Gosset polytopes in integral octonions

Woo-Nyoung Chang; Jae-Hyouk Lee; Sung Hwan Lee; Young Jun Lee

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 3, page 683-702
  • ISSN: 0011-4642

Abstract

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We study the integral quaternions and the integral octonions along the combinatorics of the 24 -cell, a uniform polytope with the symmetry D 4 , and the Gosset polytope 4 21 with the symmetry E 8 . We identify the set of the unit integral octonions or quaternions as a Gosset polytope 4 21 or a 24 -cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the E 8 or D 4 actions on the 4 21 or the 24 -cell, respectively. Moreover, we show that each level set in the unit integral numbers forms a uniform polytope, and we explain the dualities between them. In particular, the set of the pure unit integral octonions is identified as a uniform polytope 2 31 with the symmetry E 7 , and it is a dual polytope to a Gosset polytope 3 21 with the symmetry E 7 which is the set of the unit integral octonions with Re = 1 / 2 .

How to cite

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Chang, Woo-Nyoung, et al. "Gosset polytopes in integral octonions." Czechoslovak Mathematical Journal 64.3 (2014): 683-702. <http://eudml.org/doc/262177>.

@article{Chang2014,
abstract = {We study the integral quaternions and the integral octonions along the combinatorics of the $24$-cell, a uniform polytope with the symmetry $D_\{4\}$, and the Gosset polytope $4_\{21\}$ with the symmetry $E_\{8\}$. We identify the set of the unit integral octonions or quaternions as a Gosset polytope $4_\{21\}$ or a $24$-cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the $E_\{8\}$ or $D_\{4\}$ actions on the $4_\{21\}$ or the $24$-cell, respectively. Moreover, we show that each level set in the unit integral numbers forms a uniform polytope, and we explain the dualities between them. In particular, the set of the pure unit integral octonions is identified as a uniform polytope $2_\{31\}$ with the symmetry $E_\{7\}$, and it is a dual polytope to a Gosset polytope $3_\{21\}$ with the symmetry $E_\{7\}$ which is the set of the unit integral octonions with $\operatorname\{Re\}=1/2$.},
author = {Chang, Woo-Nyoung, Lee, Jae-Hyouk, Lee, Sung Hwan, Lee, Young Jun},
journal = {Czechoslovak Mathematical Journal},
keywords = {integral octonion; 24-cell; Gosset polytope; integral octonion; 24-cell; Gosset polytope},
language = {eng},
number = {3},
pages = {683-702},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Gosset polytopes in integral octonions},
url = {http://eudml.org/doc/262177},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Chang, Woo-Nyoung
AU - Lee, Jae-Hyouk
AU - Lee, Sung Hwan
AU - Lee, Young Jun
TI - Gosset polytopes in integral octonions
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 683
EP - 702
AB - We study the integral quaternions and the integral octonions along the combinatorics of the $24$-cell, a uniform polytope with the symmetry $D_{4}$, and the Gosset polytope $4_{21}$ with the symmetry $E_{8}$. We identify the set of the unit integral octonions or quaternions as a Gosset polytope $4_{21}$ or a $24$-cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the $E_{8}$ or $D_{4}$ actions on the $4_{21}$ or the $24$-cell, respectively. Moreover, we show that each level set in the unit integral numbers forms a uniform polytope, and we explain the dualities between them. In particular, the set of the pure unit integral octonions is identified as a uniform polytope $2_{31}$ with the symmetry $E_{7}$, and it is a dual polytope to a Gosset polytope $3_{21}$ with the symmetry $E_{7}$ which is the set of the unit integral octonions with $\operatorname{Re}=1/2$.
LA - eng
KW - integral octonion; 24-cell; Gosset polytope; integral octonion; 24-cell; Gosset polytope
UR - http://eudml.org/doc/262177
ER -

References

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