Kieran Roberts | Universität Bielefeld (original) (raw)

Papers by Kieran Roberts

Research paper thumbnail of A class of 2-groups admitting an action of the symmetric group of degree 3

Cornell University - arXiv, Jan 2, 2013

A biextraspecial group of rank m is an extension of a special 2group Q of the form 2 2+2m by L ∼ ... more A biextraspecial group of rank m is an extension of a special 2group Q of the form 2 2+2m by L ∼ = L 2 (2), such that the 3-element from L acts on Q fixed-point-freely. Subgroups of this type appear in at least the sporadic simple groups J 2 , J 3 , M cL, Suz, and Co 1. In this paper we completely classify biextraspecial groups, namely, we show that the rank m must be even and for each such m there exist exactly two biextraspecial groups B ε (m) up to isomorphism where ε ∈ {+, −}. We also prove that Out(B ε (m)) is an extension of the mdimensional orthogonal GF (2)-space of type ε by the corresponding orthogonal group. The extension is non-split except in a few small cases. Lemma 2.3 Let V be an L-module containing a submodule W such that both W and V /W are isomorphic to the natural L-module. Then V is completely reducible, that is, it is isomorphic to the direct sum of two natural modules. Proof. Let s, t ∈ L be of order 3 and 2, respectively. Since dim C V (t) ≥ 2, we have that C V (t) is not contained in W. Let u ∈ C V (t) \ W and define U = u, u s , u s 2. Note that s acts on V fixed-point-freely, which means that u + u s + u s 2 must be zero. It follows that dim U = 2. Also, since t fixes u, we have that u L = {u, u s , u s 2 }, which yields that U is L-invariant. Since both W and U are irreducible, they intersect trivially. Thus, V is the direct sum of W and U. By induction, every L-module whose composition factors are all natural modules, is completely reducible. In particular, we now have the exact structure of Q. Corollary 2.4 The L-module Q is the direct sum of m copies of the natural L-module. This in turn implies the following. Corollary 2.5 Every irreducible L-submodule of Q is isomorphic to the natural L-module and there are exactly 2 m − 1 irreducible submodules of Q. 3 Dents Definition 3.1 A dent in a biextraspecial group G = Q : L is a normal subgroup D of G such that Z < D < Q and D/Z is an irreducible L-submodule of Q/Z. Note that any two distinct dents intersect in Z. Also, it follows from Corollary 2.5 that there are exactly 2 m − 1 dents. Lemma 3.2 Each dent D is abelian. Moreover, D is isomorphic either to 2 4 or 4 2 .

Research paper thumbnail of Recovering the Lie algebra from its extremal geometry

Journal of Algebra, 2015

An element x of a Lie algebra L over the field F is extremal if [x, [x, L]] = F x. Under minor as... more An element x of a Lie algebra L over the field F is extremal if [x, [x, L]] = F x. Under minor assumptions, it is known that, for a simple Lie algebra L, the extremal geometry E(L) is a subspace of the projective geometry of L and either has no lines or is the root shadow space of an irreducible spherical building ∆. We prove that if ∆ is of simply-laced type, then L is a quotient of a Chevalley algebra of the same type.

Research paper thumbnail of On the definition of saturated fusion systems

Journal of Group Theory, 2009

We propose a simplification of the definition of saturation for fusion systems over p-groups and ... more We propose a simplification of the definition of saturation for fusion systems over p-groups and prove the equivalence of our definition with that of Broto, Levi, and Oliver.

Research paper thumbnail of Lie algebras and incidence geometry

An element \(\char{cmti10}{0x78}\) of a Lie algebra \(\char{cmmi10}{0x4c}\) over the field \(\cha... more An element \(\char{cmti10}{0x78}\) of a Lie algebra \(\char{cmmi10}{0x4c}\) over the field \(\char{cmmi10}{0x46}\) is extremal if [\(\char{cmti10}{0x78}\), [\(\char{cmti10}{0x78}\), \(\char{cmmi10}{0x4c}\)]] \(\subseteq\)\(\char{cmmi10}{0x46}\)\(\char{cmti10}{0x78}\). One can define the extremal geometry of \(\char{cmmi10}{0x4c}\) whose points \(\char{cmsy10}{0x45}\) are the projective points of extremal elements and lines \(\char{cmsy10}{0x46}\) are projective lines all of whose points belong to \(\char{cmsy10}{0x45}\). We prove that any finite dimensional simple Lie algebra \(\char{cmmi10}{0x4c}\) is a classical Lie algebra of type A\(_n\) if it satisfies the following properties: \(\char{cmmi10}{0x4c}\) contains no elements \(\char{cmti10}{0x78}\) such that [\(\char{cmti10}{0x78}\), [\(\char{cmti10}{0x78}\), \(\char{cmmi10}{0x4c}\)]] = 0, \(\char{cmmi10}{0x4c}\) is generated by its extremal elements and the extremal geometry \(\char{cmsy10}{0x45}\) of \(\char{cmmi10}{0x4c}\) is a...

Research paper thumbnail of On the Hurwitz action in finite Coxeter groups

Journal of Group Theory, 2016

We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure... more We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a factorization into a product of reflections that generate a parabolic subgroup. We give an unusual definition of a parabolic subgroup that we show to be equivalent to the classical one for finite Coxeter groups.

