All Questions

Consider the group algebra of the symmetric group $\mathbb{C}S_n$. Then there is a corresponding Lie algebra $\mathfrak{L}(S_n)$ defined by $$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$ ...

For a partition $\lambda$ let $\chi_\lambda$ be the corresponding irreducible representation of the symmetric group $S_n$. Let $\mathrm{Imm}_\lambda(x) = \sum\limits_{\pi \in S_n} \chi_\lambda(\pi) x_{...

This is a follow up on my earlier MO question. Let $\operatorname{Inv}(\mathfrak{S}_n):=\{\pi\in\mathfrak{S}_n: \pi^2=1\}$ be the set of involutions in the symmetric group $\mathfrak{S}_n$. Denote $...

Let $\operatorname{Inv}(\mathfrak{S}_n):=\{\pi\in\mathfrak{S}_n: \pi^2=1\}$ be the set of involutions in the symmetric group $\mathfrak{S}_n$. Denote $I_n:=\#\operatorname{Inv}(\mathfrak{S}_n)$. Let $\...

Consider the set $ X_1^n=\{1,2,...,2^n\} $. Then define $ X_2^n $ to be the set of two element subsets of $ X_1^n $. I will construct $ X_i $ by induction on $ i $. $ X_i^n $ is the set of two element ...