Symmetric Decomposition in the Column Case

where wd(J ) = J i 1J i 2 . . ., P is the insertion tableaux map, and we assume that all the evaluations are in lexicographic order and all the row words of an evaluation are in lexicographic order. The tableau P ( ∏ J wd(J )1wd(J)2 . . .wd(JJ )λJ ) has k|I| boxes and is equal to the (plactic) product of k tableaux of shape I, and these k tableaux are in lexicographic order. If H is the shape of the tableau P ( ∏ J wd(J )1wd(J)2 . . .wd(JJ )λJ ), I am trying to show that any two tableaux of shape H have an equal number of factorizations into a product of k tableaux of shape I which are in lexicographic order. If the number of such factorizations (denoted by c) is an invariant for shape H, then we can conclude that the coefficient of SH in the expansion of S(SI) is c H . The energy function h on the product of two tableaux p, p′ of shape ν, ν′ respectively is defined as h(p⊗ p′) = |ν + ν′| − |shape(p · p′) ∩ (ν + ν′)|.