scholarly journals Branching rules and commuting probabilities for Triangular and Unitriangular matrices

2021 ◽  
pp. 2250231
Author(s):  
Dilpreet Kaur ◽  
Uday Bhaskar Sharma ◽  
Anupam Singh
Keyword(s):  
Finite Field ◽  
Conjugacy Classes ◽  
Commuting Matrices ◽  
Branching Rules ◽  
Unitriangular Group ◽  
Triangular Group

This paper concerns the enumeration of simultaneous conjugacy classes of [Formula: see text]-tuples of commuting matrices in the upper triangular group [Formula: see text] and unitriangular group [Formula: see text] over the finite field [Formula: see text] of odd characteristic. This is done for [Formula: see text] and [Formula: see text], by computing the branching rules. Further, using the branching matrix thus computed, we explicitly get the commuting probabilities [Formula: see text] for [Formula: see text] in each case.

scholarly journals p-power conjugacy classes in U(n,q) and T(n,q)

2020 ◽  
pp. 2150121
Author(s):  
Silvio Dolfi ◽  
Anupam Singh ◽  
Manoj K. Yadav
Keyword(s):  
Algebraic Groups ◽  
Recursive Formula ◽  
Conjugacy Classes ◽  
Image Size ◽  
Large Size ◽  
Power Maps ◽  
Unitriangular Group ◽  
Triangular Group

Let [Formula: see text] be a [Formula: see text]-power where [Formula: see text] is a fixed prime. In this paper, we look at the [Formula: see text]-power maps on unitriangular group [Formula: see text] and triangular group [Formula: see text]. In the spirit of Borel dominance theorem for algebraic groups, we show that the image of this map contains large size conjugacy classes. For the triangular group we give a recursive formula to count the image size.

On the number of conjugacy classes of the sylow p-subgroups of GL(n,q)

1995 ◽  
Vol 52 (3) ◽  
pp. 431-439 ◽  
Author(s):  
Antonio Vera-López ◽  
J.M. Arregi ◽  
F.J. Vera-López
Keyword(s):  
Finite Field ◽  
Conjugacy Classes ◽  
Number Of Classes

If G is a finite p-group of order pn, P. Hall determined the number of conjugacy classes of G, r(G), modulo (p2 − 1)(p − 1). Namely, he proved the existence of a constant k ≥ 0 such that r(G) = n(p2 − 1) + pe + k(p2 − 1)(p − 1). In this paper, we denote by the group of the upper unitriangular matrices over , the finite field with q = pt elements, and we determine the number of classes of modulo (q − 1)5.

Torsion elements of order 𝑝2 in the Nottingham group

2020 ◽  
Vol 23 (3) ◽  
pp. 489-502
Author(s):  
Chun Yin Hui ◽  
Krishna Kishore
Keyword(s):  
Finite Field ◽  
Conjugacy Class ◽  
Upper Bound ◽  
Conjugacy Classes ◽  
Torsion Element ◽  
Exact Number

AbstractLet κ be a characteristic p finite field of q elements and {\mathfrak{N}_{\kappa}} the Nottingham group over κ. Lubin associated to every conjugacy class of torsion element of {\mathfrak{N}_{\kappa}} a type. We establish an upper bound {B(q;l,m)} on the number of conjugacy classes of order {p^{2}} torsion elements u of {\mathfrak{N}_{\kappa}} of type {\langle l,m\rangle}. In the case where {l<p}, the bound {B(q;l,m)} is the exact number of conjugacy classes. Moreover, we give a criterion on when u and {u^{n}} are conjugate.

scholarly journals Combinatorial Hopf algebra of supercharacters of type D

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Carolina Benedetti
Keyword(s):  
Hopf Algebra ◽  
Finite Field ◽  
Type A ◽  
The Other ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
Type D ◽  
International Audience ◽  
Supercharacter Theory ◽  
Triangular Group

International audience We provide a Hopf algebra structure on the supercharacter theory for the unipotent upper triangular group of type {D} over a finite field. Also, we make further comments with respect to types {B} and {C}. Type {A} was explored by M. Aguiar et. al (2010), thus this extended abstract is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types. Dotamos con una estructura de álgebra de Hopf la teoría de supercaracteres del grupo de matrices unipotentes triangulares superiores de tipo{D} sobre un cuerpo finito. Ademas, discutimos brevemente los tipos {B} y {C}. El tipo A fue explorado por M. Aguiar et al (2010), por lo tanto este resumen extendido es una contribución para entender combinatoriamente la teoría de supercaracteres de los otros tipos de Lie clásicos. Nous construisons une structure d'algèbre de Hopf sur la thérie des supercharactères du groupe de matrices unipotentes triangulaires supéieures de type {D}. Nous donnons aussi quelques commentaires à l'égard des types {B} et {C} . Le type {A} a été explorée par M. Aguiar et al. (2010), donc ce résumé étendu est une contribution à la théorie combinatoire des supercharactères pour les autres types de Lie classiques. \par

scholarly journals Conjugacy classes in projective and special linear groups

1980 ◽  
Vol 22 (3) ◽  
pp. 339-364 ◽  
Author(s):  
G.E. Wall
Keyword(s):  
Finite Field ◽  
Conjugacy Classes ◽  
Linear Groups ◽  
Commutative Field ◽  
Special Linear ◽  
Projective Special Linear Groups ◽  
Special Linear Groups ◽  
Finite Dimensional

The conjugacy classes in the finite-dimensional projective full linear, special linear and projective special linear groups over an arbitrary commutative field are determined. The results over a finite field are applied to certain enumerative problems.

Pairs of commuting matrices over a finite field

1960 ◽  
Vol 27 (1) ◽  
pp. 91-94 ◽  
Author(s):  
Walter Feit ◽  
N. J. Fine
Keyword(s):  
Finite Field ◽  
Commuting Matrices
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