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Articles 1 - 11 of 11
Full-Text Articles in Algebra
Quasisymmetric Functions Distinguishing Trees, Jean-Christophe Aval, Karimatou Djenabou, Peter R. W. Mcnamara
Quasisymmetric Functions Distinguishing Trees, Jean-Christophe Aval, Karimatou Djenabou, Peter R. W. Mcnamara
Faculty Journal Articles
A famous conjecture of Stanley states that his chromatic symmetric function distinguishes trees. As a quasisymmetric analogue, we conjecture that the chromatic quasisymmetric function of Shareshian and Wachs and of Ellzey distinguishes directed trees. This latter conjecture would be implied by an affirmative answer to a question of Hasebe and Tsujie about the P-partition enumerator distinguishing posets whose Hasse diagrams are trees. They proved the case of rooted trees and our results include a generalization of their result.
On The Ranks Of String C-Group Representations For Symplectic And Orthogonal Groups, Peter A. Brooksbank
On The Ranks Of String C-Group Representations For Symplectic And Orthogonal Groups, Peter A. Brooksbank
Faculty Journal Articles
We determine the ranks of string C-group representations of the groups PSp(4,q)=Ω(5,q), and comment on those of higher-dimensional symplectic and orthogonal groups.
On The Ranks Of String C-Group Representations For Symplectic And Orthogonal Groups, Peter A. Brooksbank
On The Ranks Of String C-Group Representations For Symplectic And Orthogonal Groups, Peter A. Brooksbank
Faculty Journal Articles
We determine the ranks of string C-group representations of 4-dimensional projective symplectic groups over a finite field, and comment on those of higher-dimensional symplectic and orthogonal groups.
Exact Sequences Of Inner Automorphisms Of Tensors, Peter A. Brooksbank
Exact Sequences Of Inner Automorphisms Of Tensors, Peter A. Brooksbank
Faculty Journal Articles
We produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor. Our sequence generalizes known sequences for associative and non-associative algebras. In a manner similar to those, our sequence facilitates inductive reasoning about, and calculation of the groups of symmetries of a tensor. The new insights these methods afford can be applied to problems ranging from understanding algebraic structures to distinguishing entangled states in particle physics.
Orthogonal Groups In Characteristic 2 Acting On Polytopes Of High Rank, Peter A. Brooksbank, Dimitri Leemans, John T. Ferrara
Orthogonal Groups In Characteristic 2 Acting On Polytopes Of High Rank, Peter A. Brooksbank, Dimitri Leemans, John T. Ferrara
Faculty Journal Articles
No abstract provided.
Testing Isomorphism Of Graded Algebras, Peter A. Brooksbank, James B. Wilson, Eamonn A. O'Brien
Testing Isomorphism Of Graded Algebras, Peter A. Brooksbank, James B. Wilson, Eamonn A. O'Brien
Faculty Journal Articles
We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that often dramatically improve the performance of the algorithm and report on an implementation in Magma.
Rank Reduction Of String C-Group Representations, Peter A. Brooksbank, Dimitri Leemans
Rank Reduction Of String C-Group Representations, Peter A. Brooksbank, Dimitri Leemans
Faculty Journal Articles
We show that a rank reduction technique for string C-group representations first used in [Adv. Math. 228 (2018), pp. 3207–3222] for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on d-dimensional modules over fields of even order greater than 2 possess string C-group representations of all ranks. The broad applicability of the rank reduction technique provides fresh impetus to construct, for suitable families of groups, string C-groups of highest possible rank. It also suggests that the alternating group Alt(11)—the only known group having “rank gaps”—is perhaps more unusual …
Polytopes Of Large Rank For Psl(4,Q), Peter A. Brooksbank, Dimitri Leemans
Polytopes Of Large Rank For Psl(4,Q), Peter A. Brooksbank, Dimitri Leemans
Faculty Journal Articles
This paper examines abstract regular polytopes whose automorphism group is the projective special linear group PSL(4,q). For q odd we show that polytopes of rank 4 exist by explicitly constructing PSL(4,q) as a string C-group of that rank. On the other hand, we show that no abstract regular polytope exists whose group of automorphisms is PSL(4,2k).
The Module Isomorphism Problem Reconsidered, Peter A. Brooksbank
The Module Isomorphism Problem Reconsidered, Peter A. Brooksbank
Faculty Journal Articles
Algorithms to decide isomorphism of modules have been honed continually over the last 30 years, and their range of applicability has been extended to include modules over a wide range of rings. Highly efficient computer implementations of these algorithms form the bedrock of systems such as GAP and MAGMA, at least in regard to computations with groups and algebras. By contrast, the fundamental problem of testing for isomorphism between other types of algebraic structures -- such as groups, and almost any type of algebra -- seems today as intractable as ever. What explains the vastly different complexity status of the …
Groups Acting On Tensor Products, Peter A. Brooksbank
Groups Acting On Tensor Products, Peter A. Brooksbank
Faculty Journal Articles
Groups preserving a distributive product are encountered often in mathematics. Examples include automorphism groups of associative and non associative rings, classical groups, and automorphisms of p-groups. While the great variety of such products precludes any realistic hope of describing the general structure of the groups that preserve them, it is reasonable to expect that insight may be gained from an examination of the universal distributive products: tensor products. We give a detailed description of the groups preserving such tensor products over semisimple and semi primary rings, and present effective algorithms to construct generators for these groups. We also discuss …
On Groups With A Class-Preserving Outer Automorphism, Peter A. Brooksbank
On Groups With A Class-Preserving Outer Automorphism, Peter A. Brooksbank
Faculty Journal Articles
Four infinite families of 2-groups are presented, all of whose members possess an outer automorphism that preserves conjugacy classes. The groups in these families are central extensions of their predecessors by a cyclic group of order 2. For each integer r>1, there is precisely one 2-group of nilpotency class r in each of the four families. All other known families of 2-groups possessing a class-preserving outer automorphism consist entirely of groups of nilpotency class 2.