The Unapologetic Mathematician

Mathematics for the interested outsider

Generalized Young Tableaux

And now we have another generalization of Young tableaux. These are the same, except now we allow repetitions of the entries.

Explicitly, a generalized Young tableau T — we write them with capital letters — of shape \lambda is an array obtained by replacing the points of the Ferrers diagram of \lambda with positive integers. Any skipped or repeated numbers are fine. We say that the “content” of T is the composition \mu=(\mu_1,\dots,\mu_m) where \mu_i is the number of i entries in T.

As an example, we have the generalized Young tableau

\displaystyle\begin{array}{ccc}4&1&4\\1&3&\end{array}

of shape (3,2) and content (2,0,1,2).

Notice that if \lambda\vdash n, then \mu\vdash n as well, since both count up the total number of places in the tableau. Given a partition \lambda and a composition \mu, both decomposing the same number n, we define T_{\lambda\mu} to be the collection of generalized Young tableaux of shape \lambda and content \mu. All the tableaux we’ve considered up until now have content (1,\dots,1)=(1^n).

Now, pick some fixed (ungeneralized) tableau t. We can use the same one we usually do, numbering the rows from 1 to n across each row and from top to bottom, but it doesn’t really matter which we use. For our examples we’ll pick

\displaystyle t=\begin{array}{ccc}1&2&3\\4&5&\end{array}

Using this “reference” tableau, we can rewrite any generalized tableau as a function; define T(i) to be the entry of T in the same place as i is in t. That is, any generalized tableau looks like

\displaystyle\begin{array}{ccc}T(1)&T(2)&T(3)\\T(4)&T(5)&\end{array}

and in our particular example above we have T(1)=T(3)=4, T(2)=T(4)=1, and T(5)=3. Conversely, any such function assigning a positive integer to each number from 1 to n can be interpreted as a generalized Young tableau. Of course the particular correspondence depends on exactly which reference tableau we use, but there will always be some such correspondence between functions and generalized tableaux.

February 2, 2011 Posted by | Algebra, Representation Theory, Representations of Symmetric Groups | 2 Comments

   

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