Understanding representations of the symmetric group in terms of matrices I'm a newbie in group theory and I can't understand linear representations of the symmetric group. Could you please expain it to me with such an example?
Let $n=5$. So, we have the symmetric group $S_5$.
What does the linear representation of this group corresponding to the Young diagram (3,2) look like?
As I can understand, the dimension of this representation is ${5\choose 3}=10$.
What are the invertible matrices that form this representation? How to get them, to generate them?
 A: This is a nontrivial question first solved by Alfred Young. You can find an account in lectures by Adriano Garsia which now appeared in book form
https://www.springer.com/gp/book/9783030583729
For an (integer) partition $\lambda$ of $n$, a filling $T$ of the Young diagram of $\lambda$ with the numbers $1,\ldots,n$ where each appears exactly once, is called an injective Young tableau. For a permutation $\sigma$, we denote by $\sigma T$ the injective tableau obtained by replacing each entry $i$, by $\sigma(i)$. We let $R(T)$ denote the subgroup of permutations which permute entries within rows. We let $C(T)$ denote the subgroup of permutations which permute entries within columns.
For $T_1,T_2$ any two injective tableaux, let $C(T_1,T_2)={\rm sgn}(\beta_1)$ if there exists $\alpha_1\in R(T_1)$ and $\beta_1\in C(T_1)$ such that $T_2=\alpha_1\beta_1 T_1$, otherwise let $C(T_1,T_2)=0$.
Let $M^{\lambda}(\sigma)$ be the matrix with rows and columns indexed by standard tableaux of shape $\lambda$ and entries
$$
M^{\lambda}(\sigma)_{T_1,T_2}=C(T_1,\sigma T_2)\ .
$$
Finally, let $R^{\lambda}(\sigma)=(M^{\lambda}({\rm Id}))^{-1}M^{\lambda}(\sigma)$.
Then $\sigma\mapsto R^{\lambda}(\sigma)$ is an explicit matrix expression for the irreducible representation indexed by the partition $\lambda$.
