Lemma 7 ([Induction from coordinate sectors]). Let a finite group G act transitively on a finite set Ω . Suppose a G -module has a direct-sum decomposition T=_A∈Ω T_A with gT_A=T_gA . Fix D∈Ω , let H be its stabilizer, and let z∈T_D . …

Proposition 6 ([Band criterion]). Let M≥9 , and let λ |L| have column heights h_{1≥h_2≥⋯} and row lengths w_{1≥w_2≥⋯} , with zeros appended. Suppose that integers d,q satisfy 1≤d≤4 , 0≤q≤8 , and ∑_i=1^d w_i≥K, dq+7+d≤K, ∑_i=1^q …

Lemma 5. The parameters satisfy r≤ (M-1)/2,, M odd,; 3M/2+2,, M even. For M≥9 , the partition with column lengths L=(M+b+s-1, M-1+b, M-2,M-3,…,3, 2^b+1 , 1^s) is self-conjugate and has size n . Consequently the support of S^L⊗S^L is …

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Choose one representative g_A with g_AD=A for each A∈Ω . The source is the direct sum of the spaces g_A⊗A_0 . Their images are g_AA_0⊆T_A , so the images for distinct A are linearly independent. On each summand the displayed map is injective, since g_A acts invertibly. Its image contains every translate of z , proving surjectivity as well.

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Lemma 7 ([Induction from coordinate sectors]). Let a finite group G act transitively on a finite set Ω . Suppose a G -module has a direct-sum decomposition T=A∈Ω TA with gTA=TgA . Fix D∈Ω , let H be its stabilizer, and let z∈TD . …

Proposition 6 ([Band criterion]). Let M≥9 , and let λ |L| have column heights h1≥h2≥⋯ and row lengths w1≥w2≥⋯ , with zeros appended. Suppose that integers d,q satisfy 1≤d≤4 , 0≤q≤8 , and ∑i=1d wi≥K, dq+7+d≤K, ∑i=1q …

Lemma 5. The parameters satisfy r≤ (M-1)/2,, M odd,; 3M/2+2,, M even. For M≥9 , the partition with column lengths L=(M+b+s-1, M-1+b, M-2,M-3,…,3, 2b+1 , 1s) is self-conjugate and has size n . Consequently the support of SL⊗SL is …

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Lemma 7 ([Induction from coordinate sectors]). Let a finite group G act transitively on a finite set Ω . Suppose a G -module has a direct-sum decomposition T=_A∈Ω T_A with gT_A=T_gA . Fix D∈Ω , let H be its stabilizer, and let z∈T_D . …

Proposition 6 ([Band criterion]). Let M≥9 , and let λ |L| have column heights h_{1≥h_2≥⋯} and row lengths w_{1≥w_2≥⋯} , with zeros appended. Suppose that integers d,q satisfy 1≤d≤4 , 0≤q≤8 , and ∑_i=1^d w_i≥K, dq+7+d≤K, ∑_i=1^q …

Lemma 5. The parameters satisfy r≤ (M-1)/2,, M odd,; 3M/2+2,, M even. For M≥9 , the partition with column lengths L=(M+b+s-1, M-1+b, M-2,M-3,…,3, 2^b+1 , 1^s) is self-conjugate and has size n . Consequently the support of S^L⊗S^L is …