho_{p , k}\) to \(y_{p , k}\). We present an application of the above classes regarding Chern subrings.Fibrations associated to smooth quotients of abelian varietieshttps://zbmath.org/1475.140862022-01-14T13:23:02.489162Z"Martinez-Nuñez, Gary"https://zbmath.org/authors/?q=ai:martinez-nunez.garySummary: Let \(A\) be a complex abelian variety and \(G\) a finite group of automorphisms of \(A\) fixing the origin such that \(A/G\) is smooth. The quotient \(A/G\) can be seen as a fibration over an abelian variety whose fibers are isomorphic to a product of projective spaces. We classify how the fibers are glued in the case when the fibers are isomorphic to a projective space and we prove that, in general, the quotient \(A/G\) is a fibered product of such fibrations.Detecting nilpotence and projectivity over finite unipotent supergroup schemeshttps://zbmath.org/1475.140892022-01-14T13:23:02.489162Z"Benson, Dave"https://zbmath.org/authors/?q=ai:benson.david-john"Iyengar, Srikanth B."https://zbmath.org/authors/?q=ai:iyengar.srikanth-b"Krause, Henning"https://zbmath.org/authors/?q=ai:krause.henning"Pevtsova, Julia"https://zbmath.org/authors/?q=ai:pevtsova.juliaLet \(k\) be a field, \(\mathrm{char}(K)=p>2\), and let \(G\) be a finite unipotent supergroup scheme over \(k\). The cohomology of \(G\) will be denoted \(H^{*,*}(G,k)\), which is isomorphic to \(\mathrm{Ext}_{kG}^{*,*}(k,k)\): the latter index in the superscript arising from the \(\mathbb Z/2\mathbb Z\)-grading. Of interest is the nilpotent elements of this cohomology group, and the authors reduce this question to one involving elementary supergroup schemes.
The main result is that \(x\in H^{*,*}(G,k)\) is nilpotent if and only if \(x_K\in H^{*,*}(G\times_k K,K)\), restricted to \(H^{*,*}(E,K)\), is nilpotent, where \(K\) is an extension of \(k\) and \(E\le G\times_k K\) is elementary. Additionally, if \(M\) is a \(kG\)-module, then \(M\) is projective if and only if the restriction of \(M\times_k K\) to \(E\) is projective. These results are then applied to finite dimensional sub-Hopf algebras of the Steenrod algebra over \(\mathbb F_p\).
Reviewer: Alan Koch (Decatur)Full level structure on some group schemeshttps://zbmath.org/1475.140902022-01-14T13:23:02.489162Z"Guan, Chuangtian"https://zbmath.org/authors/?q=ai:guan.chuangtianSummary: We give a definition of full level structure on group schemes of the form \(G\times G\), where \(G\) is a finite flat commutative group scheme of rank \(p\) over a \(\mathbb{Z}_p\)-scheme \(S\) or, more generally, a truncated \(p\)-divisible group of height 1. We show that there is no natural notion of full level structure over the stack of all finite flat commutative group schemes.Quasi-simple finite groups of essential dimension 3https://zbmath.org/1475.140912022-01-14T13:23:02.489162Z"Prokhorov, Yu."https://zbmath.org/authors/?q=ai:prokhorov.yuri-gSummary: We classify quasi-simple finite groups of essential dimension 3.On the torus quotients of Schubert varietieshttps://zbmath.org/1475.140922022-01-14T13:23:02.489162Z"Bonala, Narasimha Chary"https://zbmath.org/authors/?q=ai:bonala.narasimha-chary"Pattanayak, Santosha Kumar"https://zbmath.org/authors/?q=ai:pattanayak.santosha-kumarIn this paper, the authors consider the GIT quotients of minuscule Schubert varieties for the action of a maximal torus. Let \(G\) be a semisimple simply connected complex algebraic group, \(T\) a maximal torus, \(B\supset T\) a Borel subgroup and \(P\supset B\) a parabolic subgroup. Let \(\Phi^+\) be the set of positive roots with respect to \(B\). A fundamental weight \(\omega\) is called \textit{minuscule} if \(\langle\omega,\check{\beta}\rangle\le1\) for all \(\beta\in\Phi^+\) (Definition 4.1). For \(\omega\) a minuscule weight and \(P:=P_\omega\) the associated parabolic subgroup, the flag variety \(G/P\) and the Schubert varieties in \(G/P\) are also called \textit{minuscule}. Let \({\mathcal L}_\omega\) be the homogeneous line bundle on \(G/P\) associated to \(\omega\) and \(W^P\) the associated Weyl group. For \(w\in W^P\), denote by \(X_P(w)^{ss}_T({\mathcal L}_\omega)\) the set of semistable points in the Schubert variety \(X_P(w)\) with respect to \({\mathcal L}_\omega\) for the action of \(T\). Recall also that there is a unique minimal Schubert variety \(X_P(v)\) admitting semistable points. For any \(w\in W^P\), let \(Q_w\) be the associated quiver variety (for relevant material on quivers, see section 4).
