For any Young tableau, one can form the Young symmetrizer. I'm naturally interested in young symmetrizers coming from standard tableaux, but I'm forced to look at Young symmetrizers of non-standard tableaux. Is there a known way of writing such a Young symmetrizer as a linear combination of Young symmetrizers of standard tableaux, perhaps something akin to the Garnir relations?


Let $T$ be any (not necessarily standard) tableau of shape $\lambda$. Define $c_\lambda(T)=a_\lambda(T) b_\lambda(T)$ as usual. Let $C_i(T)$ denote the ith column of $T$. I Equation 2.4 in


says that for two columns $C_i(T)$, $C_j(T)$ with $|C_i(T)|\le |C_j(T)|$ and any $a\in C_i(T)$,

$$c_\lambda(T)\bigg(1-\sum_{x\in C_j(T)}{(a,x)}\bigg)=0,$$ where $(a,x)$ denotes the transposition switching $a$ and $x$. From here one can derive that $$c_\lambda(T)=\sum_{x\in C_j(T)}{(a,x)[(a,x)c_\lambda(T)(a,x)]}$$ where each $(a,x)c_\lambda(T)(a,x)$ is the symmetrizer of another tableaux. In fact, following the proof of Equation 2.4 shows that one can replace the sum of transpositions with any order two permutation. This gives a way to "straighten" a symmetrizer of a non-standard tableaux.


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