On the Decomposition of a Representation of SOn When Restricted to SOn-1

1992 ◽  
Vol 44 (5) ◽  
pp. 974-1002 ◽  
Author(s):  
Benedict H. Gross ◽  
Dipendra Prasad
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Local Field ◽  
Orthogonal Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Special Orthogonal Group ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Linear Maps

Let k be a local field, with char(k) ≠ 2. A quadratic space V over k is a finite dimensional vector space together with a non-degenerate quadratic form Q: V → k.The special orthogonal group SO(V) consists of all linear maps T: V → V which satisfy:Q(Tv) = Q(v) for all ν and det T = 1.

A Witt Theorem for Non-Defective Lattices

1978 ◽  
Vol 30 (03) ◽  
pp. 499-511 ◽  
Author(s):  
Karl A. Morin-Strom
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Quadratic Forms ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

In [10], Witt laid the foundation for the study of quadratic forms over fields. Suppose Q is a quadratic form defined on a finite dimensional vector space V over a field of characteristic not equal to 2. Witt showed that non-zero vectors x and y in V satisfying Q(x) = Q(y) can be mapped into each other via an isometry of the vector space V. More generally, if τ : W ⟶ W’ is an isometry between subspaces of V, then τ extends to an isometry ϕ of V.

scholarly journals Involutions and commutators in orthogonal groups

1998 ◽  
Vol 65 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Frieder Knüppel ◽  
Gerd Thomsen
Keyword(s):  
Vector Space ◽  
Orthogonal Group ◽  
Commutator Subgroup ◽  
Dimensional Vector ◽  
Orthogonal Groups ◽  
Commutative Field ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Characteristic 2

AbstractSuppose we are given a regular symmetric bilinear from on a finite-dimensional vector space V over a commutative field K of characteristic ≠ 2. We want to write given elements of the commutator subgroup ω(V) (of the orthogonal group O(V)) and also of the kernel of the spinorial norm ker(Θ) as (short) products of involutions and as products of commutators

Products of idempotent endomorphisms of an independence algebra of infinite rank

1993 ◽  
Vol 114 (2) ◽  
pp. 303-319 ◽  
Author(s):  
John Fountain ◽  
Andrew Lewin
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Common Generalization ◽  
Finite Dimensional Vector Space ◽  
Infinite Rank ◽  
Finite Dimensional ◽  
Linear Maps

AbstractIn 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (under composition) of idempotent self-maps of the same set. In 1967, J. A. Erdos considered the analogous question for linear maps of a finite dimensional vector space and in 1985, Reynolds and Sullivan solved the problem for linear maps of an infinite dimensional vector space. Using the concept of independence algebra, the authors gave a common generalization of the results of Howie and Erdos for the cases of finite sets and finite dimensional vector spaces. In the present paper we introduce strong independence algebras and provide a common generalization of the results of Howie and Reynolds and Sullivan for the cases of infinite sets and infinite dimensional vector spaces.

scholarly journals Sets of idempotents that generate the semigroup of singular endomorphisms of a finite-dimensional vector space

1982 ◽  
Vol 25 (2) ◽  
pp. 133-139 ◽  
Author(s):  
R. J. H. Dawlings
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Mathematical System

IfMis a mathematical system and EndMis the set of singular endomorphisms ofM, then EndMforms a semigroup under composition of mappings. A number of papers have been written to determine the subsemigroupSMof EndMgenerated by the idempotentsEMof EndMfor different systemsM. The first of these was by J. M. Howie [4]; here the case ofMbeing an unstructured setXwas considered. Howie showed that ifXis finite, then EndX=Sx.

On Some Invariants of a Bilinear Form

1961 ◽  
Vol 4 (3) ◽  
pp. 261-264
Author(s):  
Jonathan Wild
Keyword(s):  
Vector Space ◽  
Bilinear Form ◽  
Arbitrary Field ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Special Cases ◽  
Orthogonal Subspace

Let E be a finite dimensional vector space over an arbitrary field. In E a bilinear form is given. It associates with every sub s pa ce V its right orthogonal sub space V* and its left orthogonal subspace *V. In general we cannot expect that dim V* = dim *V. However this relation will hold in some interesting special cases.

Sign in / Sign up

Export Citation Format

Share Document