§6 · Agent 6 · Audit: integral models · 2026-10-18T03:58:08Z

Audit 6.2 · Audit of the sign data, the torus \(T^\theta\) and \(C_W(w_L)\) (claim A2-2)

proved audit: correct

Second audit: correct. Re-derived item 1 directly on the 78-dimensional E6 Lie algebra (Chevalley basis via Kac's bimultiplicative cocycle, Tits lifts as products of exp(ad)), not in the so8 model: with S4-11's sign theta0^2 = 1, theta0 fixes E_beta for all 24 beta in Phi_L, dim g^theta = 52 and dim g^-theta = 26, while the opposite sign gives 36 (type E I), so the recorded convention dependence is real; theta0'^2 = Int(w_L(t0)t0^-1) with w_L(t0)t0^-1 central was checked by hand. The Smith form (1,1) of theta*-1 on the weight lattice, the 2-torsion argument, primitivity of every root restriction to T^{theta_x} (integral basis of X_*(T)^{theta_x}, all 45 x), |C_W(w_L)| = 1152 by brute force over W together with its generation by the listed elements, and alpha_i +- beta_i not in Phi were all recomputed. The lifts n_i = sigma_{alpha_i} theta(sigma_{alpha_i}) in H and the action w -> w_L w w_L of theta on W were checked by hand; nothing material missed in Lemma 2.2.

Exact computations confirm that \(\theta_0\) is trivial on \(\mathrm{Spin}_8\) for the sign convention of S4-11, that \(T^\theta\simeq\mathbb G_m^4\) over \(\mathbb Z\), and that the listed elements generate \(C_W(w_L)\) of order 1152; moreover every root restricts to a primitive character of \(T^{\theta_x}\). Outcome: holds (the triviality on \(\mathrm{Spin}_8\) depends on the sign of \(C\), recorded).

Audit 6.2.

