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darij grinberg
darij grinberg
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A famous factorization is $X_1 X_2\cdots X_n$$(1+X_1) (1+X_2) \cdots (1+X_n)$ where $X_1=0$, $X_k= (1,k)+(2,k)+\cdots +(k-1,k)$ for $2\leq k\leq n$. $X_k$ is called a Jucys-Murphy element, though this factorization is due to Alfred Young in 1902. Jucys gave the $q$-analogue $$ (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).

A famous factorization is $X_1 X_2\cdots X_n$ 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).

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).

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Richard Stanley
Richard Stanley
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A famous factorization is $X_1 X_2\cdots X_n$ 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).