Lemma 4 ([Horizontal addition]). If c_α,β ^γ >0 and c_α',β' ^γ' >0 , then c_α+α',\,β+β' ^γ+γ' >0, where addition is rowwise. In particular, suppose a target has column heights h_i=a_i+b_i , with a_i,b_i≥0 . Form α from the …

Lemma 3 ([Young support]). For a finite list A of positive integers of sum n , the module_{∏_} a∈A S_a^S_n 1 contains exactly the S^ν whose column heights h_{1≥h_2≥⋯} satisfy ∑_j=1^k h_{j≤P_A}(k):=∑_a∈A (k,a) (k≥1). The constituent …

Lemma 2 ([Dual-polytabloid test]). Let z∈E^⊗B and θ |B| with at most D rows. Then S^θ occurs in [S_B]z if and only if z has a nonzero contraction with a dual block tensor whose blocks are the columns of θ , for some placement of the …

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The first assertion is the Littlewood--Richardson semigroup property , applied to the two triples of partitions. For the second assertion, the exterior-power case of Pieri gives c_ (1^ a_i ),(1^ b_i ) ^ (1^ h_i ) >0 for each i . Add these containments horizontally. A rowwise sum of single-column partitions is exactly the diagram specified by their column-height multiset, so sorting the individual lists changes nothing.

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Lemma 4 ([Horizontal addition]). If cα,β γ >0 and cα',β' γ' >0 , then cα+α',\,β+β' γ+γ' >0, where addition is rowwise. In particular, suppose a target has column heights hi=ai+bi , with ai,bi≥0 . Form α from the …

Lemma 3 ([Young support]). For a finite list A of positive integers of sum n , the module∏_ a∈A SaSn 1 contains exactly the Sν whose column heights h1≥h2≥⋯ satisfy ∑j=1k hj≤PA(k):=∑a∈A (k,a) (k≥1). The constituent …

Lemma 2 ([Dual-polytabloid test]). Let z∈E⊗B and θ |B| with at most D rows. Then Sθ occurs in [SB]z if and only if z has a nonzero contraction with a dual block tensor whose blocks are the columns of θ , for some placement of the …

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Lemma 4 ([Horizontal addition]). If c_α,β ^γ >0 and c_α',β' ^γ' >0 , then c_α+α',\,β+β' ^γ+γ' >0, where addition is rowwise. In particular, suppose a target has column heights h_i=a_i+b_i , with a_i,b_i≥0 . Form α from the …

Lemma 3 ([Young support]). For a finite list A of positive integers of sum n , the module_{∏_} a∈A S_a^S_n 1 contains exactly the S^ν whose column heights h_{1≥h_2≥⋯} satisfy ∑_j=1^k h_{j≤P_A}(k):=∑_a∈A (k,a) (k≥1). The constituent …

Lemma 2 ([Dual-polytabloid test]). Let z∈E^⊗B and θ |B| with at most D rows. Then S^θ occurs in [S_B]z if and only if z has a nonzero contraction with a dual block tensor whose blocks are the columns of θ , for some placement of the …