  1. With \(C(x_{\pm\alpha_i}(c))=x_{\mp\alpha_i}(-c)\) (S4-11), \(\operatorname{Int}(\sigma_{w_0})\circ C\) is the identity of the pinned split \(\mathrm{Spin}_8\); hence \(\theta_0\) fixes \(U_\beta\) pointwise for every \(\beta\in\Phi_L\), over \(\mathbb Z\). With the opposite sign \(C'(x_{\pm\alpha_i}(c))=x_{\mp\alpha_i}(c)\), \(\operatorname{Int}(\sigma_{w_0})\circ C'\) acts by \(c\mapsto -c\) on every simple root group, and \(\operatorname{Int}(\sigma_L)\circ C'\) is an involution acting by \(-w_L\) on \(T\) with \(\dim\mathfrak g^{\theta}=36\) (type E I).
  2. \(\theta^*-1\) on \(X^*(T)\) has elementary divisors \((1,1)\); so \(X^*(T^\theta)\simeq\mathbb Z^4\) and \(T^\theta\simeq\mathbb G_m^4\) over \(\mathbb Z\).
  3. Every root \(\alpha\) restricts to a primitive character of \(T^{\theta_x}\) for every \(x\in\mathcal C\); hence \(\alpha|_{T^{\theta_x}}:T^{\theta_x}\to\mathbb G_m\) is a split surjection of tori over \(\mathbb Z\).
  4. \(\langle s_{\alpha_2},\dots,s_{\alpha_5},s_{\alpha_1}s_{\beta_1},s_{\alpha_6}s_{\beta_6}\rangle\) has order \(1152=|C_W(w_L)|\), with \(\beta_1=\alpha_1+\alpha_2+2\alpha_3+2\alpha_4+\alpha_5\), \(\beta_6=\alpha_2+\alpha_3+2\alpha_4+2\alpha_5+\alpha_6\), \((\alpha_i,\beta_i)=0\), \(\alpha_i\pm\beta_i\notin\Phi\); each element lifts to \(N_H(T)(\mathbb Z[1/2])\) and \(N_H(T)/T^\theta\to C_W(w_L)\) is bijective.
  5. Items 1, 3, 4, 7 of Lemma 2.2 follow from 1 and Audit 6.1.
  1. Signs on \(\mathrm{Spin}_8\). Realize split \(\mathfrak{so}_8\) as the matrices preserving the antidiagonal form \(J\) (\(J^2=1\)), with \(e_1=E_{12}-E_{78}\), \(e_2=E_{23}-E_{67}\) (centre), \(e_3=E_{34}-E_{56}\), \(e_4=E_{35}-E_{46}\), \(f_i=e_i^{\mathsf T}\); the brackets \([h_i,e_j]\) give the Cartan matrix of \(D_4\). \(C(X)=-X^{\mathsf T}\) preserves \(\mathfrak{so}(J)\) and sends \(e_i\mapsto-f_i\), which is the convention of S4-11. Tits lifts \(\sigma_i=(1+e_i)(1-f_i)(1+e_i)\) (\(e_i^2=0\)). For three different reduced words of \(w_0\) the products \(\sigma_{w_0}\) coincide, \(\sigma_{w_0}^2=1\) in \(\mathrm{SO}_8\), and exact integer arithmetic gives \(\operatorname{Ad}(\sigma_{w_0})C(e_i)=e_i\), \(\operatorname{Ad}(\sigma_{w_0})C(f_i)=f_i\) for \(i=1,\dots,4\). The composite is trivial on \(T\) (\(w_0=-1\)), multiplies each root group by a sign read off from \(\operatorname{Ad}\), and fixes the generators \(e_i,f_i\); so it is trivial on all root groups and on \(\mathrm{Spin}_8\). The \(D_4\)-Levi of \(E_6\) with the induced pinning is isomorphic to this pinned group, and \(C\), \(\sigma_L\) restrict to the corresponding objects, so \(\theta_0|_{\mathrm{Spin}_8}=\mathrm{id}\).
  2. Control (convention). With \(C'(X)=X^{\mathsf T}\) the same computation gives \(\operatorname{Ad}(\sigma_{w_0})C'(e_i)=-e_i\) for all \(i\). Writing \(C'=\operatorname{Int}(t_0)\circ C\) with \(\alpha_i(t_0)=-1\), \(\theta_0'=\operatorname{Int}(w_L(t_0))\circ\theta_0\) acts on \(\mathfrak g_\beta\), \(\beta\in\Phi_L\), by \((-1)^{\operatorname{ht}\beta}\); \(D_4\) has \(4,3,3,1,1\) positive roots of heights \(1,\dots,5\), so 8 imaginary roots stay compact and \(\dim\mathfrak g^{\theta_0'}=4+8+24=36\). One checks \(w_L(t_0)t_0^{-1}\) is killed by every simple root, so \(\theta_0'\) is an involution. Thus item 2 of Lemma 2.2 is a statement about S4-11's \(C\); Strategy 2.1 imports exactly that \(\theta_0\), so there is no error, but "the Chevalley involution" with the other sign would give \(E_6/\mathrm{Sp}_8\).
  3. Roots. By Audit 6.1: the \(\theta\)-fixed roots are \(\Phi_L\), there are no real roots, and \(\gamma+\theta_x\gamma\notin\Phi\cup\{0\}\). The bracket argument of Lemma 2.2 step 4 is also correct in characteristic \(\ne2\) (structure constants \(\pm1\)).
  4. \(T^\theta\). The matrix of \(\theta^*=-w_L\) on the weight lattice (basis \(\omega_i\), \(s_i(\omega_j)=\omega_j-\delta_{ij}\alpha_i\)) was computed; \(\theta^*-1\) has rank 2, the gcd of its entries is 1 and the gcd of its \(2\times2\) minors is 1. So \(\operatorname{coker}(\theta^*-1)\simeq\mathbb Z^4\) and \(T^\theta=\ker(\theta-1)\simeq\mathbb G_m^4\). (In general the torsion of \(X/(\theta-1)X\) is killed by 2: if \(km=(\theta-1)y\) then \(k(\theta+1)m=0\), so \((\theta+1)m=0\) and \(2m=-(\theta-1)m\). So \(p=3\) could never enter here.)
  5. Primitivity (new, strengthens Audit 2.9 item 11). \(X_*(T^\theta)=X_*(T)^\theta\) contains \(\beta^\vee\) for every \(\beta\in\Phi_L\). For a root \(\alpha\in\Phi_L\) pick \(\beta\in\Phi_L\) at angle \(60^\circ\); for \(\alpha\notin\Phi_L\) its projection \(\alpha_\parallel\) to \(\operatorname{span}\Phi_L\) is non-zero (no real roots), so some \(\beta\in\Phi_L\) has \((\alpha,\beta)=(\alpha_\parallel,\beta)\ne0\), hence \(\langle\alpha,\beta^\vee\rangle=\pm1\). So \(\alpha|_{T^\theta}\) pairs to \(\pm1\) with a cocharacter of \(T^\theta\) and is primitive; conjugating by \(\dot w\) gives the same for \(T^{\theta_x}=\dot wT^\theta\dot w^{-1}\). (Computation agrees: no root restriction is divisible by any prime \(\le97\).)
  6. \(C_W(w_L)\). The group generated by the listed elements was enumerated: 1152 elements, all commuting with \(w_L\); \(1152=51840/45\) is \(|C_W(w_L)|\) by orbit–stabilizer. For \(i\in\{1,6\}\), \((\alpha_i,\beta_i)=0\) and \(\alpha_i\pm\beta_i\notin\Phi\) (computed), so the root groups of \(\pm\alpha_i\) and \(\pm\beta_i\) commute (Conrad, Prop. 5.1.14) and \(n_i=\sigma_{\alpha_i}\theta(\sigma_{\alpha_i})\) is \(\theta\)-fixed, lies in \(H\cap N_G(T)\) and lifts \(s_{\alpha_i}s_{\beta_i}\); Tits lifts of \(s_2,\dots,s_5\) lie in \(\mathrm{Spin}_8\subseteq H\) by step 1. Conversely \(\theta\) acts on \(W\) by \(w\mapsto w_Lww_L\), so the image of \(N_H(T)\) lies in \(C_W(w_L)\); the kernel is \(T\cap H=T^\theta\).
  7. Item 7. \(\mathfrak g^\theta=\mathfrak t^\theta\oplus\bigoplus_{\Phi_L}\mathfrak g_\beta\oplus(\text{one line per pair})\): \(4+24+24=52\); \(\mathfrak g^{-\theta}\): \(2+24=26\); the splitting of \(\mathfrak t\) uses \(1/2\).

Outcome: holds (given the sign convention of S4-11 for \(C\), which Strategy 2.1 uses).

Builds on

  • Lemma 2.2 · The involutions \(\theta_x\): compact imaginary roots, no real roots

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