Partitions of n, Young tableaux and tabloids.

This is a follow-up of my this post. There is a nice way to represent and visualize partitions of . The objects we will define will admit a natural action from and will permit us to think about the symmetric group in a very combinatorial way. Suppose is a partition of . Then the Young diagram of shape is an arrangement of boxes, there is rows and the ‘th row, starting from the top, contains boxes. For example, here is the associated Young diagram for and :

If , there is also the notion of a Young tableau of shape . This is simply a Young diagram of shape filled with the numbers in (with no repeats). A tableau of shape , or tableau, is denoted by . For example, here are all the tableaux for :

The entries in a tableau are designated as in a matrix, so the first tableau above has entries .

There is a natural action of on a tableau of shape defined as for . i.e. just permutes the entries of the tableau. This action is clearly transitive.

The next step is to define an equivalence relation on the set of tableaux which will be important for our construction of the irreducible representations of the symmetric group. If are two tableaux, define them to be row equivalent if corresponding rows contain the same numbers and note this by . A tabloid is then defined as an equivalence class of tableaux for this relation. For example, here is a tableau:

The notation without the vertical bars is to suggest that the arrangement of numbers in different rows is irrelevant. Here is the set of all the tabloids:

The action of on tableaux induces an action of on tabloids. This gives, for every , a representation of on the complex vector space generated by all tableaux . Let’s look at some examples for different partitions of :

If , then the associated diagram is just n horizontally aligned boxes so every tableaux are row equivalent. In other words, there is only one tabloid. We thus find and acts trivially. This is the trivial representation.

If , the associated diagram is n vertically aligned boxes so now the situation is that no two tableaux are row equivalent. There is thus one different tabloid for every tableau. Also, every tableau can be identified with a permutation using one line notation and since this identification preserves the action of , we find that is the regular representation. In particular, the ‘s are far from being irreducibles.

If , then a tabloid is uniquely determined by the number in its second row. So we can identity the tabloids with the set of numbers from to . Clearly, will send the tabloid identified with to the tabloid identified with if and only if . This tells us that this identification is an equivalence of representations and so we have the defining representation .

In subsequent posts, we will see a way to construct, for each , an irreducible submodule of in a way that gives us all irreducible representations of .

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