Differential Equations attached to

Generalized Flag Manifolds

一般旗多様体に付随した微分方程式

Toshio OSHIMA （大島利雄） Graduate School of Mathematical Sciences

University of Tokyo

Conference on Nilpotent Orbits and Representation Theory 2004

September 6, 2004 FujiSakura-So, Kawaguchi-Ko § Introduction Radon transform J. Radon (1917, Ber. Berh. Sächs. Akad. Wiss. Leipzig. Math.-Nat. kl.)

ℓ (r cos θ, r sin θ)  C_{0}(ℝ^{2}) ∋ f ↦ (ℓ ↦ R_{f}(ℓ) = ∫_{ℓ} f(ℓ(t))dt) ℓ(t) = (r cos θ − t sin θ, r sin θ + t cos θ)

ℝ^{2} ⟶ ℝ^{n}, {ℓ} ⟶ {k-dimensional affine subspaces} k = n − 1 : the dimension of the manifold parameterizing the subspaces = n. k = 1 : X-ray transformation, dim (the parameter space) = 2n − 2.

P. Funk (1916, Math. Ann.)

ℓ  C(S^{2}) ∋ f ↦ (ℓ ↦ R_{f}(ℓ) = ∫_{ℓ} f(ℓ(θ))dθ) e_{ℓ}  ℓ(θ) = e′_{ℓ} cos θ + e′′_{ℓ} sin θ  ({e_{ℓ}, e′_{ℓ}, e′′_{ℓ}}: orthonormal)

⟹ C^{∞}(S^{2}/ℤ_{2}) →^{∼} C^{∞}(S^{2}/ℤ_{2})

ℙ^{2}(ℝ) = S^{2}/ℤ_{2}: 2-dimensional real projective space

Grassmann manifolds

Gr_{k}(ℝ^{n}) := {V_{k} : k-dimensional subspaces  ⊂ ℝ^{n}}  (ℙ^{n−1}(ℝ) = Gr_{1}(ℝ^{n}))

M^{o}(n, k; ℝ) := \{X = (\begin{matrix}x_{11} & ⋯ & x_{1k} \\ ⋮ & ⋯ & ⋮ \\ x_{n1} & ⋯ & x_{nk} \end{matrix}) ∈ M(n, k; ℝ); rank X = k\}

= M^{o}(n, k; ℝ)/GL(k, ℝ) ∋ \binom{I_{k}}{x} (x ∈ M(n − k, k; ℝ))

dim Gr_{k}(ℝ^{n}) = nk − k^{2} = (n − k)k

R_{ℓ}^{k} : C(Gr_{k}(ℝ^{n})) ∋ f ↦ (R_{ℓ}^{k}f)(x) ∈ C(Gr_{ℓ}(ℝ^{n})) (0 < k < ℓ < n)

(R_{ℓ}^{k}f)(x) = ∫_{x∋t:k-dimensional subspaces} f(t)dt

G = GL(n, ℝ) transitively acts on Gr_{k}(ℝ^{n}) from the left.

For a point Gr_{k}(ℝ^{n}) ∋ \binom{I_{k}}{0}GL(k, ℝ) = \binom{GL(k,ℝ)}{0}, put P_{k,n} := \{g ∈ G; ^{t}g^{−1} \binom{GL(k,ℝ)}{0} = \binom{GL(k,ℝ)}{0}\}. Then

Gr_{k}(ℝ^{n}) = GL(n, ℝ)/P_{k,n}  (= O(n)/O(k) × O(n − k))
P_{k,n} = \{p = (\begin{matrix}g_{1} & 0 \\ y & g_{2} \end{matrix}); g_{1} ∈ GL(k, ℝ), g_{2} ∈ GL(n − k, ℝ), y ∈ M(n − k, k; ℝ)\}

⊃ P := \{(\begin{matrix}∗ \\ ∗ & ∗ \\ ⋮ & ⋮ & ... \\ ∗ & ∗ & ⋯ & ∗ \end{matrix}) ∈ GL(n, ℝ)\}

B(G/P_{k,n};L_{λ}) := {f ∈ B(G); f(xp) = f(x)| det g_{1}|^{λ_{1}}| det g_{2}|^{λ_{2}}, ∀p ∈ P_{k,n}}