Given this setup, the authors prove that the semistable locus is contained in the smooth locus for \(X_P(w)\) if and only if \(Q_v\) contains all the essential holes of \(Q_w\) (Theorem 4.9). From now on, suppose that \(G=SL(n,{\mathbb C})\). If \(1<r<n-1\) with \(\gcd(n,r)=1\) and \(P_r\) is the maximal parabolic subgroup of \(G\) associated to the simple root \(\alpha_r\), then \(X_P(w)^{ss}_T({\mathcal L}_{\omega_r})//T\) is smooth if \(Q_v\) contains all the essential holes of \(Q_w\) (Corollary 3.5). Now let \(P\) be the parabolic subgroup of \(SL(n,{\mathbb C})\) corresonding to the highest root \(\alpha_0\). Let \({\mathcal M}\) denote the descent of \({\mathcal L}_{\alpha_0}\) to the quotient \(X_P(w)^{ss}_T({\mathcal L}_{\alpha_0})//T\). Then (Theorem 5.1) the polarized variety \((X_P(w)^{ss}_T({\mathcal L}_{\alpha_0})//T,{\mathcal M})\) is projectively normal and (Corollary 5.2) \(X_P(w)^{ss}_T({\mathcal L}_{\alpha_0})\) is isomophic to a projective space. Theorem 5.1 is proved using standard monomials.
Reviewer: P. E. Newstead (Liverpool)Character varieties for real formshttps://zbmath.org/1475.200122022-01-14T13:23:02.489162Z"Acosta, Miguel"https://zbmath.org/authors/?q=ai:acosta.miguelIn this article, the author gives a definition for character varieties for real forms of \(\mathrm{SL}(n,\mathbb{C})\). The main result is to characterize irreducible representations arising as conjugates of representations into real forms of \(\mathrm{SL}(n,\mathbb{C})\).
In particular, the author shows that the definition agrees with the topological definition given in the compact case \(\mathrm{SU}(n)\).
In the second section, the author recalls known facts about complex character varieties. In the third section, the author introduces the definition for character varieties of real form and proves the main result.
In the fourth section, the author uses his result to describe the real character variety of \(\mathbb{Z}/3*\mathbb{Z}/3\) into \(\mathrm{SU}(2,1)\) and \(\mathrm{SU}(3)\). The author uses the parametrization of \(\chi_{\mathrm{SL}(3,\mathbb{C})}(F_2)\) given by \textit{S. Lawton} [J. Algebra 313, No. 2, 782--801 (2007; Zbl 1119.13004)] and in particular find slices of representations parametrized by \textit{J. R. Parker} and \textit{P. Will} [Contemp. Math. 639, 327--348 (2015; Zbl 1360.20017)] and \textit{E. Falbel} et al. [Exp. Math. 25, No. 2, 219--235 (2016; Zbl 1353.57007)].
Reviewer: Clément Guérin (Esch-sur-Alzette)On the Humphreys conjecture on support varieties of tilting moduleshttps://zbmath.org/1475.200732022-01-14T13:23:02.489162Z"Achar, Pramod N."https://zbmath.org/authors/?q=ai:achar.pramod-n"Hardesty, William"https://zbmath.org/authors/?q=ai:hardesty.william-d"Riche, Simon"https://zbmath.org/authors/?q=ai:riche.simonLet \(G\) be a simply-connected semisimple algebraic group over an algebraically closed field \(\Bbbk\) of characteristic \(p > 0\), such that \(p\) is greater than the Coxeter number of \(G\). Let \(T \subset B \subset G\) be a fixed pair of a Borel subgroup in \(G\) and a maximal torus in \(B\), let \(\mathbf{X}\) denote the character lattice of \(T\) and let \(\mathbf{X}^+ \subset \mathbf{X}\) be the subset of dominant weights. Furthermore, let \(\dot{G}\) denote the Frobenius twist of \(G\) and let \(G_1\) be the kernel of the Frobenius morphism \(\mathrm{Fr}: G \rightarrow \dot{G}\). In the paper under review, the authors take for any \(\lambda \in \mathbf{X}^+\) the indecomposable tilting \(G\)-module \(\mathrm{T}(\lambda)\) of highest weight \(\lambda\) and consider the support variety \(V_{G_1}(\mathrm{T}(\lambda))\), which is a closed \(\dot{G}\)-stable subvariety of the nilpotent cone \(\mathcal{N}\) of \(\dot{G}\).
