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Eigenvalues of Finite Projective Planes with an Abelian Cartesian Group

J. L. Hayden
2004 Designs, Codes and Cryptography  
Let M be an incidence matrix for a projective plane of order n.  ...  Under these conditions it is shown that G is a 2-group and n ¼ 2 r when the eigenvalues of A are real.  ...  Preliminary Discussion Let ðp; lÞ be a finite projective plane of order n and set N ¼ n 2 þ n þ 1ð¼ jpj ¼ jljÞ.  ... 
doi:10.1023/b:desi.0000035469.24046.f3 fatcat:62t5pxnugna7hc23gqycfkhpb4

Distance-regular Cayley graphs with least eigenvalue $$-2$$ - 2

Alireza Abdollahi, Edwin R. van Dam, Mojtaba Jazaeri
2016 Designs, Codes and Cryptography  
Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes.  ...  We classify the distance-regular Cayley graphs with least eigenvalue -2 and diameter at most three.  ...  Mojtaba Jazaeri thanks the Graduate Studies of University of Isfahan and Tilburg University since this paper was partly written during his visit at Tilburg University as part of his PhD program in Isfahan  ... 
doi:10.1007/s10623-016-0209-4 fatcat:v2y4byyn5ngmlk5a3g22nwluyi

Page 3773 of Mathematical Reviews Vol. , Issue 81I [page]

1981 Mathematical Reviews  
Authors’ summary: “After a brief review of matrix representations of finite abelian groups, projection operators are defined and used to compute symmetry coordinates for systems of coupled harmonic oscillators  ...  N. 811:81034 Representation theory of finite abelian groups applied to a linear diatomic crystal. Internat. J. Math. Math. Sci. 3 (1980), no. 3, 559-574.  ... 

Cayley sum graphs and eigenvalues of (3,6)-fullerenes [article]

Matt DeVos, Luis Goddyn, Bojan Mohar, Robert Samal
2007 arXiv   pre-print
In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form {λ,-λ} except for the four eigenvalues {3,-1,-1,-1}.  ...  We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, "(3,6)-fullerenes", have been studied by chemists who are interested in their energy spectra.  ...  We have constructed infinite families of Cayley sum graphs whose spectra have the form M ∪ L ∪ −L, where M is a fixed finite multiset.  ... 
arXiv:0712.1631v2 fatcat:2eft7mzuhbauhmqwn5uuwi6l6e

The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U( n)

2008 Mathematical proceedings of the Cambridge Philosophical Society (Print)  
We show that there is a homotopy cofiber sequence of spectra relating Carlsson's deformation K-theory of a group G to its "deformation representation ring," analogous to the Bott periodicity sequence relating  ...  We then apply this to study simultaneous similarity of unitary matrices.  ...  Given an n 1 -dimensional plane V and an n 2 -dimensional plane W in C n , let p be the orthogonal projection from W to V and q = p T the projection from V to W .  ... 
doi:10.1017/s0305004108001928 fatcat:j6lbkdvzdvcshjm6zf6ae4wmqy

Eigenvalues of Cayley graphs [article]

Xiaogang Liu, Sanming Zhou
2019 arXiv   pre-print
We survey some of the known results on eigenvalues of Cayley graphs and their applications, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs.  ...  It is well known that for any finite projective plane π there exists an integer n ≥ 1, called the order of π, such that each line contains n + 1 points, each point is on n + 1 lines, and the number of  ...  Ramanujan finite upper half plane graphs Let q = p r be an odd prime power and δ a non-square of the finite field F q .  ... 
arXiv:1809.09829v2 fatcat:57hq3pcaurhybgfuzzh6qi74nu

The combinatorial-motivic nature of F_1-schemes [article]

Koen Thas
2014 arXiv   pre-print
certain group actions on projective spaces, and brand new results in the latter context are described.  ...  We review Deitmar's theory of monoidal schemes to start with, and have a detailed look at the standard examples.  ...  The reader recalls a basic classical result for finite projective planes which states that if a finite projective plane Γ of order n admits the projective general linear group PGL 3 (n) as an automorphism  ... 
arXiv:1406.5447v1 fatcat:rmzkge2lrveq3pukotqzu7zatm

Page 3903 of Mathematical Reviews Vol. , Issue 83i [page]

1983 Mathematical Reviews  
The authors consider a system S of first-order differential equations describing an interaction of an arbitrary finite number of scalar and spinor fields.  ...  Gauging the model introduces a third Cartesian coordinate z.  ... 