(= B(O(n)/O(k) × O(n − k)))

= {f ∈ B(M^{o}(n, k; ℝ)); f(Xg_{1}) = f(X)| det g_{1}|^{−λ_{1}}, ∀g_{1} ∈ GL(k, ℝ)}

(x = (x_{ij})\begin{matrix}1≤i≤n \\ 1≤j≤n \end{matrix} ∈ G ⊂ ℝ^{n^{2}}, x ↦ ^{t}x^{−1} ↦ X)

⟹ (R_{ℓ}^{k}f)(x) = ∫_{O(ℓ)/O(k)×O(ℓ−k)} f(xk)dk

Studied by Gelfand, Helgason, Rubin, Grinberg, Gonzalez, Kakehi etc. Problem. Characterize the image of R_{ℓ}^{k} ! (ℓ + k < n)

Poisson transform P_{k,λ}^{n} : B(G/P_{k,n}; L_{λ}) ∋ f ↦ (P_{k,λ}^{n}f)(x) = ∫_{O(n)} f(xk)dk ∈ B(G/K) (K = O(n))

G/K: a Riemannian symmetric space

P_{λ}^{n} : B(G/P; L_{λ}) := \{f ∈ B(G); f(g (\begin{matrix}a_{11} \\ a_{21} & a_{22} \\ ⋯ & ⋯ & ... \\ a_{n1} & a_{n2} & ⋯ & a_{nn} \end{matrix}))
= f(g)|a_{11} |^{λ_{1}}|a_{22} |^{λ_{2}} ⋯ |a_{nn}|^{λ_{n}}}
∋ f ↦ (P_{λ}^{n}f)(x) = ∫_{O(n)} f(xk)dk ∈ B(G/K)

⟹ P_{k,λ}^{n} : B(G/P_{k,n}; L_{λ}) ↪ B(G/P; L_{λ}) →^{P_{λ}^{n}} B(G/K)

The inverse map of P_{λ}^{n}: the boundary value map. Problem. Characterize the image of P_{k,λ}^{n} ! (In some cases, the boundary value problem of a Hermitian symmetric space and its Shilov boundary)

Remark. (P_{λ}^{n}f)(cx) = |c|^{λ_{1}+···+λ_{n}} (P_{λ}^{n}f)(x) ⇒ We may restrict on SL(n, ℝ).

Example 1 (n = 2, k = 1) SL(2, ℝ)/SO(2) ≃ D := {z ∈ ℂ; |z| < 1}, SL(2, ℝ)/(P ∩ SL(2, ℝ)) ≃ ∂D and

P_{1,λ}^{2} = P_{λ}^{2} : B(∂D) ∋ f ↦ ∫_{0}^{2π} f(e^{iθ}) (\frac{1 − |z|^{2}}{|e^{iθ} − z|^{2}} )^{1+λ} \frac{dθ}{2π} ∈ A(D; M_{λ}),
M_{λ} : \frac{(1 − |z|^{2})^{2}}{4} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) u = λ(λ + 1)u  (z = x + iy)

λ ∉ {−1, −2, −3, …} ⇒ P_{λ}^{2} : B(∂D) →^{∼} A(D; M_{λ}) a topological G-isomorphism (Helgason, 1970).

§ Differential Equations

For X ∈ M(n, ℝ), E_{ij} := (δ_{pi}δ_{qj})\begin{matrix}1≤p≤n \\ 1≤q≤n \end{matrix}, x = (x_{ij})\begin{matrix}1≤i≤n \\ 1≤j≤n \end{matrix} ∈ G and ϕ ∈ C^{∞}(G), put

(Xϕ)(x) := \frac{d}{dt}ϕ(xe^{tX})|_{t=0} ,  E_{ij} = n ; ∑ ; ν=1 x_{νi}\frac{∂}{∂x_{νj}} ,
(π(X)ϕ)(x) := \frac{d}{dt}ϕ(e^{−tX}x)|_{t=0} = \frac{d}{dt}ϕ(xe^{−tx^{−1}Xx})|_{t=0} ,  π(E_{ij}) = − n ; ∑ ; ν=1 x_{jν} \frac{∂}{∂x_{iν}} ,
(π(X)ϕ)(x) = (−(Ad(x^{−1})X)ϕ)(x),  Ad(x)X = xXx^{−1} ∈ M(n, ℂ),
𝔤 = M(n, ℝ) is a Lie algebra,  [E_{ij}, E_{kℓ}] = δ_{jk}E_{iℓ} − δ_{ℓi}E_{kj}.

