Orthogonal Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Vector Space ◽  
Quadratic Space ◽  
Positive Dimension ◽  
Witt Index ◽  
Coxeter Diagram ◽  
Vertex Set ◽  
Dimension 2

This chapter presents a few results about certain forms of orthogonal buildings. It begins with notations stating that V is a K-vector space of positive dimension, (K, V, q) is a quadratic space of positive dimension, (K, V, q) is a regular quadratic space of positive Witt index, S is the vertex set of the Coxeter diagram, (K, V, q) is a hyperbolic quadratic space of dimension 2n for some n greater than or equal to 3, S is the vertex set of the Coxeter diagram for some n greater than or equal to 3, and Dn.l,script small l is the Tits index of absolute type Dn for n greater than or equal to 3. The chapter also considers propositions dealing with regular quadratic spaces and hyperbolic quadratic spaces.

The Other Bruhat-Tits Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
The Other ◽  
Quadratic Space ◽  
Coxeter Diagram ◽  
Vertex Set

This chapter summarizes the results about the residues of Bruhat-Tits buildings other than those associated with the exceptional Moufang quadrangles examined in previous chapters. It first considers cases, for which it assumes that Λ‎ is complete with respect to a discrete valuation in an appropriate sense. It then presents the Coxeter diagram of Ξ‎ and the vertex set S of such diagram, along with the J-residue of the building Ξ‎, which is called a gem if J is the complement in the S of a special vertex. The chapter also discusses the structure of the gems of Ξ‎ as well as cases in which the pseudo-quadratic space is defined to be ramified or unramified.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

The Integral Extension of Isometries of Quadratic Forms Over Local Fields

1966 ◽  
Vol 18 ◽  
pp. 920-942 ◽  
Author(s):  
Allan Trojan
Keyword(s):  
Vector Space ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
The Other ◽  
Quadratic Space ◽  
Integral Extension ◽  
Ring Of Integers ◽  
Extension Of Isometries ◽  
Necessary And Sufficient

Let F be a local field with ring of integers 0 and prime ideal π0. If V is a vector space over F, a lattice L in F is defined as an 0-module in the vector space V with the property that the elements of L have bounded denominators in the basis for V. If V is, in addition, a quadratic space, the lattice L then has a quadratic structure superimposed on it. Two lattices on V are then said to be isometric if there is an isometry of V that maps one onto the other.In this paper, we consider the following problem: given two elements, v and w, of the lattice L over the regular quadratic space V, find necessary and sufficient conditions for the existence of an isometry on L that maps v onto w.

scholarly journals Self-dual Leonard pairs

2019 ◽  
Vol 7 (1) ◽  
pp. 1-19
Author(s):  
Kazumasa Nomura ◽  
Paul Terwilliger
Keyword(s):  
Vector Space ◽  
The Self ◽  
The Other ◽  
Endomorphism Algebra ◽  
Comprehensive Description ◽  
Leonard Pairs ◽  
Positive Dimension ◽  
Linear Maps ◽  
Leonard Pair ◽  
Dual Case

Abstract Let F denote a field and let V denote a vector space over F with finite positive dimension. Consider a pair A, A* of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which there exists an automorphism of the endomorphism algebra of V that swaps A and A*. Such an automorphism is unique, and called the duality A ↔ A*. In the present paper we give a comprehensive description of this duality. In particular,we display an invertible F-linearmap T on V such that the map X → TXT−1is the duality A ↔ A*. We express T as a polynomial in A and A*. We describe how T acts on 4 flags, 12 decompositions, and 24 bases for V.

Spinors in Hilbert space

1976 ◽  
Vol 80 (2) ◽  
pp. 337-347 ◽  
Author(s):  
R. J. Plymen
Keyword(s):  
Hilbert Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Double Cover ◽  
Projective Representation ◽  
Complex Vector Space ◽  
Complex Vector ◽  
True Representation ◽  
Dimension 2 ◽  
Image Position

In 1913, É. Cartan discovered that the special orthogonal groupSO(k) has a ‘two-valued’ representation (i.e. a projective representation) on a complex vector spaceSof dimension 2n, wherek= 2nor 2n+ 1. The projective representation in question lifts to a true representation of the double cover Spin (k) ofSO(k). We restrict attention to the casek= 2n. Under the action of Spin (2n),Sbreaks up into 2 irreducible subspaces:The vectors inSare calledspinors(relative toSO(2n)), those inS+orS−are calledhalf-spinors(4).