Research paper thumbnail of A class of 2-groups admitting an action of the symmetric group of degree 3

Cornell University - arXiv, Jan 2, 2013

A biextraspecial group of rank m is an extension of a special 2group Q of the form 2 2+2m by L ∼ ... more A biextraspecial group of rank m is an extension of a special 2group Q of the form 2 2+2m by L ∼ = L 2 (2), such that the 3-element from L acts on Q fixed-point-freely. Subgroups of this type appear in at least the sporadic simple groups J 2 , J 3 , M cL, Suz, and Co 1. In this paper we completely classify biextraspecial groups, namely, we show that the rank m must be even and for each such m there exist exactly two biextraspecial groups B ε (m) up to isomorphism where ε ∈ {+, −}. We also prove that Out(B ε (m)) is an extension of the mdimensional orthogonal GF (2)-space of type ε by the corresponding orthogonal group. The extension is non-split except in a few small cases. Lemma 2.3 Let V be an L-module containing a submodule W such that both W and V /W are isomorphic to the natural L-module. Then V is completely reducible, that is, it is isomorphic to the direct sum of two natural modules. Proof. Let s, t ∈ L be of order 3 and 2, respectively. Since dim C V (t) ≥ 2, we have that C V (t) is not contained in W. Let u ∈ C V (t) \ W and define U = u, u s , u s 2. Note that s acts on V fixed-point-freely, which means that u + u s + u s 2 must be zero. It follows that dim U = 2. Also, since t fixes u, we have that u L = {u, u s , u s 2 }, which yields that U is L-invariant. Since both W and U are irreducible, they intersect trivially. Thus, V is the direct sum of W and U. By induction, every L-module whose composition factors are all natural modules, is completely reducible. In particular, we now have the exact structure of Q. Corollary 2.4 The L-module Q is the direct sum of m copies of the natural L-module. This in turn implies the following. Corollary 2.5 Every irreducible L-submodule of Q is isomorphic to the natural L-module and there are exactly 2 m − 1 irreducible submodules of Q. 3 Dents Definition 3.1 A dent in a biextraspecial group G = Q : L is a normal subgroup D of G such that Z < D < Q and D/Z is an irreducible L-submodule of Q/Z. Note that any two distinct dents intersect in Z. Also, it follows from Corollary 2.5 that there are exactly 2 m − 1 dents. Lemma 3.2 Each dent D is abelian. Moreover, D is isomorphic either to 2 4 or 4 2 .

Research paper thumbnail of Recovering the Lie algebra from its extremal geometry

Journal of Algebra, 2015

An element x of a Lie algebra L over the field F is extremal if [x, [x, L]] = F x. Under minor as... more An element x of a Lie algebra L over the field F is extremal if [x, [x, L]] = F x. Under minor assumptions, it is known that, for a simple Lie algebra L, the extremal geometry E(L) is a subspace of the projective geometry of L and either has no lines or is the root shadow space of an irreducible spherical building ∆. We prove that if ∆ is of simply-laced type, then L is a quotient of a Chevalley algebra of the same type.

Research paper thumbnail of On the definition of saturated fusion systems

Journal of Group Theory, 2009

We propose a simplification of the definition of saturation for fusion systems over p-groups and ... more We propose a simplification of the definition of saturation for fusion systems over p-groups and prove the equivalence of our definition with that of Broto, Levi, and Oliver.

Research paper thumbnail of Lie algebras and incidence geometry

An element \(\char{cmti10}{0x78}\) of a Lie algebra \(\char{cmmi10}{0x4c}\) over the field \(\cha... more An element \(\char{cmti10}{0x78}\) of a Lie algebra \(\char{cmmi10}{0x4c}\) over the field \(\char{cmmi10}{0x46}\) is extremal if [\(\char{cmti10}{0x78}\), [\(\char{cmti10}{0x78}\), \(\char{cmmi10}{0x4c}\)]] \(\subseteq\)\(\char{cmmi10}{0x46}\)\(\char{cmti10}{0x78}\). One can define the extremal geometry of \(\char{cmmi10}{0x4c}\) whose points \(\char{cmsy10}{0x45}\) are the projective points of extremal elements and lines \(\char{cmsy10}{0x46}\) are projective lines all of whose points belong to \(\char{cmsy10}{0x45}\). We prove that any finite dimensional simple Lie algebra \(\char{cmmi10}{0x4c}\) is a classical Lie algebra of type A\(_n\) if it satisfies the following properties: \(\char{cmmi10}{0x4c}\) contains no elements \(\char{cmti10}{0x78}\) such that [\(\char{cmti10}{0x78}\), [\(\char{cmti10}{0x78}\), \(\char{cmmi10}{0x4c}\)]] = 0, \(\char{cmmi10}{0x4c}\) is generated by its extremal elements and the extremal geometry \(\char{cmsy10}{0x45}\) of \(\char{cmmi10}{0x4c}\) is a...

Research paper thumbnail of On the Hurwitz action in finite Coxeter groups

Journal of Group Theory, 2016

We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure... more We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a factorization into a product of reflections that generate a parabolic subgroup. We give an unusual definition of a parabolic subgroup that we show to be equivalent to the classical one for finite Coxeter groups.