U(𝔤): the universal enveloping algebra (⊃ 𝔤) ≃ a subalgebra of D(G) (the ring of differential operators on G) generated by X (X ∈ 𝔤).

Definition 2. E_{n,k,λ} := {P ∈ U(𝔤); π(P)B(G/P_{k,n}; L_{λ}) = 0}.

Fact. i) The image of the Radon transform R_{k}^{n}, Poisson transform P_{k,λ}^{n} and a Penrose transform etc. (any G-homomorphisms of B(G/P_{k,n}; L_{λ}) or a certain cohomology space of sections of a line bundle over G/P_{k,n}) satisfy the differential equations given by E_{n,k,λ}.

ii) E_{n,k,λ} ≃ a(_{g∈G}^{⋂} Ad(g)J_{k,λ})

J_{k,λ} = ∑ ; \begin{matrix}i≠j \\ i>k or j≤k \end{matrix} U(𝔤)E_{ij} + k ; ∑ ; i=1 U(𝔤)(E_{ii} − λ_{1}) + n ; ∑ ; j=k+1 U(𝔤)(E_{jj} − λ_{2})

by a : U(𝔤) ∋ XY ↦ (−Y)(−X) ∋ U(𝔤) (anti-automorphism, X, Y ∈ 𝔤 )

Problem. Find a good generator system of E_{n,k,λ}.

Fact. E_{n,k,λ} is a quantization of the defining ideal of the conjugacy class of the

matrices A_{k}(λ_{1}, λ_{2}) := (\begin{matrix}λ_{1}I_{k} & 0 \\ ∗ & λ_{2}I_{n−k} \end{matrix}).

We can define quantizations of characteristic polynomials , minimal polynomials , elementary divisors in the linear algebra.

Theorem 3 (Minimal Polynomial). The quantization of the minimal polynomial (x−λ_{1})(x−λ_{2}) of A_{k}(λ_{1}, λ_{2}) equals (x−λ_{1})(x−k −λ_{2}) and a(E_{n,k,λ}) is generated by

⟨((𝔼 − λ_{1})(𝔼 − k − λ_{2}))_{ij}, n ; ∑ ; i=1 E_{ii} − kλ_{1} − (n − k)λ_{2}⟩

for a generic λ. Here 𝔼 = (E_{ij})\begin{matrix}1≤i≤n \\ 1≤j≤n \end{matrix} = X∂ ∈ M(n, U(𝔤)) with X = (x_{ij})\begin{matrix}1≤i≤n \\ 1≤j≤n \end{matrix}, ∂ =

(\frac{∂}{∂x_{ij}} )\begin{matrix}1≤i≤n \\ 1≤j≤n \end{matrix}.

λ_{1} − λ_{2} ∉ {1, 2, 3, …} ⇒ the image of P_{k,λ}^{n} is characterized by this system (a generalization of Hua’s equations. the zonal spherical function P_{1,λ}^{n}(1) is reduced to Lauricella’s F_{D}).

Remark. rank (A_{k}(λ_{1}, λ_{2}) − λ_{1}) ≤ n − k ⇒ (n − k + 1)-minors of A_{k}(λ_{1}, λ_{2}) − λ_{1} vanish.

Theorem 4 (minors, elementary divisors ). Assume 2k ≤ n (for simplicity). λ_{1} − λ_{2} ∉ {ϵ, ⋯, (n − k)ϵ} ⇒

a(E_{n,k,λ}) = ⟨ det (E_{i_{μ}j_{ν}} − (λ_{1} + (ν − n + k − 1)ϵ)δ_{i_{μ}j_{ν}})\begin{matrix}1≤μ≤n−k+1 \\ 1≤ν≤n−k+1 \end{matrix} ,

det (E_{i′_{μ}j′_{ν}} − (λ_{2} + (ν − 1)ϵ)δ_{i′_{μ}j′_{ν}})\begin{matrix}1≤μ≤k+1 \\ 1≤ν≤k+1 \end{matrix} ⟩.

