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 | 9 Comments