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Scott Murray

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Papers by Scott Murray

Computing in unipotent groups and reductive algebraic groups

The unipotent groups are an important class of algebraic groups. We show that techniques used to ... more The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on maximal unipotent subgroups of split reductive groups and show how this improves computation in the reductive group itself.

Fundamental domains for congruence subgroups of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi mathvariant="normal">SL</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> in positive characteristic

Journal of Algebra, 2011

In this work, we construct fundamental domains for congruence subgroups of SL 2 (F q [t]) and PGL... more In this work, we construct fundamental domains for congruence subgroups of SL 2 (F q [t]) and PGL 2 (F q [t]). Our method uses Gekeler's description of the fundamental domains on the Bruhat-Tits tree X = X q+1 in terms of cosets of subgroups. We compute the fundamental domains for a number of congruence subgroups explicitly as graphs of groups using the computer algebra system Magma.

Computing in Unipotent and Reductive Algebraic Groups

Lms Journal of Computation and Mathematics, 2008

The unipotent groups are an important class of algebraic groups. We show that techniques used to ... more The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on maximal unipotent subgroups of split reductive groups and show how this improves computation in the reductive group itself.

Fundamental domains for congruence subgroups of SL2 in positive characteristic

arXiv (Cornell University), Aug 31, 2009

In this work, we construct fundamental domains for congruence subgroups of SL2 (Fq[t]) and PGL2 (... more In this work, we construct fundamental domains for congruence subgroups of SL2 (Fq[t]) and PGL2 (Fq[t]). Our method uses Gekeler's description of the fundamental domains on the Bruhat-Tits tree X = Xq+1 in terms of cosets of subgroups. We compute the fundamental domains for a number of congruence subgroups explicitly as graphs of groups using the computer algebra system Magma.

Computing in groups of Lie type

Mathematics of Computation, Jul 7, 2003

We describe two methods for computing with the elements of untwisted groups of Lie type: using th... more We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction.

Representations of Parabolic and Borel Subgroups

Communications in Algebra, Jan 26, 2007

We demonstrate a relationships between the representation theory of Borel subgroups and parabolic... more

Strong integrality of inversion subgroups of Kac-Moody groups

arXiv (Cornell University), Oct 4, 2022

Let A be a symmetrizable generalized Cartan matrix with corresponding Kac-Moody algebra g over Q.... more Let A be a symmetrizable generalized Cartan matrix with corresponding Kac-Moody algebra g over Q. Let V " V λ be an integrable highest weight g-module and let V Z " V λ Z be a Z-form of V . Let G be an associated minimal representation-theoretic Kac-Moody group and let GpZq be its integral subgroup. Let ΓpZq be the Chevalley subgroup of G, that is, the subgroup that stabilizes the lattice V Z in V . For a subgroup M of G, we say that M is integral if M X GpZq " M X ΓpZq and that M is strongly integral if there exists v P V λ Z such that, for all g P M , g ¨v P V Z implies g P GpZq. We prove strong integrality of inversion subgroups U pwq of G where, for w P W , U pwq is the group generated by positive real root groups that are flipped to negative roots by w ´1. We use this to prove strong integrality of subgroups of the unipotent subgroup U of G generated by commuting real root groups. When A has rank 2, this gives strong integrality of subgroups U 1 and U 2 where U " U 1 ˚U2 and each U i is generated by 'half' the positive real roots.

Algorithm for Lang's Theorem

arXiv (Cornell University), Jun 3, 2005

We give an efficient algorithm for Lang's Theorem in split connected reductive groups defined ove... more We give an efficient algorithm for Lang's Theorem in split connected reductive groups defined over finite fields of characteristic greater than 3. This algorithm can be used to construct many important structures in finite groups of Lie type. We use an algorithm for computing a Chevalley basis for a split reductive Lie algebra, which is of independent interest.

Magma Proof of Strict Inequalities for Minimal Degrees of Finite Groups

arXiv (Cornell University), Jun 19, 2009

The minimal faithful permutation degree of a finite group G, denote by µ(G) is the least non-nega... more The minimal faithful permutation degree of a finite group G, denote by µ(G) is the least non-negative integer n such that G embeds inside the symmetric group Sym(n). In this paper, we outline a Magma proof that 10 is the smallest degree for which there are groups G and H such that µ(G × H) < µ(G) + µ(H).

