arXiv subjects

David I. Stewart

Publications and source records attributed to David I. Stewart.

32 records · Page 2Linked to original sources

arXiv2016

On the smoothness of normalisers and the subalgebra structure of modular Lie algebras

We provide results on the smoothness of normalisers in connected reductive algebraic groups $G$ over fields $k$ of positive characteristic $p$. Specifically we we give bounds on $p$ which guarantee that normalisers of subalgebras of $\mathfrak{g}$ in $G$ are smooth, i.e.\ so that the Lie algebras of these normalisers coincide with the infinitesimal normalisers. One of our main tools is to exploit cohomology vanishing of small dimensional modules. Along the way, we obtain complete reducibility results for small dimensional modules in the spirit of similar results due to Jantzen, Guralnick, Serre and Bendel--Nakano--Pillen.

math.GR↗

arXiv2015

Rigid orbits and sheets in reductive Lie algebras over fields of prime characteristic

Let $G$ be a simple simply-connected algebraic group over an algebraically closed field $k$ of characteristic $p>0$ with $\mathfrak{g}={\rm Lie}(G)$. We discuss various properties of nilpotent orbits in $\mathfrak{g}$, which have previously only been considered over $\mathbb{C}$. Using a combination of theoretical and computational methods, we extend to positive characteristic various calculations of de Graaf with nilpotent orbits in exceptional Lie algebras. In particular, we classify those orbits which are reachable, those which satisfy a certain related condition due to Panyushev, and determine the codimension in the centraliser $\mathfrak{g}_e$ of its the derived subalgebra $[\mathfrak{g}_e,\mathfrak{g}_e]$. Some of these calculations are used to show that the list of rigid nilpotent orbits in $\mathfrak{g}$, the classification of sheets of $\mathfrak{g}$ and the distribution of the nilpotent orbits amongst them are independent of good characteristic, remaining the same as in the characteristic zero case. We also give a comprehensive account of the theory of sheets in reductive Lie algebras over algebraically closed fields of good characteristic.

math.RT↗

arXiv2014

Maximal subalgebras of Cartan type in the exceptional Lie algebras

In this paper we initiate the study of the maximal subalgebras of exceptional simple classical Lie algebras \g over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for \g. In this paper we deal with what is surely the most unnatural case; that is, where the maximal subalgebra in question is a simple subalgebra of non-classical type. We show that only the first Witt algebra can occur as a subalgebra of \g and give explicit details on when it may be maximal in \g.

math.RA↗

arXiv2014

Stabilisation of the LHS spectral sequence for algebraic groups

In this note, we consider the Lyndon--Hochschild--Serre spectral sequence corresponding to the first Frobenius kernel of an algebraic group G, computing the extensions between simple $G$-modules. We state and discuss a conjecture that $E_2=E_\infty$ and provide general conditions for low-dimensional terms on the $E_2$-page to be the same as the corresponding terms on the $E_{\infty}$-page, i.e. its abutment.

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arXiv2013

On unipotent algebraic G-groups and 1-cohomology

In this paper we consider non-abelian 1-cohomology for groups with coefficients in other groups. We prove versions of the `five lemma' arising from this situation. We go on to show that a connected unipotent algebraic group Q acted on morphically by a connected algebraic group G admits a filtration with successive quotients having the structure of G-modules. From these results we deduce extensions to results due to Cline, Parshall, Scott and van der Kallen. Firstly, if G is a connected, reductive algebraic group with Borel subgroup B and Q a unipotent algebraic G-group, we show the restriction map H^1(G,Q)\to H^1(B,Q) is an isomorphism. We also show that this situation admits a notion of rational stability and generic cohomology. We use these results to obtain corollaries about complete reducibility and subgroup structure.

math.GR↗

arXiv2013

On extensions for Ree groups of type F_4

Let k be an algebraically closed field of characteristic p=2. Let G=F_4 be simply connected over k and let σ:G\to G be an endomorphism such that the fixed point set G(σ) is a Ree group. We show, using the methods of Bendel--Nakano--Pillen, that self-extensions of simple kG(σ)-modules vanish generically and that for all but the first few Ree groups of type F_4, the 1-cohomology for G(σ) with coefficients in simple kG-modules can be identified with the 1-cohomology for G with coefficients in (possibly different) simple G-modules.

math.GR↗

arXiv2012

First cohomology groups for finite groups of Lie type in defining characteristic

Let G be a finite group of Lie type, defined over a field k of characteristic p > 0. We find explicit bounds for the dimension of the first cohomology group for G with coefficients in a simple kG-module. We proceed by bounding the number of composition factors of Weyl modules for simple algebraic groups independently of p and using this to deduce bounds for the 1-cohomology of simple algebraic groups. Finally, we use this to obtain estimations for the growth rate of the maximum dimension {γ_l} of these 1-cohomology groups over all groups of Lie type of rank l. We find that log γ_l is O(l^3 log l) (or if the Lusztig conjecture holds, O(l^2 log l)).

math.RT↗

arXiv2012

Unbounding Ext

We produce examples in the cohomology of algebraic groups which answer two questions of Parshall and Scott. Specifically, if $G=SL_2$, then we show: (a) $\dim \Ext_G^2(L,L)$ can be arbitrarily large for a simple module $L$; and (b) the sequence $\max_{L-\text{irred}}\dim H^k(G,L)$ grows exponentially fast with $k$.

