Computations of Colorings 7D‐Hypercube's Hyperplanes for All Irreducible Representations

Abstract

In the present study, we compute and enumerate the colorings of 7D‐hypercube for all of its hyperplanes (q = 1–7) for all 110 irreducible representations (IRs) of the seventh‐dimensional hyperoctahedral group consisting of 645,120 symmetry operations. The computations of colorings of the 7D‐hypercube are motivated by a number of chemical and biological applications such as the 7D‐hypercube representation of the periodic table, hypercube representations of water heptamer clusters, genetic regulatory networks, isomerization graphs, massively large data representations, and so forth. We have employed the Möbius inversion technique combined with generalized character cycle indices for 110 IRs to compute the generating functions for colorings of seven different types of hyperplanes of the 7D‐hypercube varying from the vertices (q = 7) to hexeracts (q = 1) of the 7D‐hypercube. Explicit computed tables are provided for 110 IRs from q = 1 to q = 7 for the 7D‐hypercube. © 2019 Wiley Periodicals, Inc.

Nonrigid water octamer: Computations with the 8‐cube

“…We note that the correlations of rotational levels as a function of J form a cyclic series similar to the pattern obtained for the C 60 Buckminterfullerene and heteronuclear analogs such as C 48 N 12 47,49,50 . Although previously the lower dimension hypercubes were considered, 46,48 the present problem of the octamer is considerably more complex as can be seen from Tables 6 and 7. Thus, we had to carry out multiple checks for the accuracy of the results in both Tables 6 and 7, as there are several tunneling levels with varying nuclear spin statistical weights.…”

“…Two different conjugacy classes of the wreath product groups can have the same cycle types. Therefore it is necessary to compute the matrix generators 35,46 as opposed to the cycle indices 44 for the character tables as the matrix generator method preserves the conjugacy class structure of wreath products. That is, in place of simple cycle types, we replace them with 2 × 8 matrices for each term in computing the matrix generating functions.…”

“…In our previous study on the 7D‐hypercube, we have outlined the generating function techniques combined with the powerful Möbius inversion method to generate the permutations, their cycle types, character tables and for the enumeration of colorings under various irreducible representations for the seventh dimensional hypercube 46 . Although the present work treats the nonrigid water octamer with a hypercube in the eighth dimension, we shall outline only the salient points that are specific to the water octamer for generating nuclear spin species, nuclear spin statistical weights from character cycle indices and the correlation tables for the rovibronic tunneling splittings.…”

“…The group theory of nonrigid molecules, their tunneling splittings of rovibronic levels, and nuclear spin statistics and related mathematical subjects that deal with spin and symmetry algebras have received considerable attention over the years 25,31,46 . The present author 35,46 has developed a matrix algebraic generator technique for the generation of the conjugacy classes and character table of the nonrigid groups of molecules in general, and in particular, when applied to the water octamer, the technique yields 185 irreducible representations and a character table that has several multidimensional irreducible representations. However, as shown here only some of the IRs exhibit nuclear spin populations, and thus the tunneling levels are somewhat tractable compared to the entire group of the fully nonrigid octamer.…”

“…Furthermore, one could consider a semirigid model wherein the overall cubic symmetry of the octamer cluster is intact, and we show here that such a semirigid cluster can be considered in a wreath product group of 12,288 operations. The group theory of nonrigid molecules, their tunneling splittings of rovibronic levels, and nuclear spin statistics and related mathematical subjects that deal with spin and symmetry algebras have received considerable attention over the years 25,31,46 . The present author 35,46 has developed a matrix algebraic generator technique for the generation of the conjugacy classes and character table of the nonrigid groups of molecules in general, and in particular, when applied to the water octamer, the technique yields 185 irreducible representations and a character table that has several multidimensional irreducible representations.…”