This is probably elementary for experts on the representation theory of the symmetric group, but I did not find the answers I need by a cursory look at the usual textbooks (they could be there, but I gave up trying to decipher conflicting notations and conventions).
Let $\lambda$ be an integer partition of $n$. A Young tableau $T$ is a bijective filling of the corresponding Young diagram with the numbers $1,2,\ldots,n$. For a permutation $\sigma$, let $\sigma T$ denote the tableau obtained by replacing each entry $i$ by $\sigma(i)$. Standard tableaux are the ones where entries increase in each row and column. For a Young tableau $T$, let $C(T)$ denote the group of permutations which preserve the columns of $T$, and let $R(T)$ the group of permutations which preserve the rows of $T$. In the group algebra $\mathbb{C}\mathfrak{S}_n$ of the symmetric group define, as usual, the elements $$ P(T)=\sum_{\sigma\in R(T)} \sigma $$ and $$ N(T)=\sum_{\sigma\in C(T)} {\rm sgn(\sigma)}\ \sigma\ . $$ Finally, the convention for Young symmetrizer that I will use is $$ Y(T)=P(T)N(T)\ . $$
Q1: Is it always true that for two different standard Young tableaux $T,T'$, of the same shape $\lambda$, we have $Y(T)Y(T')=0$?
Q2: Let $T$ be a standard Young tableau and let $\alpha\in C(T)$, $\beta\in R(T)$ be such that $\alpha\beta T$ is also standard. Does this necessarily require $\alpha=\beta=Id$?