Constructing a Lie group analog for the Monster Lie algebra

arXiv (Cornell University), Feb 16, 2020

Let m be the Monster Lie algebra. We summarize several interrelated constructions of Lie group an... more

Constructive homomorphisms for classical groups

Journal of Symbolic Computation, Apr 1, 2011

Let Ω ≤ GL(V ) be a quasisimple classical group in its natural representation over a finite vecto... more Let Ω ≤ GL(V ) be a quasisimple classical group in its natural representation over a finite vector space V , and let ∆ = N GL(V ) (Ω). We construct the projection from ∆ to ∆/Ω and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent ∆/Ω as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Ω. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spaces with forms. Our results are useful for the matrix group recognition project, can be used to solve element conjugacy problems, and can improve algorithms to construct maximal subgroups.

Constructing a Lie group analog for the Monster Lie algebra

Letters in Mathematical Physics

We describe various approaches to constructing groups which may serve as Lie group analogs for th... more We describe various approaches to constructing groups which may serve as Lie group analogs for the monster Lie algebra of Borcherds. Another approach is to construct automorphisms of m directly. Using this method, we give a representation of the monster finite simple group M in terms of automorphisms of m ([CCJM]).

$Research paper thumbnail of Infinite dimensional Chevalley groups and Kac-Moody groups over $\mathbb{Z}$$

Infinite dimensional Chevalley groups and Kac-Moody groups over $\mathbb{Z}$

arXiv (Cornell University), Mar 29, 2018

Let A be a symmetrizable generalized Cartan matrix, which is not of finite or affine type. Let g ... more Let A be a symmetrizable generalized Cartan matrix, which is not of finite or affine type. Let g be the corresponding Kac-Moody algebra over a commutative ring R with 1. We construct an infinite-dimensional group G V (R) analogous to a finite-dimensional Chevalley group over R. We use a Z-form of the universal enveloping algebra of g and a Z-form of an integrable highest-weight module V . We construct groups G V (Z) analogous to arithmetic subgroups in the finite-dimensional case. We also consider a universal representation-theoretic Kac-Moody group G and its completion G. For the completion we prove a Bruhat decomposition G(Q) = G(Z) B(Q) over Q, and that the arithmetic subgroup Γ(Z) coincides with the subgroup of integral points G(Z).

Unipotent and Reductive Algebraic Groups

The unipotent groups are an important class of algebraic groups. We show that techniques used to ... more The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on the maximal unipotent subgroup of a split reductive group and show how this improves computation in the reductive group itself.

An automated proof theory approach to computation with permutation groups

This is an introduction to data structures for permutation groups with a proof theoretic flavour.

Computing with Root Subgroups of Twisted Reductive Groups

We present algorithms to compute with relative root subgroups of twisted reductive groups. Given ... more We present algorithms to compute with relative root subgroups of twisted reductive groups. Given a Galois cocycle specifying a twisted form, we can find the relative root datum and Tits index, and carry out operations involving root elements. We can also find a presentation of the unipotent subgroup. Given a Tits index and the anisotropic subgroup, we can determine a cocycle with that index, if one exists.

Magma Proof of Strict Inequalities for Minimal Degrees of Finite Groups

The minimal faithful permutation degree of a finite group $G$, denote by $\mu(G)$ is the least no... more The minimal faithful permutation degree of a finite group $G$, denote by $\mu(G)$ is the least non-negative integer $n$ such that $G$ embeds inside the symmetric group $\Sym(n)$. In this paper, we outline a Magma proof that 10 is the smallest degree for which there are groups $G$ and $H$ such that $\mu(G \times H) < \mu(G)+ \mu(H)$. Comment: 4 pages

Computing in groups of Lie type

Mathematics of Computation, 2003

We describe two methods for computing with the elements of untwisted groups of Lie type: using th... more We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction.

Computing in Unipotent and Reductive Algebraic Groups

LMS Journal of Computation and Mathematics, 2008

The unipotent groups are an important class of algebraic groups. We show that techniques used to ... more The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on maximal unipotent subgroups of split reductive groups and show how this improves computation in the reductive group itself.

Constructive homomorphisms for classical groups

Journal of Symbolic Computation, 2011

Let Ω ≤ GL(V ) be a quasisimple classical group in its natural representation over a finite vecto... more Let Ω ≤ GL(V ) be a quasisimple classical group in its natural representation over a finite vector space V , and let ∆ = N GL(V ) (Ω). We construct the projection from ∆ to ∆/Ω and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent ∆/Ω as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Ω. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spaces with forms. Our results are useful for the matrix group recognition project, can be used to solve element conjugacy problems, and can improve algorithms to construct maximal subgroups.