scholarly journals LEONARD PAIRS AND THE ASKEY–WILSON RELATIONS

2004 ◽  
Vol 03 (04) ◽  
pp. 411-426 ◽  
Author(s):  
PAUL TERWILLIGER ◽  
RAIMUNDAS VIDUNAS
Keyword(s):  
Vector Space ◽  
Linear Transformations ◽  
Leonard Pairs ◽  
Positive Dimension ◽  
The Matrix ◽  
Leonard Pair

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A*:V→V which satisfy the following two properties: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Referring to the above Leonard pair, we show there exists a sequence of scalars β,γ,γ*,ϱ,ϱ*,ω,η,η* taken from K such that both [Formula: see text] The sequence is uniquely determined by the Leonard pair provided the dimension of V is at least 4. The equations above are called the Askey–Wilson relations.

Quadratic Forms of Type F4

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Quadratic Forms ◽  
Quadratic Extension ◽  
Scalar Multiplication ◽  
Quadratic Space

This chapter presents various results about quadratic forms of type F₄. The Moufang quadrangles of type F₄ were discovered in the course of carrying out the classification of Moufang polygons and gave rise to the notion of a quadratic form of type F₄. The chapter begins with the notation stating that a quadratic space Λ‎ = (K, L, q) is of type F₄ if char(K) = 2, q is anisotropic and: for some separable quadratic extension E/K with norm N; for some subfield F of K containing K² viewed as a vector space over K with respect to the scalar multiplication (t, s) ↦ t²s for all (t, s) ∈ K x F; and for some α‎ ∈ F* and some β‎ ∈ K*. The chapter also considers a number of propositions regarding quadratic spaces and discrete valuations.

scholarly journals Tridiagonal pairs of type III with height one

2019 ◽  
Vol 35 ◽  
pp. 555-582 ◽  
Author(s):  
Xue Li ◽  
Bo Hou ◽  
Suogang Gao
Keyword(s):  
Vector Space ◽  
Closed Field ◽  
Positive Dimension ◽  
Algebraically Closed Field ◽  
Type Iii ◽  
Tridiagonal Pairs ◽  
Unique Integer ◽  
Tridiagonal Pair

Let K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K with finite positive dimension, and let A, A∗ denote a tridiagonal pair on V  of diameter d.  Let V0, . . . , Vd  denote a standard ordering of  the eigenspaces of A on V , and let θ0, . . . , θd denote the corresponding eigenvalues of A. It is assumed that d ≥ 3.  Let ρi  denote the dimension of Vi. The sequence ρ0, ρ1, . . . , ρd is called the shape of the tridiagonal pair. It is known that ρ0 = 1 and there  exists  a  unique  integer  h (0 ≤ h ≤ d/2)  such  that  ρi−1 < ρi  for  1 ≤ i ≤ h,  ρi−1 = ρi  for  h < i ≤ d − h,  and  ρi−1 > ρi for d − h < i ≤ d. The integer h is known as the height of the tridiagonal pair. In this paper, it is showed that the shape of a tridiagonal pair of type III with height one is either 1, 2, 2, . . ., 2, 1 or 1, 3, 3, 1.  In each case, an interesting basis is found for V and the actions of A, A∗ on this basis are described.

scholarly journals $\mathsf Q$-linear functions, functions with dense graph, and everywhere surjectivity

2008 ◽  
Vol 102 (1) ◽  
pp. 156 ◽  
Author(s):  
F. J. García-Pacheco ◽  
F. Rambla-Barreno ◽  
J. B. Seoane-Sepúlveda
Keyword(s):  
Vector Space ◽  
Linear Functions ◽  
Dense Graph ◽  
Dimension 2

Let $L$, $S$ and $D$ denote, respectively, the set of $\mathsf{Q}$-linear functions, the set of everywhere surjective functions and the set of dense-graph functions on $\mathsf{R}$. In this note, we show that the sets $D\setminus(S \cup L)$, $S \setminus L$, $S \cap L$ and $D\cap L \setminus S$ are lineable. Moreover, all these sets contain (omitting zero) a vector space of the biggest possible dimension, $2^c$.

scholarly journals Two-Dimensional Partial Cubes

10.37236/8934 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Victor Chepoi ◽  
Kolja Knauer ◽  
Manon Philibert
Keyword(s):  
Cell Structure ◽  
Low Rank ◽  
Complete Graphs ◽  
Two Dimensional ◽  
Vc Dimension ◽  
Set Systems ◽  
Sample Compression ◽  
Vertex Set ◽  
Dimension 2 ◽  
Pseudoline Arrangements

We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from being solved even for general set systems of VC-dimension 2. Moreover, we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements.

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