Group-theoretical foundations for the concept of mandalas have been formulated algebraically and diagrammatically in order
to reinforce the spread of the unit-subduced-cycle-index (USCI) approach (S. Fujita,
Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991). Thus, after the introducton of right coset representations (RCR) (
H\)
G and left coset representations (LCR)
G(/
H) for the group
G and its subgroup
H, a regular body of
G-symmetry is defined as a diagrammatical expression for a right regular representation (
C
1\)
G, which is an extreme case of RCRs. The |
G| substitution positions of the regular body as a reference are numbered in accord with the numbering of the elements of
G and segmented into |
G|/|
H| of
H-segments, which are governed by the RCR (
H\)
G. By regarding each
H-segment as a substitution position, the
H-segmented regular body is reduced into a reduced regular body, which can be regarded as a secondary skeleton for generating
a molecule. The reference regular body (or
H-segmented one) is operated by every symmetry operations of
G to generate regular bodies (or
H-segmented ones), which are placed on the vertices of a hypothetical regular body of
G-symmetry. The resulting diagram (a nested regular body) is called
a mandala (or a reduced mandala), which is a diagrammatical expression for specifying the
G-symmetry of a molecule. The effect of a
K-subduction on the regular bodies of a mandala (or a reduced mandala) results in the
K-assemblage of the mandala (or the reduced mandala), where the resulting
K-assemblies governed by the LCR
G(/
K) construct a |
G|/|
K|-membered orbit, which corresponds to a molecule of
K-symmetry. The sphericity of the RCR (or the LCR) is used to characterize symmetrical properties of substitution positions
and those of stereoisomers. The fixed-point vector for each mandala (or reduced mandala) in terms of row view and the number
of fixed points of
K-assembled mandalas (or
K-assembled reduced mandalas) in terms of column view are compared to accomplish combinatorial enumeration of stereoisomers.
The relationship between a mandala and a reordered multiplication table is discussed.