Let $S\_n$ be the symmetric group on $\{1, \ldots, n\}$. Let
\begin{align}
T=\sum\_{g\in S\_n} g.
\end{align}


Are there some references about the factorization of $T$? 


In the case of $n=3$, we have
\begin{align}
& T=1 + (12) + (23) + (12)(23) + (23)(12) + (12)(23)(12) \\
& = 1 + (12) + (23) + (12)(23) + (23)(12) + (23)(12)(23) \\
& = (1 + (12))((12) + (23) + (23)(12)) \\
& = (1 + (23))((12) + (23) + (12)(23)).
\end{align}
Has this problem been studied in some references?


Thank you very much.


Edit: the group algebra I consider is $\mathbb{C} S\_n$.



Factorization in the group algebra of symmetric groups
A famous factorization is $(1+X\_1) (1+X\_2) \cdots (1+X\_n)$ where $X\_1=0$, $X\_k=
(1,k)+(2,k)+\cdots +(k-1,k)$ for $2\leq k\leq n$. $X\_k$ is called a
*Jucys-Murphy element*, though this factorization is due to Alfred
Young in 1902. Jucys gave the $q$-analogue
 $$ (q+X\_1)(q+X\_2)\cdots(q+X\_n) =\sum\_{\pi\in S\_n} q^{c(\pi)}\pi,
 $$
where $c(\pi)$ denotes the number of cycles of $\pi$. See for instance
<https://en.wikipedia.org/wiki/Jucys-Murphy_element>.


Another useful factorization is $T\_2T\_3\cdots T\_n$, where
 $$ T\_k = \sum\_{j=1}^k (k,k-1,\dots,k-j+1). $$
This also has a $q$-analogue:
 $$ T\_2(q)T\_3(q)\cdots T\_n(q) = \sum\_{\pi\in S\_n}
 q^{\mathrm{inv}(\pi)} \pi, $$
where $\mathrm{inv}(\pi)$ is the number of inversions of $\pi$ and
 $$ T\_k(q) = \sum\_{j=1}^k q^{j-1}(k,k-1,\dots,k-j+1). $$
Zagier in 1992 found a deep factorization of $T\_k(q)$ which was used
by Philip Hanlon and me in
<http://math.mit.edu/~rstan/papers/qdef.pdf> (see Theorem 2.1).


