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?



Relations among Young symmetrizers of non-standard tableaux
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


<https://www3.nd.edu/~craicu/papers/yngsymm.pdf>


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.


