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Michael Bate

Publications and source records attributed to Michael Bate.

At least 19 recordsLinked to original sources
arXiv2026

Clique complexes of groups and order sheaves

We study the clique complex $X_G$ of a finite group: a simplicial complex whose faces are in one-to-one correspondence with the elements of the group, and arise via a decomposition of group elements into $p$-elements for primes $p$. The $1$-skeleton of this complex then has the classical Gruenberg-Kegel graph $\Gamma_G$ as a quotient and a finite simple group is determined up to isomorphism by its clique complex. The clique complex can also be endowed with a combinatorial sheaf --- called the order sheaf --- and the resulting homology can be computed for an arbitrary simplicial complex in terms of the homology of the links of vertices. The homology of the clique complex with coefficients in this order sheaf is then computed for some special classes of groups, in particular those for which the centralizers of elements are nilpotent.

math.GR↗
arXiv2024

Geometric rigidity of simple modules for algebraic groups

Let k be a field, let G be an affine algebraic k-group and V a finite-dimensional G-module. We say V is rigid if the socle series and radical series coincide for the action of G on each indecomposable summand of V; say V is geometrically rigid (resp. absolutely rigid) if V is rigid after base change of G and V to k (resp. any field extension of k). We show that all simple G-modules are geometrically rigid, though not in general absolutely rigid. More precisely, we show that if V is a simple G-module, then there is a finite purely inseparable extension kV /k naturally attached to V such that V is absolutely rigid as a G-module after base change to kV. The proof turns on an investigation of algebras of the form K otimes E where K and E are field extensions of k; we give an example of such an algebra which is not rigid as a module over itself. We establish the existence of the purely inseparable field extension kV /k through an analogous version for artinian algebras. In the second half of the paper we apply recent results on the structure and representation theory of pseudo-reductive groups to give a concrete description of kV when G is smooth and connected. Namely, we combine the main structure theorem of the Conrad-Prasad classification of pseudo-reductive G together with our previous high weight theory. For V a simple G-module, we calculate the minimal field of definition of the geometric Jacobson radical of EndG(V) in terms of the high weight of V and the Conrad-Prasad classification data; this gives a concrete construction of the field kV as a subextension of the minimal field of definition of the geometric unipotent radical of G. We also observe that the Conrad-Prasad classification can be used to hone the dimension formula for V we had previously established; we also use it to give a description of EndG(V) which includes a dimension formula.

math.RT↗
arXiv2024

On good $A_1$ subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups

Suppose $G$ is a simple algebraic group defined over an algebraically closed field of good characteristic $p$. In 2018 Korhonen showed that if $H$ is a connected reductive subgroup of $G$ which contains a distinguished unipotent element $u$ of $G$ of order $p$, then $H$ is $G$-irreducible in the sense of Serre. We present a short and uniform proof of this result under an extra hypothesis using so-called good $A_1$ subgroups of $G$, introduced by Seitz. In the process we prove some new results about good $A_1$ subgroups of $G$ and their properties. We also formulate a counterpart of Korhonen's theorem for overgroups of $u$ which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of $u$ under a mild condition on $p$ depending on the rank of $G$, and we present an analogue of Korhonen's theorem for Lie algebras.

math.GR↗
arXiv2024

The subgroup structure of pseudo-reductive groups

Let $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of $G$, and let $\pi':G_{k'} \to G'$ be the quotient map. We first characterise those smooth subgroups $H$ of $G$ for which $\pi'(H_{k'})=G'$. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup $H$ of $G$ such that $\pi'(H_{k'})=H'$, and if $H'$ is smooth can we find such a $H$ that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup $H$, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of $G$ with those of $G'$.

math.GR↗
arXiv2024

$G$-complete reducibility and saturation

Let $H \subseteq G$ be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic $p> 0$. In our first main theorem we show that if a closed subgroup $K$ of $H$ is $H$-completely reducible, then it is also $G$-completely reducible in the sense of Serre, under some restrictions on $p$, generalising the known case for $G = GL(V)$. Our proof uses R.W. Richardson's notion of reductive pairs to reduce to the $GL(V)$ case. We study Serre's notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if $K$ is $H$-completely reducible, then the saturation of $K$ in $G$ is completely reducible in the saturation of $H$ in $G$ (which is again a connected reductive subgroup of $G$), under suitable restrictions on $p$, again generalising the known instance for $G = GL(V)$. We also study saturation of finite subgroups of Lie type in $G$. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in case $G = GL(V)$.

math.RT↗
arXiv2023

Homology of matching complexes and representations of symmetric groups

We compute the homology of the matching complex $M(\Gamma)$, where $\Gamma$ is the complete hypergraph on $n\geq 2$ vertices, and analyse the $S_n$-representations carried by this homology. These results are achieved using standard techniques in combinatorial topology, such as the theory of shellings. We then broaden the scope to the larger class of fibre-closed families of simplicial complexes and consider these through the lens of representation stability. This allows us to prove a number of results of an asymptotic nature, such as an analysis of the growth of Betti numbers and the kinds of irreducible $S_n$-representations that appear.

math.GR↗
arXiv2023

Edifices: Building-like spaces associated to linear algebraic groups

Given a semisimple linear algebraic $k$-group $G$, one has a spherical building $\Delta_G$, and one can interpret the geometric realisation $\Delta_G(\mathbb R)$ of $\Delta_G$ in terms of cocharacters of $G$. The aim of this paper is to extend this construction to the case when $G$ is an arbitrary connected linear algebraic group; we call the resulting object $\Delta_G(\mathbb R)$ the spherical edifice of $G$. We also define an object $V_G(\mathbb R)$ which is an analogue of the vector building for a semisimple group; we call $V_G(\mathbb R)$ the vector edifice. The notions of a linear map and an isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on $V_G(\mathbb R)$ and show they are all bi-Lipschitz equivalent to each other; with this extra structure, $V_G(\mathbb R)$ becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture.

