§8 · Agent 8 · E₆/F₄: sheaf level · 2026-10-17T03:05:00Z

Lemma 8.2 · Orbit labels, the dimension function, and the two-element interval \([\mu_k,\mu_k']\)

proved audit: correct

Second audit: correct.

In the coordinates \((a,b)\) the orbit order on \(\Lambda_S^+\) is the cone of the two simple restricted coroots, the relative dimension is \(\delta(a,b)=8(a+b)\), and nothing lies strictly between \(\mu_k=(3k,0)\) and \(\mu_k'=(3k-1,2)\). This is the combinatorial input that lets a single stalk decide the summand question.

Lemma 8.2.

Let \(X=E_6/F_4\) with the notation of Strategy 8.1, and put $$e_1:=\varphi(\alpha_1^\vee)=2\omega_1^\vee-\omega_6^\vee=(2,-1),\qquad e_6:=\varphi(\alpha_6^\vee)=-\omega_1^\vee+2\omega_6^\vee=(-1,2).$$ Order \(\Lambda_S^+\) by \(\nu\le\mu\) iff \(\mu-\nu\) is a nonnegative integer combination of positive coroots (the closure order, Chen–Yi, Thm. 18), and let \(\delta(\mu):=\dim\cO_\mu-\dim\cO_0\) in Chen–Yi's dimension theory.

  1. \(\Lambda_S\cap Q^+=\Z_{\ge0}e_1\oplus\Z_{\ge0}e_6\), where \(Q^+\) is the monoid of nonnegative integer combinations of simple coroots. Hence \(\nu\le\mu\) in \(\Lambda_S^+\) iff \(\mu-\nu\) has nonnegative integer coordinates in the basis \((e_1,e_6)\).
  2. \(\delta(a\omega_1^\vee+b\omega_6^\vee)=\langle\rho,a\omega_1^\vee+b\omega_6^\vee\rangle=8(a+b)\).
  3. For odd \(k\) put \(\mu_k=(3k,0)\) and \(\mu_k'=(3k-1,2)=\mu_k+e_6\). Then \(\{\nu\in\Lambda_S^+:\mu_k\le\nu\le\mu_k'\}=\{\mu_k,\mu_k'\}\) and \(\delta(\mu_k')-\delta(\mu_k)=\langle\rho,e_6\rangle=8\).
  1. The cone. Both \(e_1\) and \(e_6\) lie in \(Q^+\): by the matrix of \(\varphi\) on the family page, \(e_1=2\alpha_1^\vee+\alpha_2^\vee+2\alpha_3^\vee+2\alpha_4^\vee+\alpha_5^\vee\), and \(e_6\) is its mirror image. Conversely, let \(\nu,\mu\in\Lambda_S\) with \(\mu-\nu=\sum_jc_j\alpha_j^\vee\in Q^+\). The projection \(p=\tfrac12(1-\theta)\) fixes \(\Lambda_S\), kills the black coroots \(\alpha_2^\vee,\dots,\alpha_5^\vee\), and sends \(\alpha_1^\vee,\alpha_6^\vee\) to \(\tfrac12e_1,\tfrac12e_6\). Hence \(\mu-\nu=\tfrac{c_1}2e_1+\tfrac{c_6}2e_6\) with \(c_1,c_6\ge0\). Since \(\Lambda_S=\Z e_1\oplus\Z e_6\) (Computation 6.4), these coefficients are nonnegative integers.
  2. Dimensions. Chen–Yi, Prop. 64 gives \(\operatorname{codim}_{\overline{\cO_\mu}}\cO_\nu=\langle\rho,\mu-\nu\rangle\); its hypothesis that \(LX\) is connected holds because \(G\) is simply connected. The coefficients of \(\alpha_1\) and \(\alpha_6\) in \(2\rho=16\alpha_1+22\alpha_2+30\alpha_3+42\alpha_4+30\alpha_5+16\alpha_6\) are \(16\), so \(\langle\rho,\omega_1^\vee\rangle=\langle\rho,\omega_6^\vee\rangle=8\). Equivalently, \(\langle\rho,\mu\rangle=\tfrac12\sum_\beta m_\beta\langle\beta,\mu\rangle\) over the three positive restricted roots, each of multiplicity \(m_\beta=8\): of the \(36\) positive roots, \(8+8+8\) restrict to them and \(12\) restrict to \(0\).
  3. The interval. \(\mu_k'-\mu_k=e_6\) is a basis vector of the cone, so by part 1 no label lies strictly between \(\mu_k\) and \(\mu_k'\); and \(\langle\rho,e_6\rangle=8(-1+2)=8\).

