Define the operator $s_i$ on tableaux:
Consider letters $i$ and $i + 1$ in row reading word of the tableau.
Successively “bracket” pairs of the form (i + 1, I ).
Left with word of the form $i^r (i + 1)^s$. Then
$$s_i(i^r(i+1)^s)=i^s(i+1)^r$$
Show that:
(a) $s_i^2(b)=b$,
(b) $s_is_j(b)=s_js_i(b)$ if $|i-j|>1$
(c) $s_is_{i+1}s_i(b)=s_{i+1}s_is_{i+1}(b)$
Part (a) is clear. Beacuse after two operations, powers remain unchanged. Part (b) is also clear because $s_i$ and $s_j$ do not interfere with each other as long as $|i-j|>1$.
Part(c) is known as braid property. It seems to be a little messy to show this property.