math.GR↗
arXiv2023

Complete reducibility for Lie subalgebras and semisimplification

Let $G$ be a connected reductive linear algebraic group over a field $k$. Using ideas from geometric invariant theory, we study the notion of $G$-complete reducibility over $k$ for a Lie subalgebra $\mathfrak h$ of the Lie algebra $\mathfrak g = Lie(G)$ of $G$ and prove some results when $\mathfrak h$ is solvable or $char(k)= 0$. We introduce the concept of a $k$-semisimplification $\mathfrak h'$ of $\mathfrak h$; $\mathfrak h'$ is a Lie subalgebra of $\mathfrak g$ associated to $\mathfrak h$ which is $G$-completely reducible over $k$. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that $\mathfrak h'$ is unique up to $Ad(G(k))$-conjugacy in $\mathfrak g$. Moreover, we prove that the two concepts are compatible: for $H$ a closed subgroup of $G$ and $H'$ a $k$-semisimplification of $H$, the Lie algebra $Lie(H')$ is a $k$-semisimplification of $Lie(H)$.

math.GR↗
arXiv2021

Overgroups of regular unipotent elements in reductive groups

We study reductive subgroups $H$ of a reductive linear algebraic group $G$ -- possibly non-connected -- 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 generalizes 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.

math.GR↗
arXiv2020

Semisimplification for Subgroups of Reductive Algebraic Groups

Let $G$ be a reductive algebraic group---possibly non-connected---over a field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a degeneration process for obtaining from $H$ a completely reducible subgroup $H'$ of $G$; one takes a limit of $H$ along a cocharacter of $G$ in an appropriate sense. We generalise this idea to arbitrary reductive $G$ using the notion of $G$-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a $G$-completely reducible subgroup $H'$ of $G$, unique up to $G(k)$-conjugacy, which we call a $k$-semisimplification of $H$. This gives a single unifying construction which extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL_n$ and with Serre's "$G$-analogue" of semisimplification for subgroups of $G(k)$). We also show that under some extra hypotheses, one can pick $H'$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf and Rousseau.

math.GR↗
arXiv2019

Irreducible modules for pseudo-reductive groups

We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive groups to the split reductive case and the pseudo-split pseudo-reductive commutative case. Moreover, we give the first results on the latter, including a rather complete description of the rank one case.

math.RT↗
arXiv2019

On relative complete reducibility

Let $K$ be a reductive subgroup of a reductive group $G$ over an algebraically closed field $k$. The notion of relative complete reducibility, introduced in previous work of Bate-Martin-Roehrle-Tange, gives a purely algebraic description of the closed $K$-orbits in $G^n$, where $K$ acts by simultaneous conjugation on $n$-tuples of elements from $G$. This extends work of Richardson and is also a natural generalization of Serre's notion of $G$-complete reducibility. In this paper we revisit this idea, giving a characterization of relative $G$-complete reducibility which directly generalizes equivalent formulations of $G$-complete reducibility. If the ambient group $G$ is a general linear group, this characterization yields representation-theoretic criteria. Along the way, we extend and generalize several results from the aforementioned work of Bate-Martin-Roehrle-Tange.

math.GR↗
arXiv2018

Orbit Closures and Invariants

Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let Y denote the set of fixed points of H in X, and N(H) the normalizer of H in G. In this paper we study the natural map from the quotient of Y by N(H) to the quotient of X by G induced by the inclusion of Y in X. We show that, given G and H, this map is a finite morphism for all G-varieties X if and only if H is G-completely reducible (in the sense defined by J-P. Serre); this was proved in characteristic zero by Luna in the 1970s. We discuss some applications and give a criterion for the map of quotients to be an isomorphism. We show how to extend some other results in Luna's paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; then the double coset HgK is closed for generic g in G if and only if the intersection of generic conjugates of H and K is reductive.

math.AG↗
arXiv2018

On unipotent radicals of pseudo-reductive groups

We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let k' be a purely inseparable field extension of k of degree p^e and let G denote the Weil restriction of scalars R_{k'/k}(G') of a reductive k'-group G'. When G= \R_{k'/k}(G') we also provide some results on the orders of elements of the unipotent radical \RR_u(G_{\bar k}) of the extension of scalars of G to the algebraic closure \bar k of k.

math.GR↗
arXiv2016

Cocharacter-closure and the rational Hilbert-Mumford Theorem

For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V, we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure. When k is perfect, we give a criterion in terms of closed orbits for G to be k-anisotropic, answering a question of Borel.

math.AG↗
arXiv2016

Asymptotic Bounds for the Size of Hom$(A,{\rm GL}_n(q))$

Fix an arbitrary finite group $A$ of order $a$, and let $X(n,q)$ denote the set of homomorphisms from $A$ to the finite general linear group ${\rm GL}_n(q)$. The size of $X(n,q)$ is a polynomial in $q$. In this note it is shown that generically this polynomial has degree $n^2(1-a^{-1}) - ε_r$ and leading coefficient $m_r$, where $ε_r$ and $m_r$ are constants depending only on $r := n \mod a$. We also present an algorithm for explicitly determining these constants.

math.GR↗
arXiv2016

Closed H-orbits in G-varieties

Let G be a linear algebraic group, H a subgroup of G and X a G-variety. This paper explores the connection between G-orbits and H-orbits in X, concentrating in particular on the question of when we have the implications G.x closed in X implies H.x closed in X, and vice versa. Some of the general results found in this paper are then used in the special case where G and H are reductive and X is affine to extend two classical results of Luna from characteristic zero to arbitrary characteristic.

math.GR↗