Gröbner–Shirshov bases for non-crystallographic Coxeter groups

Abstract

For the group algebra of the finite non-crystallographic Coxeter group of type H 4, its Gröbner–Shirshov basis is constructed as well as the corresponding standard monomials, which describe explicitly all symmetries acting on the 120-cell and produce a natural operation table between the 14400 elements for the group.

Some Cases on Cozy Partitions

“…Also, Kang et al established an explicit isomorphic between cozy and standard tableaux of shape 𝜔. These relations are commonly used for connecting to Hecke algebras and Specht modules; (see [2], [3], [4], [5], [6], and [7]). To the best of our knowledge, there is no rule for counting the number of cozy 𝜔-tableaux.…”

Modeling of Nanomolecular and Reticular Architectures with 6-fold Grooved, Programmable Interlocking Disks

“…By introducing strain during the cyclization of ILD 35 which has relevance in understanding non-crystallographic groups 36 and the need of existence of Higgs particles. 37…”

“…As such, it is very convenient for use as an octahedral connector or for building primitive cubic crystal systems. On the other hand, the dodecahedral {[ILD 6 -(1,3,5)] 20 [ILD 6 -(1,3)] 30 } SBU may not have a direct utility in classical stereochemistry (Figure S9); however, it is worth noting that it can be used as a stepstone for modeling the 120-cell Coxeter polytope as shown recently by us, which has relevance in understanding noncrystallographic groups …”

Gröbner-Shirshov bases for the Coxeter groups of the types $G_2,F_4,E_6,E_7$ and $E_8$

“…In 2009, Yuqun Chen and Cihua Liu [12] gave an counter-example for the general hypothesis by Bokut in [8]. For non-crystallographic Coxeter groups, Jeong-Yup Lee and Dong-IL Lee [16] calculated the Gröbner-Shirshov bases in 2018.…”