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.

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 Reality properties of conjugacy classes in algebraic groups

2008 ◽  
Vol 165 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Anupam Singh ◽  
Maneesh Thakur
Keyword(s):  
Algebraic Groups ◽  
Conjugacy Classes

Classes of unipotent elements in simple algebraic groups. I

1976 ◽  
Vol 79 (3) ◽  
pp. 401-425 ◽  
Author(s):  
P. Bala ◽  
R. W. Carter
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Algebraic Groups ◽  
Adjoint Action ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Nilpotent Elements

LetGbe a simple adjoint algebraic group over an algebraically closed fieldK. We are concerned to describe the conjugacy classes of unipotent elements ofG. Goperates on its Lie algebra g by means of the adjoint action and we may consider classes of nilpotent elements of g under this action. It has been shown by Springer (11) that there is a bijection between the unipotent elements ofGand the nilpotent elements ofgwhich preserves theG-action, provided that the characteristic ofKis either 0 or a ‘good prime’ forG. Thus we may concentrate on the problem of classifying the nilpotent elements of g under the adjointG-action.

scholarly journals ON RELATIVE COMPLETE REDUCIBILITY

2020 ◽  
Vol 71 (1) ◽  
pp. 321-334 ◽  
Author(s):  
Christopher Attenborough ◽  
Michael Bate ◽  
Maike Gruchot ◽  
Alastair Litterick ◽  
Gerhard Röhrle
Keyword(s):  
Lie Algebras ◽  
Algebraic Groups ◽  
Reductive Group ◽  
Natural Generalization ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Complete Reducibility ◽  
Algebraic Description ◽  
Reductive Subgroup

Abstract Let $K$ be a reductive subgroup of a reductive group $G$ over an algebraically closed field $k$. The notion of relative complete reducibility, introduced in [M. Bate, B. Martin, G. Röhrle, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z.269 (2011), no. 1, 809–832], gives a purely algebraic description of the closed $K$-orbits in $G^n$, where $K$ acts by simultaneous conjugation on $n$-tuples of elements from $G$. This extends work of Richardson and is also a natural generalization of Serre’s notion of $G$-complete reducibility. In this paper we revisit this idea, giving a characterization of relative $G$-complete reducibility, which directly generalizes equivalent formulations of $G$-complete reducibility. If the ambient group $G$ is a general linear group, this characterization yields representation-theoretic criteria. Along the way, we extend and generalize several results from [M. Bate, B. Martin, G. Röhrle, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z.269 (2011), no. 1, 809–832].

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