λ_{1} − λ_{2} ∈ {kϵ, ⋯, (n − k)ϵ} ⇒

a(E_{n,k,λ}) = ⟨\frac{d}{dt} det (E_{i_{μ}j_{ν}} − (t + λ_{1} + (ν − n + k − 1)ϵ)δ_{i_{μ}j_{ν}})\begin{matrix}1≤μ≤n−k+1 \\ 1≤ν≤n−k+1 \end{matrix} |_{t=0} ,

det (E_{i′_{μ}j′_{ν}} − (λ_{2} + (ν − 1)ϵ)δ_{i′_{μ}j′_{ν}})\begin{matrix}1≤μ≤k+1 \\ 1≤ν≤k+1 \end{matrix} ⟩,

Here

ϵ = \begin{cases}0 & (classical ) \\ 1 & (quantum ) \end{cases}
det (A_{ij}) = _{σ}^{∑} sign (σ)A_{σ(1)1} A_{σ(2)2} ⋯, I = {i_{1}, …, i_{n−k−1}} etc.

This also characterizes the image of P_{k,λ}^{n} under the same condition.

Fact. R_{ℓ}^{k} is lifted to the G-map

R_{ℓ}^{k} : B(G/P_{k,n}; L_{ℓ,0}) → B(G/P_{ℓ,n}; L_{k,0}).

Theorem 5 ([O, 1996]). 0 < k < k +ℓ < n ⇒ R_{ℓ}^{k} is a topological G-isomorphism onto

\{Φ((x_{ij})\begin{matrix}1≤i≤ℓ \\ 1≤j≤n \end{matrix}) ∈ B(M^{0}(n, ℓ; ℝ));
Φ(xg ) = | det g|^{−k}Φ(x) for g ∈ GL(ℓ, ℝ),
det (\frac{∂}{∂x_{i_{μ}j_{ν}}} )\begin{matrix}1≤μ≤k+1 \\ 1≤ν≤k+1 \end{matrix} Φ(x) = 0 (Capell type )
for 1 ≤ i_{1} < ⋯ < i_{k+1} ≤ n, 1 ≤ j_{1} < ⋯ < j_{k+1} ≤ ℓ}.

(1)

Theorem 6 ([O 1996, Generalized Capelli identity ])

det ( n ; ∑ ; ν=1 x_{νi_{k}} \frac{∂}{∂x_{νj_{ℓ}}} + (m − ℓ)δ_{i_{k}j_{ℓ}})\begin{matrix}1≤k≤m \\ 1≤ℓ≤m \end{matrix}

= _{1≤ν_{1}<···<ν_{m}≤n}^{∑} det (x_{ν_{p}i_{q}})\begin{matrix}1≤p≤m \\ 1≤q≤m \end{matrix} · det (\frac{∂}{∂x_{ν_{p}i_{q}}} )\begin{matrix}1≤p≤m \\ 1≤q≤m \end{matrix}

for I = {i_{1}, …, i_{m}} and J = {j_{1}, …, j_{m}}. Remark. m = n ⇒ the usual Capelli identity (1887).

Theorem 7. H ⊂ GL(n, ℝ): (H_{ℂ}×GL(k, ℂ), ℂ^{n}×ℂ^{k}) is a prehomogeneous vector space (⇔ ∃ an open orbit). ⟹ R_{ℓ}^{k}(the relative invariants ) , which we call hypergeometric functions , coincides with the solution space of the system of hypergeometric equations defined by the action of H together with the system (1) .

B(G/P_{k,n}; L_{ℓ,0})  →^{∼^{Rk ℓ}} The solution space of the system (1)
∪  ∪
H-action ↷ {relative invariants } →^{∼}  {Hypergeometric functions }

Example 8. (Hypergeometric functions by Gelfand-Aomoto) k = 1. H = GL(1, ℝ)_{+} × ⋯ × GL(1, ℝ)_{+} ， μ_{1} = (α_{1}, ⋯, α_{n}) with ∑_{j=1}^{n} α_{j} = −ℓ.

