The bracket complement operators Ci's, 1 ≤ i ≤ 4, that
initially acted purely on the bracketings of the product expressions were
turned into linear operators (matrices) that acted directly on vectors in
the vector space. They were derived for left/right LC-algebras with both
left-to-right (denoted by Cli) and right-to-left
(Cri) orderings of the tetron ALM-words. The objective now
is to construct operators, Si's, with special properties.
Specifically, they are to act trivially on the internal LC, ILC. Let U be
the ILC of 2:282: U = 107 A + 66 L + 31 M = (107, 66, 31). Then
Si U = U, (1 ≤ i ≤ 4).
Stated differently, U is the fixed point of Si's. These
operators will be referred to as the stabilizers of U. One such operator
that is immediately available is the identity operator (Id) that acts
trivially on all vectors: Id V = V, for all V; all vectors are
fixed points. The nontrivial operators Si's will be
constructed as the following wedged commutators of the complement
operators:
Si = [Cli,
(Ui), Cri] = Cli Ui
Cri − Cri Ui
Cli,
where, Ui = (a, l, m) is the
vector to be determined via application of the preceding triviality
constraint:
Si U = (Cli Ui
Cri − Cri Ui Cli) U
= Cli Ui Cri U −
Cri Ui Cli U = U.
Si is a linear operator acting on U from the left. Its terms
are products of alternating matrices and vectors which act (via
multiplication) in succession from right to left. Subsequently, the
implied parenthesization is:
Cli (Ui
(Cri U)) − Cri (Ui (Cli
U)).
Within the innermost parenthesis, Cli and
Cri act on U via matrix-vector multiplications resulting in
another vector which is then multiplied with vector Ui. This is
a vector-vector multiplication and is carried out in the LC-algebras. The
product vector is then acted another time by the complement operators. In
the end, the above constraint equation will lead to a linear system (LSYS)
of three equations whose three unknowns are a, l, m, the components of the
Ui.
The solutions to the LSYS uniquely determine the Si's, for 1
≤ i ≤ 4. Since, Si's are linear operators they are
represented by matrices similar to the complement operators. Their
matrices are determined via the actions of Si's on the standard
basis {A, L, M} = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}:
Si A = Si (1, 0, 0) =
1st col of Si
Si L = Si (0, 1, 0) =
2nd col of Si
Si M = Si (0, 0, 1) =
3rd col of Si,
where the basis vectors are treated as column vectors. In general,
the product of a matrix with the ith standard basis
column-vector (1 in ith row and 0 elsewhere) is simply
the ith column of the matrix. Once, the columns of a matrix
are known the matrix is fully determined. The following is the list
of Si matrices, for 1 ≤ i ≤ 4 in both LC-algebras:
Left-LC-Algebra:
Right-LC-Algebra: