scholarly journals Partitions on Finite Projective Lines

2019 ◽  
Vol 54 (6) ◽  
Author(s):  
Najm Abdulzahra Makhrib Al-Seraji ◽  
Ahmed Bakheet ◽  
Zainab Sadiq Jafar
Keyword(s):  
Fixed Points ◽  
Prime Number ◽  
Projective Line ◽  
Fundamental Theory ◽  
Primitive Polynomials ◽  
Nonsingular Matrices ◽  
Stabilizer Group ◽  
Projective Lines

The goal of this paper is to split the finite projective line into disjoint sublines by method of subgeometries where the order of line is not a prime number. The correspondence between the points on a line and the points on a conic has been described. The stabilizer group of some lines has been constructed using the fundamental theory of projective lines. All calculations are done using the GAP program. Also primitive polynomials over Galois filed are classified. Some examples with groups which are the fixed points of lines and study the properties of these groups are introduced. The nonsingular matrices which generate the points of conic and belong to groups of projectivities have been constructed.

Fixed Points of Automorphisms of Free Pro-p Groups of Rank 2

1995 ◽  
Vol 47 (2) ◽  
pp. 383-404 ◽  
Author(s):  
Wolfgang N. Herfort ◽  
Luis Ribes ◽  
Pavel A. Zalesskii
Keyword(s):  
Fixed Points ◽  
Prime Number ◽  
Finite Order ◽  
Finite Power ◽  
Rank 2

AbstractLet p be a prime number, and let F be a free pro-p group of rank two. Consider an automorphism α of F of finite order m, and let FixF(α) = {x ∈ F | α(x) = x} be the subgroup of F consisting of the elements fixed by α. It is known that if m is prime to p and α = idF, then the rank of FixF(α) is infinite. In this paper we show that if m is a finite power pr of p, the rank of FixF(α) is at most 2. We conjecture that if the rank of F is n and the order of a is a power of α, then rank (FixF(α)) ≤ n.

Erdös-Ko-Rado theorem in some linear groups and some projective special linear group

2014 ◽  
Vol 51 (1) ◽  
pp. 83-91
Author(s):  
Milad Ahanjideh ◽  
Neda Ahanjideh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Prime Number ◽  
Linear Group ◽  
Projective Line ◽  
Special Linear Group ◽  
Linear Groups ◽  
Projective Special Linear Group ◽  
Special Linear ◽  
Derangement Graph

Let V be the 2-dimensional column vector space over a finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document} (where q is necessarily a power of a prime number) and let ℙq be the projective line over \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. In this paper, it is shown that GL2(q), for q ≠ 3, and SL2(q) acting on V − {0} have the strict EKR property and GL2(3) has the EKR property, but it does not have the strict EKR property. Also, we show that GLn(q) acting on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( {\mathbb{F}_q } \right)^n - \left\{ 0 \right\}$$ \end{document} has the EKR property and the derangement graph of PSL2(q) acting on ℙq, where q ≡ −1 (mod 4), has a clique of size q + 1.

scholarly journals Picard groups of higher real -theory spectra at height

2017 ◽  
Vol 153 (9) ◽  
pp. 1820-1854 ◽  
Author(s):  
Drew Heard ◽  
Akhil Mathew ◽  
Vesna Stojanoska
Keyword(s):  
Spectral Sequence ◽  
Fixed Points ◽  
Galois Extension ◽  
Finite Subgroup ◽  
Picard Group ◽  
Picard Groups ◽  
Ring Spectra ◽  
Morava Stabilizer Group ◽  
Stabilizer Group ◽  
Homotopy Fixed Points

Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real $K$-theory spectra of Hopkins and Miller at height $n=p-1$, for $p$ an odd prime. More generally, we determine the Picard groups of the homotopy fixed points spectra $E_{n}^{hG}$, where $E_{n}$ is Lubin–Tate $E$-theory at the prime $p$ and height $n=p-1$, and $G$ is any finite subgroup of the extended Morava stabilizer group. We find that these Picard groups are always cyclic, generated by the suspension.

scholarly journals Results on the Projective Plane over a Finite Field of Order Seventeen

2020 ◽  
Vol 55 (2) ◽  
Author(s):  
Najm A. M. AL-Seraji ◽  
Hussam. H. Jawad
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Projective Line ◽  
Projective Mapping ◽  
Cubic Curve ◽  
Inflection Points ◽  
Stabilizer Group

The main goal of this research is to find the projective mapping that transforms a geometric formation called an i -set onto an arc such that the domain of the mapping is a subset of the projective line PG (1,q), q=17 , such that a5-set is called a pentad, a6-set is a hexad, a7-set is a heptad, a8-set is an octad, and a9 -set is a nonad, mapped onto a conicY2-XZ. The research also aims to find the stabilizer group of points on a non-singular cubic curve, with or without rational inflection points, on the projective plane over a finite field of order seventeen, and to give some examples.

scholarly journals Weighted Projective Lines and Rational Surface Singularities

2020 ◽  
Vol Volume 3 ◽  
Author(s):  
Osamu Iyama ◽  
Michael Wemyss
Keyword(s):  
Rational Surface ◽  
Projective Line ◽  
Degree Zero ◽  
Dual Graphs ◽  
Canonical Algebra ◽  
Tilting Bundle ◽  
Surface Singularities ◽  
Necessary And Sufficient ◽  
Projective Lines ◽  
Rational Surface Singularities

