14
votes
1answer
162 views

Involutions, RSK and Young Tableaux

Let $S_n$ be the symmetric group on $n$ elements. The Robinson-Schensted-Knuth (RSK) correspondence sends a permutation $\pi\in S_n$ to a pair of Standard Young Tableaux $(P,Q)$ with equal shapes ...
4
votes
3answers
338 views

Matrix which commutes with permutation matrix

I'm trying to show that if $A$ commutes with all $3\times 3$ permutation matrices, then $A$ has to be of the following form: $ A = \begin{pmatrix} a & b & b \\ b & a & b \\ b & b ...
6
votes
0answers
115 views

Expression of basis vectors of permutation modules in different bases.

Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
1
vote
2answers
664 views

Irreducibility of the standard representation of $S_n$.

The permutation representation of $S_n$ is $\mathbb C^n$ with elements of $S_n$ permuting the basis vectors $\{e_1, e_2, \ldots, e_n\}$. It has a trivial subrepresentation spanned by the vector $v = ...
2
votes
2answers
595 views

How does one decompose the regular representation of $S_3$?

I need to decompose the regular representation of $S_3$ into irreducible ones. What I know so far is this: $S_3$ is generated by $\tau = (12)$ and $\sigma = (123)$. If $v$ is an eigenvector of ...
1
vote
1answer
164 views

Categorical definition of permutation representation

In Mac Lane's, Categories for the Working Mathematician, on p.15 ex.3c) it asks to interpret "functor" when F: (group G)-->Set is a permutation representation of G. Here is where I get stuck, G is ...