Four-band non-Abelian topological insulator and its experimental realization [article]

Tianshu Jiang, Qinghua Guo, Ruo-Yang Zhang, Zhao-Qing Zhang, Biao Yang, C. T. Chan
2021 arXiv   pre-print
Furthermore, a system with even or odd number of bands will exhibit significant difference in non-Abelian topological classifications.  ...  We show the evolution of bulk topology by extending the 1D Hamiltonian onto a 2D plane and provide the accompanying edge state distributions following an analytical method.  ...  Eigenvalue of such an edge state is determined from Eq. (S25).  ... 
arXiv:2106.16080v1 fatcat:djnwmo5k3jhutm5dwx6r4hpb3m

Fake projective planes, automorphic forms, exceptional collections [article]

Sergey Galkin, Ilya Karzhemanov, Evgeny Shinder
2019 arXiv   pre-print
The novelty stems from the idea that if some line bundle is non-linearizable with respect to a finite abelian group, then it should be linearized by a finite, non-abelian, Heisenberg group.  ...  In the present note we give two short proofs of this statement and show acyclicity of some other line bundles on the fake projective planes with at least 9 automorphisms.  ...  with an action of a finite group G : X (not necessarily faithful).  ... 
arXiv:1602.06107v2 fatcat:qbqafqycurdljpiq2rsnv353mm

Distance-regular Cayley graphs with small valency

Edwin R. van Dam, Mojtaba Jazaeri
2019 Ars Mathematica Contemporanea  
We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs.  ...  We consider the problem of which distance-regular graphs with small valency are Cayley graphs.  ...  Let G be a finite group and S be an inverse-closed subset of G not containing the identity element e of G.  ... 
doi:10.26493/1855-3974.1964.297 fatcat:pu2ispk3drfhpfj6mr3ar4yhby

Distance-regular Cayley graphs with small valency [article]

Edwin R. van Dam, Mojtaba Jazaeri
2019 arXiv   pre-print
We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs.  ...  We consider the problem of which distance-regular graphs with small valency are Cayley graphs.  ...  Acknowledgements The research of Mojtaba Jazaeri was in part supported by a grant from School of Mathematics, Institute for Research in Fundamental Sciences (IPM) (No. 95050039).  ... 
arXiv:1808.01428v2 fatcat:acxku3se6ndihgakxsrkdpazaq

Applications of the Weyl-Wigner formalism to noncommutative geometry [article]

Alessandro Zampini
2005 arXiv   pre-print
It is presented an extension of this formalism to the case of a more general classical phase space, namely one whose configuration space is a compact simple Lie group.  ...  In this dissertation the Weyl-Wigner approach is presented as a map between functions on a real cartesian symplectic vector space and a set of operators on a Hilbert space, to analyse some aspects of the  ...  L is identified with the manifold representing the abelian Lie group of translations, then a Weyl system can be seen as a unitary, projective representation of the translation group.  ... 
arXiv:hep-th/0505271v1 fatcat:hmmk5ptygvhmdewukzfgwtlqzy

Integral form of Yang-Mills equations and its gauge invariant conserved charges

L. A. Ferreira, G. Luchini
2012 Physical Review D  
Our results are important in the understanding of global properties of non-abelian gauge theories.  ...  Despite the fact that the integral form of the equations of classical electrodynamics is well known, the same is not true for non-abelian gauge theories. The aim of the present paper is threefold.  ...  It might connect to the so-called Abelian projection and arguments for confinement. B.  ... 
doi:10.1103/physrevd.86.085039 fatcat:zokygymp6jelfajjcn5qvpo7ve

Remarks on endomorphisms and rational points

E. Amerik, F. Bogomolov, M. Rovinsky
2011 Compositio Mathematica  
AbstractLetXbe an algebraic variety and letf:X−−→Xbe a rational self-map with a fixed pointq, where everything is defined over a number fieldK.  ...  As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold, originally proved by Claire Voisin and the first author in 2007.  ...  Acknowledgements: The starting ideas for this article have emerged during the special trimester "Groups and Geometry" at Centro Di Giorgi, Pisa, where the three authors were present in October 2008.  ... 
doi:10.1112/s0010437x11005537 fatcat:awv72cmgqvgjvm4ute5tnkxmp4
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