§2 · Agent 2 · Integral models · 2026-10-18T03:20:57Z

Computation 2.6 · The \(45\times6\) pairs: 90 of type \(G\), 180 of type \(U\)

proved audit: correct

Second audit: correct. Every item reproduced by an enumeration of my own, written before reading the run's scripts (W(E6) as 6×6 integer matrices on root coordinates; Bruhat order by Deodhar's Z-property recursion instead of his parabolic criterion): |C|=45 with w₀∈C; dimensions 14,…,26 with multiplicities 1,2,3,4,5,5,5,5,5,4,3,2,1; θ_xΦ⁺=Φ⁺ only for x=w₀; 90 G, 90 U-open, 90 U-small, none with θ_xα=−α; w_L of type U at α₁, α₆ and G at α₂,…,α₅; item 5 for all 90 open type-U pairs; Σ_x independent of the choices at every step and equal to the Bruhat up-set in C, 821 pairs. The dimension distribution is also confirmed by the point count |E6(q)|/|F4(q)|=q¹²(q⁵−1)(q⁹−1), and the quoted word for w_L is reduced of length 12. The numbers A2 reports from its own scripts (root permutations, Deodhar's criterion) agree with mine.

Exact enumeration over \(\mathcal C\) (45 elements) and \(\Delta\): types, orbit dimensions, the unique closed orbit \(x=w_0\), and the closure order, which equals the Bruhat order on \(\mathcal C\) and does not depend on the field.

Computation 2.6.

  1. \(|\mathcal C|=45\), \(w_0\in\mathcal C\). Open orbit \(x=w_L\) (\(c=0\)): type U (\(Y\) open) for \(\alpha_1,\alpha_6\), type \(G\) for \(\alpha_2,\dots,\alpha_5\).
  2. \(\dim Y_x=26-c_x=32-\frac12\ell(x)\) takes the values \(14,\dots,26\) with multiplicities \(1,2,3,4,5,5,5,5,5,4,3,2,1\).
  3. \(\theta_x\Phi^+=\Phi^+\) only for \(x=w_0\) (dimension 14); every other \(x\) has a simple \(\alpha\notin\Phi_x\) with \(x\alpha>0\).
  4. Of \(270\) pairs: \(90\) with \(\alpha\in\Phi_x\) (G), \(90\) with \(x\alpha\in\Phi^+\smallsetminus\Phi_x\) (U, \(Y\) open), \(90\) with \(x\alpha\in\Phi^-\smallsetminus\Phi_x\) (U, \(Y\) small); none with \(\theta_x\alpha=-\alpha\).
  5. In the 90 open type-U pairs: \(\Phi(P_\alpha)\cap\theta_x\Phi(P_\alpha)\cap\{\pm\alpha\}=\{-\alpha\}\), \(-\alpha-\theta_x\alpha\notin\Phi\cup\{0\}\), \(c_{s_\alpha xs_\alpha}=c_x+1\), \(\dim(P_\alpha)_y=\dim B_y+1\).
  6. \(\Sigma_{w_0}=\{w_0\}\), \(\Sigma_x=\Sigma_{x'}\cup\{s_\alpha vs_\alpha:v\in\Sigma_{x'},\alpha\notin\Phi_v\}\) (\(x'=s_\alpha xs_\alpha\), \(\alpha\) as in 3) is independent of the choices and equals \(\{z\in\mathcal C:x\le z\}\) (Bruhat order; 821 pairs).

Computed. Roots of \(E_6\) from the Cartan matrix; \(W\) as permutations of the \(72\) roots; \(w_L=s_2s_3s_4s_2s_3s_4s_5s_4s_2s_3s_4s_5\), \(\theta=-w_L\) (\(\theta\alpha_1=-(\alpha_1+\alpha_2+2\alpha_3+2\alpha_4+\alpha_5)\)); \(\mathcal C\) by conjugation-closure with representatives \(w\). For each \(x\): \(-x=w\theta w^{-1}\), \(24\) imaginary roots, no real roots, no \(\gamma+\theta_x\gamma\in\Phi\), \(\ell(x)=12+2c_x\), the type of each \(\alpha\) (agreeing with \(w^{-1}\alpha\in\Phi_L\) for type \(G\)), and the partner \(s_\alpha xs_\alpha\) (\(=x\) iff type \(G\); otherwise open/small exchanged, \(c\) changes by \(\pm1\)). Item 5 from the fixed-vector count \(4+\#(\Psi\cap\Phi_x)+\frac12\#(\Psi\smallsetminus\Phi_x)\). Item 6 by recursion for every choice rule, and Deodhar's criterion for the Bruhat order.

Why it suffices. Items 1–5 are the finite inputs of Proposition 2.3, Proposition 2.5, Lemma 2.7; item 6 gives the closure sets of Lemma 2.7 (which fixes one choice). All are statements about \(W(E_6)\), valid over every field. Scripts e6_orbits.py, e6_checks.py (under one second).

Builds on

  • Lemma 2.2 · The involutions \(\theta_x\): compact imaginary roots, no real roots
  • Proposition 2.3 · Borel orbits of \(E_6/F_4\) over an algebraically closed field of characteristic \(\ne2\)

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