TFPT scientific source excerpt
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000072 | \begin{theorem}[Internal carrier stabilizer]
000073 | \label{thm:global-sm-theorem}
000074 | The determinant-preserving connected stabilizer of the minimal carrier split is
000075 | \[
000076 | G_{\mathrm{car}}
000077 | =
000078 | S(U(3)\times U(2))
000079 | \cong
000080 | \frac{SU(3)\times SU(2)\times U(1)_Y}{\mathbb Z_6}.
000081 | \]
000082 | The corresponding Lie algebra factorization
000083 | \[
000084 | \mathfrak g_{\SM}=\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak u(1)
000085 | \]
000086 | is only the differential, local form of the same statement; throughout the manuscript this quotient
000087 | is intended as the internal carrier-stabilizer form whenever a group symbol is written, and the
000088 | local algebra notation is used only when explicitly stated.
000089 | \end{theorem}
000090 | 
000091 | \begin{proof}
000092 | An automorphism preserving the split $E_3\oplus E_2$ acts by a pair
000093 | $(u_3,u_2)\in U(3)\times U(2)$. Requiring determinant preservation on the full
000094 | carrier gives
000095 | \[
000096 | \det(u_3)\det(u_2)=1,
000097 | \]
000098 | so the connected stabilizer is exactly $S(U(3)\times U(2))$.