math.RT↗

arXiv2012

Bounding extensions for finite groups and Frobenius kernels

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ:G->G be a strict endomorphism (i. e., the subgroup G(σ) of σ-fixed points is finite). Also, let G_σ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the degree m cohomology H^m(G(σ),L) of any irreducible kG(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. A similar result holds for the degree m cohomology of G_σ. These bounds are actually established for the degree m extension groups Ext^m_{G(σ)}(L,L') between irreducible kG(σ)-modules L and L', with again a similar result holding for G_σ. In these Ext^m results, of interest in their own right, the bounds depend also on L, or, more precisely, on length of the p-adic expansion of the highest weight associated to L. All bounds are, nevertheless, independent of the characteristic p. These results extend earlier work of Parshall and Scott for rational representations of algebraic groups G. We also show that one can find bounds independent of the prime for the Cartan invariants of G(σ) and G_σ, and even for the lengths of the underlying PIMs. These bounds, which depend only on the root system of G and the "height" of σ, provide in a strong way an affirmative answer to a question of Hiss, for the special case of finite groups G(σ) of Lie type in the defining characteristic.

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arXiv2012

Shifted generic cohomology

The idea that the cohomology of finite groups might be fruitfully approached via the cohomology of ambient semisimple algebraic groups was first shown to be viable in the papers [CPS75] and [CPSvdK77]. The second paper introduced, through a limiting process, the notion of generic cohomology, as an intermediary between finite Chevalley group and algebraic group cohomology. The present paper shows that, for irreducible modules as coefficients, the limits can be eliminated in all but finitely many cases. These exceptional cases depend only on the root system and cohomological degree. In fact, we show that, for sufficiently large r, depending only on the root system and m, and not on the prime p or the irreducible module L, there are isomorphisms H^m(G(p^r),L) -> H^m(G(p^r),L') -> H^m_gen(G,L') -> H^m(G,L'), where the subscript "gen" refers to generic cohomology and L' is a constructibly determined irreducible "shift" of the (arbitrary) irreducible module L for the finite Chevalley group G(p^r). By a famous theorem of Steinberg, both L and L' extend to irreducible modules for the ambient algebraic group G with p^r-restricted highest weights. This leads to the notion of a module or weight being "shifted m-generic," and thus to the title of this paper. Our approach is based on questions raised by the third author in [Stea], which we answer here in the cohomology cases. We obtain many additional results, often with formulations in the more general context of Ext^m_G(q) with irreducible coefficients.

math.RT↗

arXiv2012

Non-G-completely reducible subgroups of the exceptional algebraic groups

Let G be an exceptional algebraic group defined over an algebraically closed field k of characteristic p>0 and let H be a subgroup of G. Then following Serre we say H is G-completely reducible or G-cr if, whenever H is contained in a parabolic subgroup P of G, then H is in a Levi subgroup of that parabolic. Building on work of Liebeck and Seitz, we find all triples (X,G,p) such that there exists a closed, connected, simple non-G-cr subgroup H<G with root system X.

math.GR↗

arXiv2010

G-complete reducibility and the exceptional algebraic groups

Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p>0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is contained in a Levi subgroup of that parabolic. A subgroup $X$ of $G$ is said to be $G$-irreducible if $X$ is in no parabolic subgroup of $G$; and $G$-reducible if it is in some parabolic of $G$. In this thesis, we consider the case that $G$ is of exceptional type. When $G$ is of type $G_2$ we find all conjugacy classes of closed, connected, reductive subgroups of $G$. When $G$ is of type $F_4$ we find all conjugacy classes of closed, connected, reductive $G$-reducible subgroups $X$ of $G$. Thus we also find all non-$G$-completely reducible closed, connected, reductive subgroups of $G$. When $X$ is closed, connected and simple of rank at least two, we find all conjugacy classes of $G$-irreducible subgroups $X$ of $G$. Together with the work of Amende in [Ame05] classifying irreducible subgroups of type $A_1$ this gives a complete classification of the simple subgroups of $G$. Amongst the classification of subgroups of $G=F_4(K)$ we find infinite collections of subgroups $X$ of $G$ which are maximal amongst all reductive subgroups of $G$ but not maximal subgroups of $G$; thus they are not contained in any maximal reductive subgroup of $G$. The connected, semisimple subgroups contained in no maximal reductive subgroup of $G$ are of type $A_1$ when $p=3$ and of semisimple type $A_1^2$ or $A_1$ when $p=2$. Some of those which occur when $p=2$ act indecomposably on the 26-dimensional irreducible representation of $G$. We also use this classification to find all subgroups of $G=F_4$ which are generated by short root elements of $G$, by utilising and extending the results of [LS94].

math.GR↗

arXiv2009

The second cohomology of simple SL_2-modules

Let $G$ be the simple algebraic group $SL_2$ defined over an algebraically closed field $K$ of characteristic $p>0$. In this paper, we find the second cohomology of all irreducible representations of $G$

math.RT↗

arXiv2009

The Reductive Subgroups of G_2

Let $G:=G_2(K)$ be a simple algebraic group of type $G_2$ defined over an algebraically closed field $K$ of characteristic $p>0$. Let $σ$ denote a standard Frobenius automorphism of $G$ such that $G_σ\cong G_2(q)$ with $q\geq 4$. In this paper we find all reductive subgroups of $G$ and quasi-simple subgroups of $G_σ$ in the defining characteristic.

math.GR↗