2×2

Δ^{2} × Δ^{2}2Δ^{2}

Δ^{2} × Δ^{2} = 9p_{i,j} ∈ R^{4}(i, j = 1, ⋯ 3),

p_{1,1} = (0, 0, 0, 0), p_{1,2} = (0, 0, 1, 0), p_{1,3} = (0, 0, 0, 1), p_{2,1} = (1, 0, 0, 0), p_{2,2} = (1, 0, 1, 0), p_{2,3} = (1, 0, 0, 1), p_{3,1} = (0, 1, 0, 0), p_{3,2} = (0, 1, 1, 0), p_{3,3} = (0, 1, 0, 1). 5p_{i_{k},j_{k}}k = 1, ⋯, 5

{i_{1}, j_{1}}, ⋯, {i_{5}, j_{5}}.

{1, 1}, {1, 2}, {1, 3}, {2, 3}, {3, 3}4P_{1,1}, p_{1,2}, p_{1,3}, p_{2,3}, p_{3,3}Δ^{2} × Δ^{2}465 T Δ^{2} × Δ^{2}23S_{3}S_{3} × S_{3}Δ^{2} × Δ^{2}T S_{3} × S_{3} T 5T_{i}(i = 1, ⋯, 9)5

T_{1} = \begin{matrix}\{1, 1\} & \{1, 2\} & \{1, 3\} & \{2, 3\} & \{3, 3\} \\ \{1, 1\} & \{1, 2\} & \{2, 2\} & \{2, 3\} & \{3, 1\} \\ \{1, 1\} & \{2, 1\} & \{2, 2\} & \{2, 3\} & \{3, 1\} \\ \{1, 2\} & \{2, 2\} & \{2, 3\} & \{3, 1\} & \{3, 3\} \\ \{1, 2\} & \{2, 2\} & \{3, 1\} & \{3, 2\} & \{3, 3\} \\ \{1, 1\} & \{1, 2\} & \{2, 3\} & \{3, 1\} & \{3, 3\} \end{matrix}

T_{2} = \begin{matrix}\{1, 1\} & \{1, 2\} & \{1, 3\} & \{2, 3\} & \{3, 2\} \\ \{1, 1\} & \{1, 2\} & \{2, 2\} & \{2, 3\} & \{3, 2\} \\ \{1, 1\} & \{2, 1\} & \{2, 2\} & \{2, 3\} & \{3, 2\} \\ \{1, 1\} & \{2, 1\} & \{2, 3\} & \{3, 1\} & \{3, 2\} \\ \{1, 1\} & \{2, 3\} & \{3, 1\} & \{3, 2\} & \{3, 3\} \\ \{1, 1\} & \{1, 3\} & \{2, 3\} & \{3, 2\} & \{3, 3\} \end{matrix}

T_{5} = \begin{matrix}\{1, 1\} & \{1, 2\} & \{1, 3\} & \{2, 3\} & \{3, 1\} \\ \{1, 1\} & \{1, 2\} & \{2, 2\} & \{2, 3\} & \{3, 1\} \\ \{1, 1\} & \{2, 1\} & \{2, 2\} & \{2, 3\} & \{3, 1\} \\ \{1, 2\} & \{2, 2\} & \{2, 3\} & \{3, 1\} & \{3, 2\} \\ \{1, 3\} & \{2, 3\} & \{3, 1\} & \{3, 2\} & \{3, 3\} \\ \{1, 2\} & \{1, 3\} & \{2, 3\} & \{3, 1\} & \{3, 2\} \end{matrix}

T_{6} = \begin{matrix}\{1, 1\} & \{1, 2\} & \{1, 3\} & \{2, 3\} & \{3, 1\} \\ \{1, 1\} & \{1, 2\} & \{2, 2\} & \{2, 3\} & \{3, 1\} \\ \{1, 1\} & \{2, 1\} & \{2, 2\} & \{2, 3\} & \{3, 1\} \\ \{1, 2\} & \{2, 2\} & \{2, 3\} & \{3, 1\} & \{3, 2\} \\ \{1, 2\} & \{2, 3\} & \{3, 1\} & \{3, 2\} & \{3, 3\} \\ \{1, 2\} & \{1, 3\} & \{2, 3\} & \{3, 1\} & \{3, 3\} \end{matrix}

T_{9} = \begin{matrix}\{1, 1\} & \{1, 2\} & \{1, 3\} & \{2, 3\} & \{3, 1\} \\ \{1, 1\} & \{1, 2\} & \{2, 2\} & \{2, 3\} & \{3, 1\} \\ \{1, 1\} & \{2, 1\} & \{2, 2\} & \{2, 3\} & \{3, 1\} \\ \{1, 2\} & \{2, 2\} & \{2, 3\} & \{3, 1\} & \{3, 3\} \\ \{1, 2\} & \{2, 2\} & \{3, 1\} & \{3, 2\} & \{3, 3\} \\ \{1, 2\} & \{1, 3\} & \{2, 3\} & \{3, 1\} & \{3, 3\} \end{matrix} .

T_{1}T_{2}T_{5}12618T_{i}(i = 6, 9)36Δ^{2} × Δ^{2}108 T_{5} T 2

Δ_{2} × Δ_{2} = ( \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \end{matrix} ) =: A.

T 108

T Δ_{2} × Δ_{2}ϕ_{T} = ((ϕ_{T})_{ij}) , (1 ≤ i ≤ 3, 1 ≤ j ≤ 3)Z^{3×3} (ϕ_{T})_{ij} = (i, j)T

ϕ_{T_{2}} = ( \begin{matrix}3 & 5 & 1 \\ 1 & 4 & 5 \\ 5 & 1 & 4 \end{matrix} )

T_{2} Σ(Δ_{2} × Δ_{2})108{ϕ_{T} | τ ∈ T } Σ(Δ_{2} × Δ_{2})R^{9}1081081, 2, 322214430 Σ(Δ_{2} × Δ_{2})2

1, 236, 95 1, 234, 64 ϕ_{T_{2}}ϕ_{T_{i}}(i = 1, 5, 6, 9)

N(Σ(Δ_{2} × Δ_{2}))Σ(Δ_{2} × Δ_{2})f

N(Σ(Δ_{2} × Δ_{2}), f) = {v | ⟨v, p − q⟩ ≥ 0p ∈ Σ(Δ_{2} × Δ_{2})q ∈ f}.

N(Σ(Δ_{2} × Δ_{2}))Σ(Δ_{2} × Δ_{2})