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I am reading the book "Young tableaux with application" by Fulton W. There is claim says that

"The product operation makes the set of tableaux into an associative monoid"

Where the product operation between two tableaux $T$ and $U$ is defined as follows:

$T*U$ is determined by starting with $T$ and row-insert the left-most entry in the bottom row of $U$ into T. Row-insert into the result the next entry of the bottom row $U$, and continue until all entries in the bottom row of $U$ have been inserted. Then insert the next bottom row of $U$ in the same way and we can get $T*U$ once we insert all of the boxes in $U$

And my question is that how can I prove that this operation is associative?

Eric Wofsey
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  • If you want to ask a new question, click "Ask Question" and make a new post. Do not edit your old questions to change what they ask. – Eric Wofsey Jul 09 '19 at 01:32

1 Answers1

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Proving that this operation is associative is not straightforward and you should not spend too much time trying to prove it on your own. It is explained later in the book. Relations are derived for the plactic monoid and shown to characterize it. The insertion algorithm is governed by these relations.

Matt Samuel
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  • Thank you sir. Is there any chance you can tell me which part of the book showed the associativity? I have tried to find it on my own but I just can't :-( –  Nov 09 '16 at 04:18
  • @Tort Don't currently have the book handy, but it should be in the section on the plactic monoid and Knuth equivalence. – Matt Samuel Nov 09 '16 at 04:20