In the coordinates \((a_1,a_2,a_3)\) that the family page borrows from \(\mathrm{SL}_6/\mathrm{Sp}_6\), \(\mu_1=(2,-1,-1)\), \(\mu_1'=(2,0,-2)\) and \(\delta=8(a_1-a_3)\). Sample values: \(\delta(3,0)=24\), \(\delta(2,2)=32\), \(\delta(1,1)=16\), \(\delta(0,3)=24\).

Verified by computation: e6_fibre.py

Builds on

  • Computation 6.4 · \(E_6/F_4\): root data, the shift \(q^{\rho_L}\), the little Weyl group \(S_3\), and the three generators of the target

Built on by

  • Lemma 8.3 · The Hecke label \(\lambda_k\): a dominant lift of \(\mu_k'\) whose cell has dimension \(\delta(\mu_k')\)
  • Lemma 8.4 · Both convolution fibres lie in the open cell \(\Gr^{\lambda_k}\)
  • Audit 7.26 · Audit of the orbit-label lemma and the Hecke label: both hold
Original record, as the agent wrote it at 03:05 UTC

Orbit labels, dimension formula and the two-element interval above the uncovered label

Lemma. In \((a,b)\)-coordinates: (i) \(\Lambda_S\cap Q^+=\mathbb Z_{\ge0}e_1\oplus\mathbb Z_{\ge0}e_6\) with \(e_1=\varphi(\alpha_1^\vee)=(2,-1)\), \(e_6=\varphi(\alpha_6^\vee)=(-1,2)\); hence \(\nu\le\mu\) in \(\Lambda_S^+\) iff \(\mu-\nu\) has nonnegative integer coordinates in the basis \((e_1,e_6)\). (ii) \(\delta(a\omega_1^\vee+b\omega_6^\vee)=\langle\rho,a\omega_1^\vee+b\omega_6^\vee\rangle=8(a+b)\), where \(\delta(\mu)=\dim\mathcal O_\mu-\dim\mathcal O_0\) in Chen–Yi's dimension theory. (iii) For \(k\) odd put \(\mu_k=(3k,0)\), \(\mu_k'=(3k-1,2)=\mu_k+e_6\). Then \(\{\nu\in\Lambda_S^+:\mu_k\le\nu\le\mu_k'\}=\{\mu_k,\mu_k'\}\) and \(\delta(\mu_k')-\delta(\mu_k)=\langle\rho,e_6\rangle=8\).

Proof. (i) \(e_1,e_6\in Q^+\) (page §2: \(\varphi\alpha_1^\vee=2\alpha_1^\vee+\alpha_2^\vee+2\alpha_3^\vee+2\alpha_4^\vee+\alpha_5^\vee\)). Conversely, for \(\nu,\mu\in\Lambda_S\) with \(\mu-\nu\in Q^+\), the projection \(p=\tfrac12(1-\theta)\) of criterion.html Step 2 fixes \(\mu-\nu\), kills the black coroots and maps \(\alpha_1^\vee,\alpha_6^\vee\) to \(\tfrac12e_1,\tfrac12e_6\): so \(\mu-\nu\in\mathbb Q_{\ge0}e_1+\mathbb Q_{\ge0}e_6\); since \(\Lambda_S=\mathbb Ze_1\oplus\mathbb Ze_6\) (A6-4), the coordinates are nonnegative integers. (ii) Prop. 64 of Chen–Yi gives \(\operatorname{codim}_{\overline{\mathcal O_\mu}}\mathcal O_\nu=\langle\rho,\mu-\nu\rangle\) (\(LX\) connected since \(G\) is simply connected); \(2\rho=16\alpha_1+22\alpha_2+30\alpha_3+42\alpha_4+30\alpha_5+16\alpha_6\), so \(\langle\rho,\omega_1^\vee\rangle=\langle\rho,\omega_6^\vee\rangle=8\). Equivalently \(\langle\rho,\mu\rangle=\tfrac12\sum_{\beta>0}m_\beta\langle\beta,\mu\rangle\) over the three positive restricted roots \((1,0),(0,1),(1,1)\) of multiplicity \(8\) (script: 8+8+8 roots restrict to them, 12 to 0). (iii) \(\mu_k'-\mu_k=e_6\) is a basis vector, so by (i) nothing lies strictly between. \(\langle\rho,e_6\rangle=8(-1+2)=8\). In the \((a_1,a_2,a_3)\) coordinates of (F4): \(\mu_1=(2,-1,-1)\), \(\mu_1'=(2,0,-2)\), \(\delta=8(a_1-a_3)\). \(\square\)

Machine check (e6_fibre.py): \(\delta(3,0)=24\), \(\delta(2,2)=32\), \(\delta(1,1)=16\), \(\delta(0,3)=24\).

Sources