The Unapologetic Mathematician

Mathematics for the interested outsider


Given any collection H\subseteq S_n of permutations, we define two group algebra elements.

\displaystyle\begin{aligned}H^+&=\sum\limits_{\pi\in H}\pi\\H^-&=\sum\limits_{\pi\in H}\mathrm{sgn}(\pi)\pi\end{aligned}

Notice that H doesn’t have to be a subgroup, though it often will be. One particular case that we’ll be interested in is

\displaystyle\kappa_t=C_t^-=\sum\limits_{\pi\in C_t}\mathrm{sgn}(\pi)\pi

where C_t is the column-stabilizer of a Young tableau t. If t has columns C_1,\dots,C_k, then C_t=S_{C_1}\times\dots\times S_{C_k}. Letting \pi run over C_t is the same as letting \pi_i run over S_{C_i} for each i from 1 to k. That is,

\displaystyle\begin{aligned}\kappa_t&=\sum\limits_{\pi_1\in S_{C_1}}\dots\sum\limits_{\pi_k\in S_{C_k}}\mathrm{sgn}(\pi_1\dots\pi_k)\pi_1\dots\pi_k\\&=\sum\limits_{\pi_1\in S_{C_1}}\dots\sum\limits_{\pi_k\in S_{C_k}}\mathrm{sgn}(\pi_1)\dots\mathrm{sgn}(\pi_k)\pi_1\dots\pi_k\\&=\left(\sum\limits_{\pi_1\in S_{C_1}}\mathrm{sgn}(\pi_1)\pi_1\right)\dots\left(\sum\limits_{\pi_k\in S_{C_k}}\mathrm{sgn}(\pi_k)\pi_k\right)\end{aligned}

so we have a nice factorization of this element.

Now if t is a tableau, we define the associated “polytabloid”

\displaystyle e_t=\kappa_t\{t\}

Now, as written this doesn’t really make sense. But it does if we move from just considering Young tabloids to considering the vector space of formal linear combinations of Young tabloids. This means we use Young tabloids like basis vectors and just “add” and “scalar multiply” them as if those operations made sense.

As an example, consider the tableau

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

Our factorization lets us write


And so we calculate

\displaystyle e_t=\begin{array}{ccc}\cline{1-3}4&1&2\\\cline{1-3}3&5&\\\cline{1-2}\end{array}-\begin{array}{ccc}\cline{1-3}3&1&2\\\cline{1-3}4&5&\\\cline{1-2}\end{array}-\begin{array}{ccc}\cline{1-3}4&5&2\\\cline{1-3}3&1&\\\cline{1-2}\end{array}+\begin{array}{ccc}\cline{1-3}3&5&2\\\cline{1-3}4&1&\\\cline{1-2}\end{array}

Now, the nice thing about e_t is that if we hit it with any permutation \pi\in C_t, we get \pi e_t=\mathrm{sgn}(\pi)e_t.

December 23, 2010 - Posted by | Algebra, Representation Theory, Representations of Symmetric Groups


  1. […] defined a bunch of objects related to polytabloids. Let’s see how they relate to […]

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  2. […] 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 […]

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  3. […] is the alternating sum of the elements of . The proof basically runs the same as when we showed that where has shape […]

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  4. […] of shapes and , respectively, where and . If — where is the group algebra element we’ve defined — then dominates […]

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  5. […] and columns are all increasing sequences. In this case, we also say that the Young tabloid and the polytabloid are […]

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  6. […] to the dominance order on tabloids. Specifically, if is standard and appears as a summand in the polytabloid , then […]

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  7. […] Polytabloids are Independent Now we’re all set to show that the polytabloids that come from standard tableaux are linearly independent. This is half of showing that they form a […]

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  8. […] We 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 that standard polytabloids […]

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

  9. This looks plagiarized from Bruce E. Sagan’s book “The Symmetric Group,” pp. 61-62. Just how unapologetic are you?

    Comment by John Doe | August 27, 2013 | Reply

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