Φ(α, x) = ∫_{t_{1}^{2}+···+t_{ℓ}^{2} =1} n ; ∏ ; j=1 | ℓ ; ∑ ; ν=1 t_{ν}x_{jν}|_{±}^{α_{j}} ω (Hypergeometric function ).

System of Hypergeometric equations:

ℓ ; ∑ ; j=1 x_{ij} \frac{∂Φ}{∂x_{ij}} = α_{j}Φ for 1 ≤ i ≤ n (the left action of H)
n ; ∑ ; ν=1 x_{νi}\frac{∂Φ}{∂x_{νj}} = −ℓδ_{ij}Φ for 1 ≤ i, j ≤ k (the right action of GL(k, ℝ))
\frac{∂^{2}Φ}{∂x_{i_{1}j_{1}}x_{i_{2}j_{2}}} = \frac{∂^{2}Φ}{∂x_{i_{2}j_{1}}x_{i_{1}j_{2}}} for 1 ≤ i_{1} < i_{2} ≤ n, 1 ≤ j_{1} < j_{2} ≤ ℓ (Capelli type ).

§ Remarks

i) k = 1, ℓ = 2, n = 4 in the above Example 8 ⇒ Gauss’ hypergeometric functions and its differential equation.

ii) # orbits of the prehomogeneous vector space in M^{o}(n, ℓ; ℂ) < ∞ ⇒ The system of differential equations is holonomic (Tanisaki)

iii) A quantization of elementary divisors for every element A of M(n, ℂ) is defined and using it we can construct a generator system of the annihilator of any generalized Verma module of the scalar type for 𝔤𝔩(n) together with its classical limit ([O, to appear in Adv. in Math.]).

In the classical case for the nilpotent conjugacy class, a generator system of its defining ideal, is obtained by Kostant (regular nilpotent) and Weyman (any nilpotent, 1989; conjectured by Tanisaki).

A ∼ A_{Θ,λ} := \{(\begin{matrix}λ_{1}I_{n′_{1}} & & & & 0 \\ A_{21} & λ_{2}I_{n′_{2}} \\ A_{31} & A_{32} & λ_{3}I_{n′_{3}} \\ ⋮ & ⋮ & ⋮ & ... \\ A_{L1} & A_{L2} & A_{L3} & ⋯ & λ_{L}I_{n′_{L}} \end{matrix}) ∈ M(n, ℂ) ; A_{ij} ∈ M(n′_{i}, n′_{j}; ℂ)\} ,

J_{Θ} (λ) = L ; ∑ ; ν=1 ∑ ; \begin{matrix}n_{ν−1}<i≤n_{L} \\ n_{ν−1}<j≤n_{ν} \end{matrix} U(𝔤)(E_{ij} − λ_{ν}δ_{ij})  (n_{0} = 0, n_{ν} := n′_{1} + ⋯ + n′_{ν}),

E_{Θ} (λ) : = _{g∈GL(n,ℂ)}^{⋂} Ad(g)J_{Θ} (λ)  (← Construct its generator system!  ),

z^{(ℓ)} : = \begin{cases}z(z − ϵ) ⋯ (z − (ℓ − 1)ϵ) & if ℓ > 0, \\ 1 & if ℓ ≤ 0, \end{cases}
d_{m}^{ϵ}(x) = L ; ∏ ; j=1 (x − λ_{j} − n_{j−1}ϵ)^{(n′_{j}+m−n)} (d_{n}^{ϵ}(x): characteristic polynomial)  ,
e_{m}^{ϵ}(x) = d_{m}^{ϵ}(x)/d_{m−1}^{ϵ}(x) (elementary divisors) ,
q^{ϵ}(x) = L ; ∏ ; j=1 (x − λ_{j} − n_{j−1}ϵ) (minimal polynomial)

iv) The quantization of the minimal polynomial q_{Θ} (x) is defined and calculated for any simple Lie algebra 𝔤 and its non-trivial finite-dimensional representation (π, ℂ^{N}) and for the generalized Verma module M_{Θ} (λ) of the scalar type ([O, O-Oda]).

