Proposition. Finite identities for the binary tetrahedral spinor [fdip-00A0]
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  • November 2, 2026

Let \(2T\) be the group of 24 Hurwitz units in the binary tetrahedral quotient. In this model the central group element is the quaternion \(-1\). The distinguished central involution of the abstract group is the preimage of that quaternion under the comparison isomorphism, not the quaternion itself. Let \(\pi :SU(2)\to SO(3)\) be the conjugation homomorphism of Proposition [fdip-007G], and write \(R_T=\pi (2T)\). Let \(\rho _s\) be the defining representation on \(\mathbb C^2\) through the matrix map \(M\) of the same statement, and let \(\tau \) be the three-dimensional representation on \(\operatorname {span}_{\mathbb R}\{i,j,k\}\) given by \(\tau (q)v=qvq^{-1}\). Both are the fibers of the two-fiber example. The substrate action \(\alpha \) is trivial. On the sum \(F=\mathbb C^2\oplus \mathbb R^3\), with \(\rho =\rho _s\oplus \tau \), the selector \(\sigma \) of the compatibility datum is the idempotent \[P=\frac {I-\rho (-1)}2.\] The abstract involution \(z\) maps to the quaternion \(-1\), so \(P\) is the image of the central idempotent \(e_-=(1_A-[z])/2\) under \(\rho \). That idempotent is the one in the real group algebra. The reference comparison \(C\) is the pair of homomorphisms \(\pi :SU(2)\to SO(3)\) and the vertex identification \(R_T\cong A_4\) proved below.

Then \(\ker (\pi |_{2T})=\{1,-1\}\), the vertex action induces \(R_T\cong A_4\), and \[P=I_2\oplus 0.\] Left multiplication by the quaternion \(-1\) is \(-1\) on the quaternion module, so that module is fixed by \(e_-\) and annihilated by \(e_+=(1_A+[z])/2\). The products of the \(d=3\) Clifford matrices of the spatial Dirac operator reproduce the quaternion multiplication table. None of these identities supplies a covariance law for that Dirac operator.

Proof. 

  • November 2, 2026

The kernel statement is the restriction of \(\ker \pi =\{1,-1\}\) from Proposition [fdip-007G]. The same source identifies \(2T\) with the full preimage \(\pi ^{-1}(R_T)\) and gives \(|R_T|=12\). The matrix \(M(-1)=-I_2\), so \(\rho _s(-1)=-I_2\). Conjugation by the quaternion \(-1\) is the identity, so \(\tau (-1)=I_3\). Therefore \[P=I_2\oplus 0.\] On the quaternion module, left multiplication by the quaternion \(-1\) is \(-1\), so that module's algebra action sends \(e_-\) to \(1\) and \(e_+\) to \(0\). The abstract involution maps to that quaternion, and \(\rho =\rho _s\oplus \tau \), so \[\rho (z)=-I_2\oplus I_3.\] Hence \(\rho (e_-)=(I-\rho (z))/2=P\), while \[\rho (e_+)=(I+\rho (z))/2=0\oplus I_3.\] The scalar \(0\) is the quaternion-module action of \(e_+\), not this operator.

Label the 4 imaginary vectors of Proposition [fdip-007G] by \[v_1=(1,1,1),\; v_2=(1,-1,-1),\; v_3=(-1,1,-1),\; v_4=(-1,-1,1).\] Their differences \[v_2-v_1=(0,-2,-2),\; v_3-v_1=(-2,0,-2),\; v_4-v_1=(-2,-2,0)\] are linearly independent: after removing the common factor \(-2\), the determinant of \[\begin {pmatrix}0&1&1\\1&0&1\\1&1&0\end {pmatrix}\] equals \(2\). A rotation fixing all 4 vertices therefore fixes a basis of \(\mathbb R^3\) and is the identity. The vertex action gives an injective homomorphism \(R_T\to S_4\).

Conjugation by \(i\) is the coordinate map \((x,y,z)\mapsto (x,-y,-z)\). Indeed, for \(v=xi+yj+zk\), \[iv=(-x)+(yk-zj),\qquad (iv)(-i)=xi-yj-zk,\] where the last step uses \(ji=-k\) and \(ki=j\). The same expansion with cycled units gives \[j:(x,y,z)\mapsto (-x,y,-z),\qquad k:(x,y,z)\mapsto (-x,-y,z).\] On the 4 vertices these are the permutations \[(1\;2)(3\;4),\qquad (1\;3)(2\;4),\qquad (1\;4)(2\;3).\] Each is even. Let \(u=-1+i+j+k\) and \(\omega =u/2\in 2T\). Direct multiplication yields \[\begin {array}{c|c} v&uv\overline u/4\\\hline v_1&v_1\\ v_2&v_4\\ v_3&v_2\\ v_4&v_3 \end {array}\] so \(\pi (\omega )\) fixes \(v_1\) and acts by the cycle \((2\;4\;3)\). This cycle is even. Conjugation by it permutes the three double transpositions: \[\begin {aligned} (2\;4\;3)(1\;2)(3\;4)(2\;4\;3)^{-1}&=(1\;4)(2\;3),\\ (2\;4\;3)(1\;4)(2\;3)(2\;4\;3)^{-1}&=(1\;3)(2\;4),\\ (2\;4\;3)(1\;3)(2\;4)(2\;4\;3)^{-1}&=(1\;2)(3\;4). \end {aligned}\] The cycle has order \(3\), while every non-identity element of that Klein subgroup has order \(2\), so the two subgroups meet only at the identity. These 12 permutations are the vertex permutations of products in \(R_T\). They are distinct, and \(|R_T|=12\), so they exhaust the vertex image. All 12 are even. Combined with injectivity, the vertex map is an isomorphism \(R_T\cong A_4\). The composite \(2T\to R_T\to A_4\) has kernel \(\{1,-1\}\).

For the \(d=3\) matrices of Definition [fdip-000R], \(\gamma _a^2=I\) and \(\gamma _a\gamma _b=-\gamma _b\gamma _a\) whenever \(a\ne b\). Set \[I\mapsto 1,\quad \gamma _1\gamma _2\mapsto i,\quad \gamma _2\gamma _3\mapsto j,\quad \gamma _1\gamma _3\mapsto k.\] Then \[(\gamma _1\gamma _2)^2=-\gamma _1^2\gamma _2^2=-I,\] and the same sign occurs for the other two products. Also \[(\gamma _1\gamma _2)(\gamma _2\gamma _3)=\gamma _1\gamma _3,\] and the two cyclic products match \(jk=i\) and \(ki=j\). These are the quaternion relations. They identify an action of the same abstract group on the even products. They do not state an intertwiner between left multiplication and the spatial operator \(\sum _j\gamma _j\nabla _j\).