Representation of coxeter group and orthogonal group

Research Article
Open access

Representation of coxeter group and orthogonal group

Di Cai 1* , Yiding Tian 2
  • 1 5240 Fiore Terr, Apt J411, 92122, San Diego, CA, USA    
  • 2 5240 Fiore Terr, Apt J411, 92122, San Diego, CA, USA    
  • *corresponding author bruceyidingtian@gmail.com
TNS Vol.43
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-83558-537-5
ISBN (Online): 978-1-83558-538-2

Abstract

The paper is primarily divided into two parts. The main focus of the first part is the construction of a representation of Coxeter groups. This begins with the definition of the Coxeter system and connected components, followed by the introduction of the length function and subsequent theorems. The faithfulness of this representation is then proven, allowing for the identification of isomorphisms that enable the final classification of finite Coxeter groups. This classification is achieved by leveraging the established relationship between irreducible representations of Coxeter groups and positive definite quadratic forms. Given the strong connection between Coxeter groups and orthogonal groups, the primary objective of the second part is to create a specific representation of orthogonal groups. This is accomplished through an examination of the decomposition of harmonic polynomials into subspaces of homogeneous harmonic polynomials, using the action of O(2) on these subspaces. The paper concludes by drawing connections to results in Invariant Theory, demonstrating the applicability of the presented concepts in a more general duality context.

Keywords:

Coxeter Group, Reflection Representation, Orthogonal Group, Laplace Operator, Invariant Theory

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References

[1]. Mukherjee S 2021 Classification of finite coxeter groups [Online; accessed August 2021] URL https://math. uchicago.edu/˜may/REU2019/REUPapers/Mukherjee.pdf

[2]. Cohen A M 2008 Eindhoven University of Technology

[3]. Cohen A M 1994 Recent results on coxeter groups Polytopes: Abstract, Convex and Computational (Springer) pp 1–19

[4]. Humphreys J E 1992 Reflection Groups and Coxeter Groups 29 (Cambridge University Press)

[5]. Vinberg E B 1985 Russian Mathematical Surveys 40 31

[6]. Qmechanic What are the irreducible representations of the o(2) group? Physics Stack Exchange uRL:https://physics.stackexchange.com/q/759619 (version: 2023-04-18) (Preprint https://physics. stackexchange.com/q/759619) URL https://physics.stackexchange.com/q/759619

[7]. Tung W K 1985 Group Theory in Physics vol 1 (World Scientific)

[8]. 2023 Spherical harmonics Wikipedia, The Free Encyclopedia accessed on: August 2023 URL https://en. wikipedia.org/wiki/Spherical_harmonics

[9]. Axler S and Ramey W 1995 Proceedings of the American Mathematical Society 123 3765–3773

[10]. Vilenkin N I 1978 Special Functions and the Theory of Group Representations vol 22 (American Mathematical Soc.)

[11]. Goodman R and Wallach N R 2009 Symmetry, Representations, and Invariants vol 255 (Springer)

Cite this article

Cai,D.;Tian,Y. (2024). Representation of coxeter group and orthogonal group. Theoretical and Natural Science,43,298-311.

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About volume

Volume title: Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics

ISBN：978-1-83558-537-5(Print) / 978-1-83558-538-2(Online)
Editor：Yazeed Ghadi
Conference website: https://www.confciap.org/
Conference date: 27 January 2024
Series: Theoretical and Natural Science
Volume number: Vol.43
ISSN：2753-8818(Print) / 2753-8826(Online)

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References

[1]. Mukherjee S 2021 Classification of finite coxeter groups [Online; accessed August 2021] URL https://math. uchicago.edu/˜may/REU2019/REUPapers/Mukherjee.pdf

[2]. Cohen A M 2008 Eindhoven University of Technology

[3]. Cohen A M 1994 Recent results on coxeter groups Polytopes: Abstract, Convex and Computational (Springer) pp 1–19

[4]. Humphreys J E 1992 Reflection Groups and Coxeter Groups 29 (Cambridge University Press)

[5]. Vinberg E B 1985 Russian Mathematical Surveys 40 31

[6]. Qmechanic What are the irreducible representations of the o(2) group? Physics Stack Exchange uRL:https://physics.stackexchange.com/q/759619 (version: 2023-04-18) (Preprint https://physics. stackexchange.com/q/759619) URL https://physics.stackexchange.com/q/759619

[7]. Tung W K 1985 Group Theory in Physics vol 1 (World Scientific)

[8]. 2023 Spherical harmonics Wikipedia, The Free Encyclopedia accessed on: August 2023 URL https://en. wikipedia.org/wiki/Spherical_harmonics

[9]. Axler S and Ramey W 1995 Proceedings of the American Mathematical Society 123 3765–3773

[10]. Vilenkin N I 1978 Special Functions and the Theory of Group Representations vol 22 (American Mathematical Soc.)

[11]. Goodman R and Wallach N R 2009 Symmetry, Representations, and Invariants vol 255 (Springer)

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