scholarly journals Characteristic-dependent linear rank inequalities in 21 variables

2019 ◽  
Vol 43 (169) ◽  
pp. 764-770
Author(s):  
Victor Peña-Macias ◽  
Humberto Sarria - Zapata
Keyword(s):  
Finite Field ◽  
Linear Inequality ◽  
Binary Matrix ◽  
The Third ◽  
Linear Rank ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Characteristic 2 ◽  
Characteristic 3 ◽  
New Inequalities

In Linear Algebra over finite fields, a characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of spans of vector subspaces of a finite dimensional vector space over a finite field of determined characteristic, and does not in general hold over fields with other characteristic. This paper shows a preliminary result in the production of these inequalities. We produce three new inequalities in 21 variables using as guide a particular binary matrix, with entries in a finite field, whose rank is 8, with characteristic 2; 9 with characteristic 3; or 10 with characteristic neither 2 nor 3. The first inequality is true over fields whose characteristic is 2; the second inequality is true over fields whose characteristic is 2 or 3; the third inequality is true over fields whose characteristic is neither 2 nor 3.

scholarly journals Irreducible subgroups of symplectic groups in characteristic 2

2002 ◽  
Vol 73 (1) ◽  
pp. 85-96 ◽  
Author(s):  
Christopher Parker ◽  
Peter Rowley
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Symplectic Group ◽  
Symplectic Groups ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Characteristic 2 ◽  
Zero Vector

AbstractSuppose that V is a finite dimensional vector space over a finite field of characteristic 2, G is the symplectic group on V and a is a non-zero vector of V. Here we classify irreducible subgroups of G containing a certain subgroup of O2(StabG(a)) all of whose non-trivial elements are 2-transvections.

scholarly journals On nonzero component graph of vector spaces over finite fields

2017 ◽  
Vol 16 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Angsuman Das
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Clique Number ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Nonzero Component ◽  
Component Graph

In this paper, we study nonzero component graph [Formula: see text] of a finite-dimensional vector space [Formula: see text] over a finite field [Formula: see text]. We show that the graph is Hamiltonian and not Eulerian. We also characterize the maximal cliques in [Formula: see text] and show that there exists two classes of maximal cliques in [Formula: see text]. We also find the exact clique number of [Formula: see text] for some particular cases. Moreover, we provide some results on size, edge-connectivity and chromatic number of [Formula: see text].

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

scholarly journals On the Index of a Symmetric Form

1960 ◽  
Vol 3 (3) ◽  
pp. 293-295
Author(s):  
Jonathan Wild
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Dimensional Vector ◽  
Symmetric Form ◽  
Characteristic P ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

Let E be a finite dimensional vector space over a finite field of characteristic p > 0; dim E = n. Let (x,y) be a symmetric bilinear form in E. The radical Eo of this form is the subspace consisting of all the vectors x which satisfy (x,y) = 0 for every y ϵ E. The rank r of our form is the codimension of the radical.

scholarly journals Combinatorial polarization, code loops, and codes of high level

2004 ◽  
Vol 2004 (29) ◽  
pp. 1533-1541 ◽  
Author(s):  
Petr Vojtechovský
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Binary Codes ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Moufang Loops ◽  
Hamming Weights ◽  
High Level

We first find the combinatorial degree of any mapf:V→F, whereFis a finite field andVis a finite-dimensional vector space overF. We then simplify and generalize a certain construction, due to Chein and Goodaire, that was used in characterizing code loops as finite Moufang loops that possess at most two squares. The construction yields binary codes of high divisibility level with prescribed Hamming weights of intersections of codewords.

scholarly journals EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

2020 ◽  
pp. 1-14
Author(s):  
NICOLÁS ANDRUSKIEWITSCH ◽  
DIRCEU BAGIO ◽  
SARADIA DELLA FLORA ◽  
DAIANA FLÔRES
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Group Algebras ◽  
Finite Abelian Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Characteristic 2 ◽  
Nichols Algebras ◽  
Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

scholarly journals Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic

1985 ◽  
Vol 38 (3) ◽  
pp. 330-350 ◽  
Author(s):  
D. F. Holt ◽  
N. Spaltenstein
Keyword(s):  
Finite Field ◽  
Lie Algebras ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Reductive Algebraic Group ◽  
Exceptional Lie Algebras ◽  
Closed Fields ◽  
Characteristic 3 ◽  
Algebraically Closed Fields

AbstractThe classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

Several classes of permutation trinomials from Niho exponents over finite fields of characteristic 3

2019 ◽  
Vol 18 (04) ◽  
pp. 1950069
Author(s):  
Qian Liu ◽  
Yujuan Sun
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Coding Theory ◽  
Permutation Polynomials ◽  
Combinatorial Designs ◽  
Dickson Polynomial ◽  
Niho Exponents ◽  
Characteristic 3 ◽  
Permutation Trinomials ◽  
Algebraic Forms

Permutation polynomials have important applications in cryptography, coding theory, combinatorial designs, and other areas of mathematics and engineering. Finding new classes of permutation polynomials is therefore an interesting subject of study. Permutation trinomials attract people’s interest due to their simple algebraic forms and additional extraordinary properties. In this paper, based on a seventh-degree and a fifth-degree Dickson polynomial over the finite field [Formula: see text], two conjectures on permutation trinomials over [Formula: see text] presented recently by Li–Qu–Li–Fu are partially settled, where [Formula: see text] is a positive integer.

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.

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