In this paper we study rational surface singularities R with star shaped dual graphs, and under very mild assumptions on the self-intersection numbers we give an explicit description of all their special Cohen-Macaulay modules. We do this by realising R as a certain Z-graded Veronese subring S^x of the homogeneous coordinate ring S of the Geigle-Lenzing weighted projective line X, and we realise the special CM modules as explicitly described summands of the canonical tilting bundle on X. We then give a second proof that these are special CM modules by comparing qgr S^x and coh X, and we also give a necessary and sufficient combinatorial criterion for these to be equivalent categories. In turn, we show that qgr S^x is equivalent to qgr of the reconstruction algebra, and that the degree zero piece of the reconstruction algebra coincides with Ringel's canonical algebra. This implies that the reconstruction algebra contains the canonical algebra, and furthermore its qgr category is derived equivalent to the canonical algebra, thus linking the reconstruction algebra of rational surface singularities to the canonical algebra of representation theory. Comment: Final version

On the Theorems of Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk

1984 ◽  
Vol 27 (2) ◽  
pp. 192-204 ◽  
Author(s):  
H. Steinlein
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Fixed Points ◽  
Prime Number ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Asymptotic Fixed Point ◽  
Asymptotic Fixed Point Theory

AbstractLet p ≥ 3 be a prime number and m a positive integer, and let S be the sphere S(m-1)(p-1)-1. Let f:S→S be a map without fixed points and with fp = idS. We show that there exists an h: S→ℝm with h(x) ≠ h(f(x)) for all x ∈ S. From this we conclude that there exists a closed cover U1,…, U4m of S with Uinf(Ui) = Ø for i = 1,…, 4m. We apply these results to Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk theorems in the framework of the sectional category and to a problem in asymptotic fixed point theory.

scholarly journals Applications of Mutations in the Derived Categories of Weighted Projective Lines to Lie and Quantum Algebras

2018 ◽  
Vol 2020 (19) ◽  
pp. 5814-5871
Author(s):  
Bangming Deng ◽  
Shiquan Ruan ◽  
Jie Xiao
Keyword(s):  
Projective Line ◽  
Derived Categories ◽  
Line Bundles ◽  
Hall Algebra ◽  
Moody Algebra ◽  
Weighted Projective Line ◽  
Coherent Sheaves ◽  
Enveloping Algebra ◽  
The Right ◽  
Projective Lines

Abstract Let $\textrm{coh}\ \mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\textrm{coh}\ \mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\textrm{coh}\ \mathbb{X})$ attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star-shaped quiver ${Q}$ associated with $\mathbb{X}$. By further dealing with the Ringel–Hall algebra of $\mathbb{X}$, we show that these functors provide a realization for Tits’ automorphisms of the Kac–Moody algebra ${\mathfrak g}_{Q}$ associated with ${Q}$, as well as for Lusztig’s symmetries of the quantum enveloping algebra of ${\mathfrak g}_{Q}$.

Inequivalent representations of geometric relation algebras

2003 ◽  
Vol 68 (1) ◽  
pp. 267-310 ◽  
Author(s):  
Steven Givant
Keyword(s):  
Automorphism Group ◽  
Collineation Group ◽  
Relation Algebra ◽  
Projective Line ◽  
Finite Geometries ◽  
Relation Algebras ◽  
Geometric Relation ◽  
Image Position ◽  
Projective Lines ◽  
Inequivalent Representations

AbstractIt is shown that the automorphism group of a relation algebra constructed from a projective geometry P is isomorphic to the collineation group of P. Also, the base automorphism group of a representation of over an affine geometry D is isomorphic to the quotient of the collineation group of D by the dilatation subgroup. Consequently, the total number of inequivalent representations of , for finite geometries P, is the sum of the numberswhere D ranges over a list of the non-isomorphic affine geometries having P as their geometry at infinity. This formula is used to compute the number of inequivalent representations of relation algebras constructed over projective lines of order at most 10. For instance, the relation algebra constructed over the projective line of order 9 has 56,700 mutually inequivalent representations.

scholarly journals Hecke invariants of knot groups

1974 ◽  
Vol 15 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Robert Riley
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Integral Domain ◽  
Unique Fixed Point ◽  
Projective Line ◽  
Transcendence Degree ◽  
Quotient Field ◽  
Prime Field ◽  
Projective Transformations ◽  
The Right

For each characteristic p, let Fp be the prime field and let Ώp be a fixed universal field which is algebraically closed and of infinite transcendence degree over Fp. When p = 0 we take Ώp = ℂ. Let F be a subfield of Ώp and let R be an integral domain whose quotient field is F. We abbreviate SL(2, R), PGL(2, R), PSL(2, R) to SL(R), PGL(R), PSL(R) respectively, and we cohsider PSL(R) as a group of projective transformations of the projective line ℘(Ώp) and of the “subline” ℘(F) ⊂ ℘(ΏP). The elements of PSL(R) are classified by the number of fixed points they have on ℘(F). If x ∈ PSL(R) has one such fixed point P, then P is the unique fixed point of x on ℘(ΏP) and x is called parabolic. All other x (except the identity E) have two distinct fixed points on ℘(Ώp) and x is called hyperbolic if these are on ℘(F), and elliptic otherwise. We put symbols for operators on the right.

scholarly journals A uniqueness theorem for Frobenius manifolds and Gromov–Witten theory for orbifold projective lines

2015 ◽  
Vol 2015 (702) ◽  
Author(s):  
Yoshihisa Ishibashi ◽  
Yuuki Shiraishi ◽  
Atsushi Takahashi
Keyword(s):  
Uniqueness Theorem ◽  
Initial Conditions ◽  
Projective Line ◽  
Frobenius Manifolds ◽  
Witten Theory ◽  
Projective Lines

AbstractWe prove that the Frobenius structure constructed from the Gromov–Witten theory for an orbifold projective line with at most three orbifold points is uniquely determined by the Witten–Dijkgraaf–Verlinde–Verlinde equations with certain natural initial conditions.

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