I was reading the book on Young Tableaux by Fulton. On first page of notations, he defined Young diagram to be left justified rows of boxes, weakly decreasing downwards. Then, he defines Young Tableaux to be filling positive integer entries in a Young diagram with condition
(i) Entries are weakly increasing in rows
(ii) Entries are strictly increasing in columns.
Q.1 Is there any technical reason to put strong condition (ii) for columns as compared to (i)?
I am looking this subject first time. But I have heard that these Tableaux are useful in construction of irreducible representations of $S_n$, finding their dimensions, etc. So, regarding condition (ii), a question came to my mind:
Q.2 If we allow entries in columns also weakly increasing, what kind of information about representations of $S_n$ can be obtained from such filling in Diagrams? Does it help us to get certain other invarients associated to irreducible representations of $S_n$?
For example, there are also Schur polynomials associated to a Young Tableaux, which are certain class of symmetric polynomials, and these polynomials are also useful for representations of $S_n$. If we allow entries in columns weakly increasing, then associating corresponding polynomials to such filling, can we get some other information about representations of $S_n$?