ϖ: the projection of M(N; ℂ) to 𝔤 with respect to ⟨X, Y ⟩ = Trace XY (X, Y ∈ M(N, ℂ))

𝔼_{π}; = (ϖ(E_{ij}))\begin{matrix}1≤i≤N, \\ 1≤j≤N \end{matrix} ∈ M(N; U(𝔤)),

q_{π,Θ} (𝔼_{π})M_{Θ} (λ) = 0.

Theorem 9 (Oda-O). π̄: the lowest weight of π. 𝔭_{Θ} = 𝔩_{Θ} + 𝔫_{Θ} W(π): the set of the weights of (π, ℂ^{N}). \overline{W}_{𝔭}(π): the set of the lowest weights of (π|_{𝔩_{Θ}} , ℂ^{N}).

q′_{π,Θ} (x) := ′ ; ∏ ; ϖ∈\overline{W}_{𝔭}(π) (x − ⟨λ, ϖ⟩ − \frac{1}{2}⟨π̄ − ϖ, π̄ + ϖ − 2ρ⟩)

Then q′_{π,Θ} (x) = q_{π,Θ} (x) for a generic λ.

π = \begin{cases}the natural representation if 𝔤 is classical , \\ the minimal dimensional representation if 𝔤 is exceptional. \end{cases}

\begin{matrix}2 & 2 & 2 & 2 \\ ∘—— & ∘— ⋯ — & ∘—— & ∘ \end{matrix}

\begin{matrix}3 & 3 & 2 \\ •— ⋯ — & •—— & •—— & •— ⋯ — & • \\ ↖ & ↗ \\ ——— & —— & ——— & —— —— \end{matrix}  \begin{matrix}3 & 3 \\ •— ⋯ — & •—— & •— ⋯ — & • \\ ↖ & ↗ \\ ——— & —— & ———— \end{matrix}

\begin{matrix}3 & 3 & 3 & 3 \\ ∘—— & •— ⋯ — & ∘⟹ & ∘ \end{matrix}  \begin{matrix}3 & 3 & 3 & 2 \\ ∘—— & ∘— ⋯ — & ∘ ⟸ & • \end{matrix}

\begin{aligned}3 & 3 & 3 & 2 \\ ∘—— & •— ⋯ — & ∘—— & • \\ & & • \\ & & 2 \end{aligned}  \begin{aligned}3 & 3 & 3 & 3 \\ ∘—— & •— ⋯ — & ∘—— & • \\ & & ↗ ↙ \\ & & • \end{aligned}

\begin{aligned}3 & 4 & 5 & 4 & 3 \\ ∘—— & ∘—— & ∘—— & ∘—— & ∘ \\ & & ∘ \\ & & 3 \end{aligned}  \begin{matrix}— & —— & ———— & —— \\ ↙ & ↘ \\ 6 & 8 & 5 \\ •—— & ∘—— & ∘—— & ∘—— & • \\ ∘ \\ 3 \end{matrix}

\begin{aligned}3 & 5 & 7 & 6 & 5 & 4 \\ ∘—— & ∘—— & ∘—— & ∘—— & ∘—— & • \\ & & ∘ \\ & & 4 \end{aligned}  \begin{aligned}6 & 11 & 16 & 13 & 11 & 9 & 6 \\ ∘—— & ∘—— & ∘—— & ∘—— & ∘—— & ∘—— & • \\ & ∘ \\ & 8 \end{aligned}

\begin{matrix}3 & 5 & 8 & 6 \\ ∘—— & ∘⟹ & ∘—— & ∘ \end{matrix}  \begin{matrix}5 & 3 \\ ∘⇚ & ∘ \end{matrix}

0 100000
↘ 1
1  ¯110000
↘ 3
2  0¯11000
↘ 4
3  00¯1 110
↙ 2  ↘ 5
4  000 ¯110  0001¯11
↘ 5  ↙ 2  ↘ 6
5  001 ¯1¯11  00010¯1
↙ 4  ↘ 6  ↙ 2
6  01¯1 001  001 ¯10¯1
↙ 3  ↘ 6  ↙ 4
7  1¯10001  01¯1 01¯1
↙ 1  ↘ 6  ↙ 3  ↘ 5
8 ¯100001  1¯1001¯1  0100¯10
↘ 6  ↙ 1  ↘ 5  ↙ 3
9  ¯10001¯1  1¯110¯10
↘ 5  ↙ 1  ↘ 4
10  ¯1010¯10  10¯1 100
↘ 4  ↙ 1  ↘ 2
11  ¯11¯1 100  100 ¯100
↙ 3  ↘ 2  ↙ 1
12  0¯10100  ¯110 ¯100
↘ 2  ↙ 3
13  0¯11 ¯100
↙ 4
14  00¯1 010
↙ 5
15  0000¯11
↙ 6
16 00000¯1

