Example. All real types in a binary tetrahedral Cayley graph [fdip-0053]

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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.