Example. All real types in a binary tetrahedral Cayley graph [fdip-0053]
AGENTDRAFTED
Let \(T=2T\) be the 24 Hurwitz units, with
\(\omega =(-1+i+j+k)/2\), and take the inverse-closed generating set
\[S=\{i,-i,j,-j,k,-k,\omega ,\bar \omega \}.\]
Give each unordered edge \(\{g,gs\}\) weight one, counting it once.
There are no loops, the degree is 8, and left multiplication
acts by signed graph symmetries. On \(\mathbb R^T\) the Laplacian is
\[(Lf)(g)=8f(g)-\sum _{s\in S}f(gs).\]
The underlying group construction is
[voight2021quaternion, Section 11.2, pp. 166-168].
The following spectrum and blocks are calculations for this
particular generating set.
The quotient \(T/Q_8\cong C_3\) gives the character
\(\lambda (\omega )=\zeta =e^{2\pi i/3}\), \(\lambda |_{Q_8}=1\).
Let \(\sigma \) be the defining two-dimensional complex quaternion
representation, with \(\chi _\sigma (q)=2\operatorname {Re}q\), and
let \(\tau \) act by quaternion conjugation on
\(\operatorname {span}_{\mathbb R}\{i,j,k\}\).
Its character is \(4(\operatorname {Re}q)^2-1\).
The 7 conjugacy classes and their characters are
\[\begin {array}{c|rrrrrrr}
q&1&-1&i&\omega &-\omega &\omega ^2&-\omega ^2\\\hline
|[q]|&1&1&6&4&4&4&4\\
\lambda ^j&1&1&1&\zeta ^j&\zeta ^j&\zeta ^{2j}&\zeta ^{2j}\\
\sigma \lambda ^j&2&-2&0&-\zeta ^j&\zeta ^j&-\zeta ^{2j}&\zeta ^{2j}\\
\tau &3&3&-1&0&0&0&0
\end {array}\]
with \(j=0,1,2\). The class of \(i\) comprises
\(\{\pm i,\pm j,\pm k\}\); each remaining class is determined
by its \(Q_8\)-coset and real part. These descriptions follow by
conjugating the explicit Hurwitz units.
Using the displayed sizes, the character inner products are
\(\delta _{\alpha \beta }\) and the sum of squared degrees is
\(3\cdot 1^2+3\cdot 2^2+3^2=24\). Thus the 7 characters are
irreducible, inequivalent and complete. Squaring sends the first
three classes to \(1,1,-1\), the two \(\pm \omega \) classes to
\([\omega ^2]\), and the two \(\pm \omega ^2\) classes to \([\omega ]\).
Substitution into \(|T|^{-1}\sum _g\chi (g^2)\) gives indicators
\(1,0,0\) for \(1,\lambda ,\lambda ^2\), \(-1,0,0\) for
\(\sigma ,\sigma \lambda ,\sigma \lambda ^2\), and \(1\) for \(\tau \).
For example the numerator for \(\sigma \) is
\(2+2-12-16=-24\); for \(\sigma \lambda \) it is
\(4-12-8(\zeta +\zeta ^2)=0\).
The real-type bridge therefore gives
\[\operatorname {End}_T(\mathbb R^T)
\cong \mathbb R\times \mathbb C\times M_3(\mathbb R)
\times \mathbb H\times M_2(\mathbb C).\]
Here the regular real multiplicity is \(m_\Xi =\dim _{D_\Xi }W_\Xi \):
a real simple matrix algebra \(M_r(D)\) has \(r\) column copies of
its defining module, as in Proposition [fcap-0042].
Fourier decomposition of right translation gives, up to the
chosen dual-basis convention, the block
\(8I-\sum _{s\in S}\pi (s)\). Its adjoint or transpose gives the
same spectra and determinants, since \(S=S^{-1}\).
The multiplicity matrices are as follows; in the table
\(m\) is multiplicity, not the mass parameter:
\[\begin {array}{c|c|c|c|c}
W\text { type}&D&m&\dim _{\mathbb R}W&L_\Xi \\\hline
1&\mathbb R&1&1&0\\
\lambda ,\bar \lambda &\mathbb C&1&2&3\\
\tau &\mathbb R&3&3&11I_3-J_3\\
\sigma &\mathbb H&1&4&9\\
\sigma \lambda ,\sigma \bar \lambda &\mathbb C&2&4&\operatorname {diag}(6,9)
\end {array}\]
Here \(J_3\) is the all-ones matrix. For \(\tau \), the 6
\(\pm i,\pm j,\pm k\) contribute \(-2I_3\), while
\(\tau (\omega )+\tau (\bar \omega )=J_3-I_3\); the block has
eigenvalues \(8,11,11\). For \(\sigma \) the 6 contributions
cancel and \(\sigma (\omega )+\sigma (\bar \omega )=-I_2\), giving
\(9\). For \(\sigma \lambda \), diagonalize \(\sigma (\omega )\)
with eigenvalues \(\zeta ,\zeta ^2\). The twisted adjacency sum has
eigenvalues \(-1,2\), giving \(9,6\). The nonreal linear pair has
adjacency eigenvalue \(6+\zeta +\zeta ^2=5\).
The real sector dimensions are \(1,2,9,4,8\), summing to \(24\).
Consequently
\[\operatorname {spec}(L)=\{0,3^{[2]},6^{[4]},8^{[3]},9^{[8]},11^{[6]}\},\]
where brackets denote eigenvalue multiplicity. For a mass squared
\(a>0\), set \(Q=L+aI\); every block receives the same shift and
\[\begin {aligned}
\det _{\mathbb R}Q={}&a(a+3)^2(a+6)^4\\
&\cdot (a+8)^3(a+9)^8(a+11)^6.
\end {aligned}\]
The complex and quaternionic exponents are real dimensions, not
complex degrees copied without descent. The commutant dimension
is \(24\); the symmetric dimension is \(13\), also the number of
inversion orbits in \(T\), since only \(1,-1\) are self-inverse.
These are finite graph calculations; the possible physical
interpretations discussed in Remark [fgap-0018] are separate hypotheses.