0 0000001
↘ 7
1  000001¯1
↘ 6
2  00001¯10
↘ 5
3  0010¯100
↘ 4
4  01¯1 1000
↙ 2  ↘ 3
5  010 ¯1000  1¯101000
↘ 3  ↙ 2  ↘ 1
6  1¯11 ¯1000  ¯1001000
↙ 4  ↘ 1  ↙ 2
7  10¯1 0100  ¯101 ¯1000
↙ 5  ↘ 1  ↙ 4
8  1000¯100  ¯11¯1 0100
↙ 6  ↘ 1  ↙ 5  ↘ 3
9  10000¯11  ¯1100¯110  0¯100100
↙ 7  ↘ 1  ↙ 6  ↘ 3  ↙ 5
10 100000¯1  ¯11000¯11  0¯110¯110
↘ 1  ↙ 7  ↘ 3  ↙ 6  ↘ 4
11  ¯110100¯1  00¯1 11¯11  00¯1 1010
↘ 3  ↙ 7  ↘ 4  ↙ 6 ↓2
12  0¯11000¯1  00¯1 11¯11 000 ¯1010
↘ 4  ↙ 7  ↘ _{5} ↙ ^{6} ↓2
13  00¯1 110¯1 000 ¯11¯11 0001¯101
↙ ^{7} ↘ _{5} 2↓  {↙ ^{7} \atop ↘ _{5}} ↓2
14  000 ¯110¯1 000110¯1 001 ¯1¯101
{↙ ^{6} \atop ↘ _{5}}  ↙ 7 ↓2  ↘ 4
15  00010¯10 001 ¯1¯11¯1  01¯1 0001
↙ 6 2↓  ↘ 4  ↙ 7  ↘ 3
16  001 ¯10¯10  01¯1 001¯1  1¯100001
↘ 4  ↙ 6  ↘ 3  ↙ 7  ↘ 1
17  01¯1 01¯10  0¯10001¯1  ¯1000001
↙ 5  ↘ 3  ↙ 6  ↘ 1  ↙ 7
18  0100¯100  1¯1001¯10  ¯100001¯1
↘ 3  ↙ 5  ↘ 1  ↙ 6
19  1¯110¯100  ¯10001¯10
↙ 4  ↘ 1  ↙ 5
20  10¯1 1000  ¯1010¯100
↙ 2  ↘ 1  ↙ 4
21  100 ¯1000  ¯11¯1 1000
↘ 1  ↙ 2  ↘ 3
22  ¯110 ¯1000  0¯101000
↘ 3  ↙ 2
23  0¯11 ¯1000
↘ 4
24  00¯1 0100
↘ 5
25  0000¯110

↘ 6

26  00000¯11
↘ 7
27  000000¯1

q_{π,Θ_{4}} (x; λ) = (x − \frac{1}{2})(x − \frac{9}{20} )(x − \frac{7}{15} )(x − \frac{5}{12} )(x − \frac{1}{10} λ − \frac{9}{10} )
· (x − \frac{1}{12} λ − \frac{5}{6})(x − \frac{1}{15} λ − \frac{11}{15} )(x − \frac{1}{20} λ − \frac{13}{20} )(x − \frac{1}{30} λ − \frac{17}{30} )
· (x − \frac{1}{60} λ − \frac{1}{2})(x + \frac{1}{60} λ − \frac{7}{20} )(x + \frac{1}{30} λ − \frac{4}{15} )(x + \frac{1}{20} λ − \frac{1}{5})
· (x + \frac{1}{15} λ − \frac{2}{15} )(x + \frac{1}{12} λ − \frac{1}{12} )(x + \frac{1}{10} λ).