A conjecture of \textit{J. E. Humphreys}, stated in [AMS/IP Stud. Adv. Math. 4, 69--80 (1997; Zbl 0919.17013)] gives a conjectural description of the support varieties \(V_{G_1}(\mathrm{T}(\lambda))\) as closures of certain \(\dot{G}\)-orbits in \(\mathcal{N}\). This conjecture uses Lusztig's bijection between the set of \(\dot{G}\)-orbits in \(\mathcal{N}\) and the set of two-sided cells in the affine Weyl group of \(G\).
In an earlier work of the second author [Adv. Math. 329, 392--421 (2018; Zbl 1393.14043)], Humphreys conjecture was proved for \(G = \mathrm{SL}_n(\Bbbk)\) and \(p > n+1\) . The main result of the paper under review is to show that for any \(G\), the support variety \(V_{G_1}(\mathrm{T}(\lambda))\) always contains the variety predicted by Humphreys. Furthermore, there is an integer \(N > 0\) (depending only on the root system of \(G\)) such that if \(p > N\) the two varieties coincide (i.e., Humphreys conjecture is true). At the moment \(N\) cannot be determined explicitly, except for \(G = \mathrm{SL}_n(\Bbbk)\).
In addition, the authors also state and prove a variant of Humphreys conjecture involving ``relative support varieties''.
Reviewer: Elitza Hristova (Sofia)Total positivity in Springer fibreshttps://zbmath.org/1475.200802022-01-14T13:23:02.489162Z"Lusztig, G."https://zbmath.org/authors/?q=ai:lusztig.georgeSummary: Let \(u\) be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at \(u\) with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold.Loop of formal diffeomorphisms and Faà di Bruno coloop bialgebrahttps://zbmath.org/1475.200962022-01-14T13:23:02.489162Z"Frabetti, Alessandra"https://zbmath.org/authors/?q=ai:frabetti.alessandra"Shestakov, Ivan P."https://zbmath.org/authors/?q=ai:shestakov.ivan-pIn this paper, the authors study two generalizations of proalgebraic groups, on one side to representable functors on categories of non-commutative algebras, on the other side to functors taking divisions that is loops. The main motivation for the authors comes from two proalgebraic groups of formal series appearing in the renormalization in quantum field theory namely: The group of invertible series with constant term equal to 1, represented by the Hopf algebra of symmetric functions; and that of formal diffeomorphisms tangent to the identity represented by the Faà di Bruno Hopf algebra. The authors are interested in the relationship between the non-commutative algebras and sets of series. In Section 2.1 of Part 2, Loops and functors in loops, the authors describe coloops in an axiomatic way. In Section 2.2, Coloops in general categories, the authors give some easy examples of algebraic and non-algebraic loops on associative and non-associative algebras, and extensively study the loops of invertible series and that of formal diffeomorphisms. In Section 2.3, the authors study (Pro)algebraic loops. Part 3, Coloops of invertible and unitary elements. Section 3.1, Loop of invertible elements. In this section, the authors give an example of an abelian algebraic group which can be extended to associative algebras as a group, to alternative algebras as a loop, but not to non-associative algebras, even as a loop. In Section 3.2, the authors study loops of unitary elements. Section 3.3, Unitary Cayley-Dickson loops. In this section, they give an example of a loop which is not algebraic on associative algebras. Part 4, Coloops of invertible series. They give a definition of groups of invertible series \(\operatorname{Inv}(A)\), where \(A\) is a commutative algebra. And they show that the functors \(\operatorname{Inv}\) can be extended to non-associative algebras, as a proalgebraic loop. In Section 4.1, the authors study loops of invertible series. In Section 4.2, the authors study coloop bialgebras of invertible series. In Section 4.3, they study properties of the loop of invertible series. Part 5, Coloop of formal diffeomorphisms. Initially, they define the group of formal diffeomorphisms \(\operatorname{Diff}(A)\), where \(A\) is a commutative algebra. In Section 5.1, the authors study loops of formal diffeomorphisms, and define formal diffeomorphisms \(\operatorname{Diff}(A)\) in \(\lambda\) with coefficients in \(A\), where \(\lambda\) is a formal variable and \(A\) a unital associative algebra, non-necessarily commutative. In Section 5.2, the authors study Faà di Bruno coloop bialgebras and define it. In Section 5.3, they study Faà di Bruno co-operations in terms of recursive operators. In Section 5.4, the authors study functoriality of the diffeomorphism loops. In Section 5.5, they study properties of the diffeomorphism loops. In Appendix A, the authors presents categorical proofs with tangles. Tangle diagrams are an efficient tool to prove formal (categorical) properties. In the context of non-associative algebras, tangle diagrams have been used to code deformations of the enveloping algebra of a Malcev algebra, seen as the infinitesimal structure of a Moufang loop. In this Appendix, the authors present the list of the tangles needed to represent all the operations and the co-operations in coloops, with their defining identities.
Reviewer: C. Pereira da Silva (Curitiba)