The Terwilliger algebra of the halved n-cube from the viewpoint of its automorphism group action

2021 ◽  
pp. 103480
Author(s):  
Lihang Hou ◽  
Bo Hou ◽  
Na Kang ◽  
Suogang Gao
Keyword(s):  
Automorphism Group ◽  
Group Action ◽  
Terwilliger Algebra

scholarly journals On the automorphism group of a symplectic half-flat 6-manifold

2019 ◽  
Vol 31 (1) ◽  
pp. 265-273
Author(s):  
Fabio Podestà ◽  
Alberto Raffero
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Group Action ◽  
Cohomogeneity One ◽  
Cohomogeneity One Action

Abstract We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat {\mathrm{SU}(3)} -structure has Abelian Lie algebra with dimension bounded by {\min\{5,b_{1}(M)\}} . Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on {T\mathbb{S}^{3}} which are invariant under a cohomogeneity one action of {\mathrm{SO}(4)} .

Kleinian characterization of asymptotic symmetries of real general relativity

1993 ◽  
Vol 441 (1913) ◽  
pp. 465-471 ◽  
Keyword(s):  
General Relativity ◽  
Automorphism Group ◽  
Group Action ◽  
Radon Transform ◽  
Symmetry Groups ◽  
Asymptotic Symmetry ◽  
The Real ◽  
The Given ◽  
Asymptotic Symmetries

According to Klein’s Erlanger programme, one may (indirectly) specify a geometry by giving a group action. Conversely, given a group action, one may ask for the corresponding geometry. Recently, I showed that the real asymptotic symmetry groups of general relativity (in any signature) have natural ‘projective’ classical actions on suitable ‘Radon transform’ spaces of affine 3-planes in flat 4-space. In this paper, I give concrete models for these groups and actions. Also, for the ‘atomic’ cases, I give geometric structures for the spaces of affine 3-planes for which the given actions are the automorphism group.

scholarly journals Computation of Zagreb and atom-bond connectivity indices of certain families of dendrimers by using automorphism group action

2017 ◽  
Vol 82 (2) ◽  
pp. 151-162
Author(s):  
Uzma Ahmad ◽  
Sarfraz Ahmad ◽  
Rabia Yousaf
Keyword(s):  
Automorphism Group ◽  
Carbon Atom ◽  
Physicochemical Properties ◽  
Group Action ◽  
Chemical Compounds ◽  
Topological Indices ◽  
Zagreb Indices ◽  
Connectivity Indices ◽  
Atom Bond

In QSAR/QSPR studies, topological indices are utilized to predict the bioactivity of chemical compounds. In this paper, the closed forms of different Zagreb indices and atom?bond connectivity indices of regular dendrimers G[n] and H[n] in terms of a given parameter n are determined by using the automorphism group action. It was reported that these connectivity indices are correlated with some physicochemical properties and are used to measure the level of branching of the molecular carbon-atom skeleton.

scholarly journals The Terwilliger Algebra of the Twisted Grassmann Graph: the Thin Case

10.37236/9873 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Hajime Tanaka ◽  
Tao Wang
Keyword(s):  
Automorphism Group ◽  
Adjacency Matrix ◽  
Matrix Algebra ◽  
Simple Graph ◽  
Characteristic Vector ◽  
Terwilliger Algebra ◽  
Grassmann Graph ◽  
Complex Semisimple ◽  
Vertex Set ◽  
Twisted Grassmann Graph

The Terwilliger algebra $T(x)$ of a finite connected simple graph $\Gamma$ with respect to a vertex $x$ is the complex semisimple matrix algebra generated by the adjacency matrix $A$ of $\Gamma$ and the diagonal matrices $E_i^*(x)=\operatorname{diag}(v_i)$ $(i=0,1,2,\dots)$, where $v_i$ denotes the characteristic vector of the set of vertices at distance $i$ from $x$. The twisted Grassmann graph $\tilde{J}_q(2D+1,D)$ discovered by Van Dam and Koolen in 2005 has two orbits of the automorphism group on its vertex set, and it is known that one of the orbits has the property that $T(x)$ is thin whenever $x$ is chosen from it, i.e., every irreducible $T(x)$-module $W$ satisfies $\dim E_i^*(x)W\leqslant 1$ for all $i$. In this paper, we determine all the irreducible $T(x)$-modules of $\tilde{J}_q(2D+1,D)$ for this "thin" case.

scholarly journals From semigroup action to homomorphic encryption and some related problems in cryptography

2015 ◽  
Vol 18 (3) ◽  
pp. 254-259
Author(s):  
Thuong Tuan Dang ◽  
Tuan Anh Nguyen ◽  
Tran Thi Bao Ngo
Keyword(s):  
Automorphism Group ◽  
Group Action ◽  
Homomorphic Encryption ◽  
Key Exchange ◽  
Public Key ◽  
Key Exchange Protocol ◽  
Semigroup Action ◽  
Exact Sequences

In some recent papers, the authors have showed some homomorphic cryptosystems which are particular cases of split exact sequences of groups. By connecting the relation between these ideas to the concept of group action, in this paper, we build a public key exchange protocol based on actions to a group, from its automorphism group and semigroup ℤ under usual multiplication.

scholarly journals The cycle index of the automorphism group of Zn

2017 ◽  
Vol 101 (115) ◽  
pp. 99-108
Author(s):  
Vladimir Bozovic ◽  
Zana Kovijanic-Vukicevic
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Group Action ◽  
Building Blocks ◽  
Cycle Index ◽  
Residue Classes

We consider the group action of the automorphism group Un = Aut(Zn) on the set Zn, that is the set of residue classes modulo n. Clearly, this group action provides a representation of Un as a permutation group acting on n points. One problem to be solved regarding this group action is to find its cycle index. Once it is found, there appears a vast class of related enumerative and computational problems with interesting applications. We provide the cycle index of specified group action in two ways. One of them is more abstract and hence compact, while another one is basically a procedure of composing the cycle index from some building blocks. However, those building blocks are also well explained and finally presented in a very detailed fashion.

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