scholarly journals THREE-DIMENSIONAL ISOLATED QUOTIENT SINGULARITIES IN EVEN CHARACTERISTIC

2017 ◽  
Vol 60 (2) ◽  
pp. 435-445
Author(s):  
VLADIMIR SHCHIGOLEV ◽  
DMITRY STEPANOV
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Three Dimensional ◽  
Finite Subgroup ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Quotient Singularities ◽  
Characteristic 2

AbstractThis paper is a complement to the work of the second author on modular quotient singularities in odd characteristic. Here, we prove that if V is a three-dimensional vector space over a field of characteristic 2 and G < GL(V) is a finite subgroup generated by pseudoreflections and possessing a two-dimensional invariant subspace W such that the restriction of G to W is isomorphic to the group SL2(𝔽2n), then the quotient V/G is non-singular. This, together with earlier known results on modular quotient singularities, implies first that a theorem of Kemper and Malle on irreducible groups generated by pseudoreflections generalizes to reducible groups in dimension three, and, second, that the classification of three-dimensional isolated singularities that are quotients of a vector space by a linear finite group reduces to Vincent's classification of non-modular isolated quotient singularities.

scholarly journals Identities Generalizing the Theorems of Pappus and Desargues

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1382
Author(s):  
Roger D. Maddux
Keyword(s):  
Projective Plane ◽  
Vector Space ◽  
Invariant Theory ◽  
Lattice Theory ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

Classification of distance-transitive orbital graphs of overgroups of the Jevons group

2020 ◽  
Vol 30 (1) ◽  
pp. 7-22
Author(s):  
Boris A. Pogorelov ◽  
Marina A. Pudovkina
Keyword(s):  
Vector Space ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Isometry Group ◽  
Complete Bipartite Graph ◽  
Dimensional Vector ◽  
Translation Group ◽  
Dimensional Vector Space ◽  
Hamming Metric

AbstractThe Jevons group AS̃n is an isometry group of the Hamming metric on the n-dimensional vector space Vn over GF(2). It is generated by the group of all permutation (n × n)-matrices over GF(2) and the translation group on Vn. Earlier the authors of the present paper classified the submetrics of the Hamming metric on Vn for n ⩾ 4, and all overgroups of AS̃n which are isometry groups of these overmetrics. In turn, each overgroup of AS̃n is known to define orbital graphs whose “natural” metrics are submetrics of the Hamming metric. The authors also described all distance-transitive orbital graphs of overgroups of the Jevons group AS̃n. In the present paper we classify the distance-transitive orbital graphs of overgroups of the Jevons group. In particular, we show that some distance-transitive orbital graphs are isomorphic to the following classes: the complete graph 2n, the complete bipartite graph K2n−1,2n−1, the halved (n + 1)-cube, the folded (n + 1)-cube, the graphs of alternating forms, the Taylor graph, the Hadamard graph, and incidence graphs of square designs.

scholarly journals Modular representations of abelian groups with regular rings of invariants

1982 ◽  
Vol 86 ◽  
pp. 229-248 ◽  
Author(s):  
Haruhisa Nakajima
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Finite Subgroup ◽  
Dimensional Vector ◽  
Modular Representations ◽  
Invariant Polynomials ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Regular Rings

Let k be a field of characteristic p and G a finite subgroup of GL(V) where V is a finite dimensional vector space over k. Then G acts naturally on the symmetric algebra k[V] of V. We denote by k[V]G the subring of k[V] consisting of all invariant polynomials under this action of G. The following theorem is well known.Theorem 1.1 (Chevalley-Serre, cf. [1, 2, 3]). Assume that p = 0 or (|G|, p) = 1. Then k[V]G is a polynomial ring if and only if G is generated by pseudo-reflections in GL(V).

A Combinatorial Determinant Dual to the Group Determinant

2015 ◽  
Vol 29 ◽  
pp. 17-29
Author(s):  
Murali Srinivasan ◽  
Ashish Mishra
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Group Action ◽  
Dimensional Vector ◽  
Finite Group Action ◽  
Irreducible Polynomials ◽  
Dimensional Vector Space ◽  
Finite Set ◽  
A Finite Group

We define the commuting algebra determinant of a finite group action on a finite set, a notion dual to the group determinant of Dedekind. We give the following combinatorial example of a commuting algebra determinant. Let $\Bq(n)$ denote the set of all subspaces of an $n$-dimensional vector space over $\Fq$. The {\em type} of an ordered pair $(U,V)$ of subspaces, where $U,V\in \Bq(n)$, is the ordered triple $(\mbox{dim }U, \mbox{dim }V, \mbox{dim }U\cap V)$ of nonnegative integers. Assume that there are independent indeterminates corresponding to each type. Let $X_q(n)$ be the $\Bq(n)\times \Bq(n)$ matrix whose entry in row $U$, column $V$ is the indeterminate corresponding to the type of $(U,V)$. We factorize the determinant of $X_q(n)$ into irreducible polynomials.

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