The Unapologetic Mathematician

Mathematics for the interested outsider

Permutation Representations from Partitions

Now that we have an action of S_n on the Young tabloids of shape \lambda\vdash n, we can consider the permutation representation M^\lambda that corresponds to it. Let’s consider a few examples.

First, let \lambda=(n). This is a pretty trivial “partition”, consisting of one piece of length n. The Ferrers diagram of \lambda looks like

\displaystyle\begin{array}{cccc}\bullet&\bullet&\cdots&\bullet\end{array}

Any Young tableau thus contains all n numbers on the single row, so they’re all row-equivalent. There is only one Young tabloid:

\displaystyle\begin{array}{cccc}\cline{1-4}1&2&\cdots&n\\\cline{1-4}\end{array}

We conclude that M^{(n)} is a one-dimensional vector space with the trivial action of S_n.

Next, let \lambda=(1^n) — another simple partition with n parts of length 1 each. The Ferrers diagram looks like

\displaystyle\begin{array}{c}\bullet\\\bullet\\\vdots\\\bullet\end{array}

Now in every Young tableau each number is on a different line, so no two tableaux are row-equivalent. They each give rise to their own Young tabloid, such as

\displaystyle\begin{array}{c}\cline{1-1}1\\\cline{1-1}2\\\cline{1-1}\vdots\\\cline{1-1}n\\\cline{1-1}\end{array}

These tabloids correspond to permutations; a generic one looks like

\displaystyle\begin{array}{c}\cline{1-1}\pi(1)\\\cline{1-1}\pi(2)\\\cline{1-1}\vdots\\\cline{1-1}\pi(n)\\\cline{1-1}\end{array}

The action of S_n on these tabloids is basically the same as left-multiplication on the underlying set S_n. And so we find the left regular representation.

Finally, consider the partition \lambda=(n-1,1). This time the Ferrers diagram looks like

\displaystyle\begin{array}{cccc}\bullet&\bullet&\cdots&\bullet\\\bullet&&&\end{array}

and a sample Young tabloid looks like

\displaystyle\begin{array}{cccc}\cline{1-4}1&2&\cdots&n-1\\\cline{1-4}n&&&\\\cline{1-1}\end{array}

Any Young tabloid of shape \lambda is uniquely specified by the single entry on the second row. Any permutation shuffles them around exactly like it does these entries, and so M^{(n-1,1)} is isomorphic to the defining representation.

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December 14, 2010 - Posted by | Algebra, Representation Theory, Representations of Symmetric Groups

7 Comments »

  1. […] try to calculate the characters of the Young tabloid modules we’ve constructed. Since these come from actions of on various sets, we have our usual […]

    Pingback by Characters of Young Tabloid Modules (first pass) « The Unapologetic Mathematician | December 15, 2010 | Reply

  2. […] we’re interested in. For any partition , the Specht module is the submodule of the Young tabloid module spanned by the polytabloids where runs over the Young tableaux of shape […]

    Pingback by Specht Modules « The Unapologetic Mathematician | December 27, 2010 | Reply

  3. […] space with the trivial group action. This is the only possibility anyway, since , and we’ve seen that is itself a one-dimensional vector space with the trivial action of […]

    Pingback by Examples of Specht Modules « The Unapologetic Mathematician | December 28, 2010 | Reply

  4. […] be a submodule of one of the Young tabloid modules. Then I say that either contains the Specht module , or it is contained in the orthogonal […]

    Pingback by The Submodule Theorem « The Unapologetic Mathematician | January 4, 2011 | Reply

  5. […] in the case we care about the space is the Young tabloid module , with the basis of Young tabloids having the dominance ordering. In particular, we consider for […]

    Pingback by Standard Polytabloids are Independent « The Unapologetic Mathematician | January 13, 2011 | Reply

  6. […] defined the Specht module as the subspace of the Young tabloid module spanned by polytabloids of shape . But these polytabloids are not independent. We’ve seen […]

    Pingback by Standard Polytabloids Span Specht Modules « The Unapologetic Mathematician | January 21, 2011 | Reply

  7. […] of generalized tableaux of the vector space is in bijection with the basis of -tabloids of the vector space . And this space carries an action of — the linear extension of the action on tabloids. We […]

    Pingback by Modules of Generalized Young Tableaux « The Unapologetic Mathematician | February 3, 2011 | Reply


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