A question on Young tableau. I am reading Fulton's book representation theory. My question occurs in the proof of Lemma 4.23. I will introduce my question concisely without letting you read that book.
The book introduces an order:
Let $\lambda$ and $\mu$ be two partitions of $d$, $\lambda > \mu$ if the first nonvanishing $\lambda_i-\mu_i$ is positive.
Now denote the Young tableau of $\lambda$ and $\mu$ by $T$ and $T'$ respectively.
Then in that proof it says "$\lambda > \mu$ implies that there are two integers in the same row of $T$ and the same column of $T'$".
Why can we get this claim? I have no idea. Could you share your solution? Thank you.
 A: Let $k\geq 1$ be the number of rows of $T$: we argue by induction on $k$. If $k=1$ then we are done. Now we suppose $k>1$ and distinguish between two cases.

*

*If $\lambda_1>\mu_1$ the integers on the first row of $T$ are more than the number of columns of $T'$, thus one of the columns of $T'$ must contain two numbers of the first row of $T$.


*If $\lambda_1=\mu_1$ we can assume that the integers in the first row of $T$ are in different columns of $T'$, otherwise we are done. Then take $\sigma\in Q_{T'}$ such that the first row of $\sigma T'$ contains the same integers of the first row of $T$ (as in Fulton's book, $\sigma\in Q_{T'}$ is a permutation of the symmetric group over $d$ elements which preserves each column of $T'$). Observe that two integers are in the same column of $T'$ if and only if they are in the same column of $\sigma T'$, so that we can replace $T'$ with $\sigma T'$ for the purpose of the proof. Consider the Young tableaux obtained by deleting the first row of $T$ and of $\sigma T'$ and apply the induction hypothesis.
