# Relations among Young symmetrizers of non-standard tableaux

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.