The Jucys-Murphy elements of the symmetric group algebra

Let $n\ge 1$ and let $R$ be a commutative ring. In the group ring $R[S_n]$ of the symmetric group on $\{1,\dots ,n\}$ (Partitions, English diagrams, and conjugation, The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition, The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements), where $(jk)$ denotes the transposition of the distinct entries $j,k\in \{1,\dots ,n\}$ (The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition), the Jucys-Murphy elements are $X_1:=0,X_k:={\sum }_{j=1}^{k-1}(jk)\in R[S_n](2\le k\le n)\mathrm{.}$ Each $X_k$ is a finite sum of group elements with coefficient $1$, so the formula already defines an element of the integral group ring $\mathbb{Z}[S_n]$, and its image under the base change $\mathbb{Z}[S_n]\to R[S_n]$ is the displayed element (Restriction of scalars and extension of scalars $S{\otimes }_{R}M$ along a ring homomorphism $R\to S$); for $1\le k\le m\le n$ the element $X_k$ of $R[S_m]$, computed in the subgroup $S_m=\mathrm{Sym}(\{1,\dots ,m\})\subseteq S_n$ of permutations fixing $m+1,\dots ,n$, maps to $X_k$ under the inclusion $S_m\hookrightarrow S_n$. Equivalently $X_k=T_k-T_{k-1}$, where $T_k={\sum }_{1\le i<j\le k}(ij)$ is the sum of all transpositions in $S_k$.

• Well-definedness. The sum defining $X_k$ has the $k-1$ terms $(1k),\dots ,(k-1k)$, each of which is an element of the subgroup $S_k\subseteq S_n$, hence a basis element of the free $R$-module $R[S_n]$; a finite sum of basis elements is an element of $R[S_n]$ independent of any ordering of the summands. The formula uses only the labels $1,\dots ,k$, so it is preserved by the inclusion $S_m\hookrightarrow S_n$ for $k\le m\le n$ and by the base change $\mathbb{Z}[S_n]\to R[S_n]$. The group-ring multiplication is bilinear and sends basis elements $g,h$ to $gh$ (The group ring $R[G]$ is a unital $R$-algebra with basis $G$, and each $g\in G$ is a unit of $R[G]$). Thus the subgroup inclusions and the coefficient map ${\sum }_g{a}_{g}g\mapsto {\sum }_g(a_g{1}_R)g$ preserve products and the identity.

• The identity $X_k=T_k-T_{k-1}$. The transpositions of $S_k$ are $(ij)$ with $1\le i<j\le k$; those with $j<k$ are exactly the transpositions of $S_{k-1}$, and the remaining ones are $(ik)$ with $i<k$. Hence $T_k-T_{k-1}={\sum }_{j=1}^{k-1}(jk)=X_k$. For $k=2$ this reads $X_2=(12)=T_2-T_1$, where $T_1=0$. Since each $T_k$ is a sum of all transpositions of the subgroup $S_k$, it is a sum of full conjugacy classes of $S_k$.

• Integral normalization. Every coefficient in $X_k$ is $1$, not $\pm 1$ or a fraction; no characteristic is inverted, so $X_k$ is defined over $\mathbb{Z}$ and over every commutative ring. This is the normalization used throughout this page; the spectral statements below specialize the coefficient ring to $\mathbb{C}$, but no integral identity of this page uses division.

• Centrality. The elements $X_k$ need not be central in $R[S_n]$. The element $X_1=0$ is always central, and for $n=2$ the algebra $R[S_2]$ is commutative, so $X_2$ is central as well. For $n\ge 3$ and $R\mathrm{\ne }0$, conjugation by $(23)$ sends $X_2=(12)$ to $(13)\mathrm{\ne }{X}_2$: these are distinct basis elements and $1_R\mathrm{\ne }0$. Thus $X_2$ is not central in this case. Pairwise commutativity of the $X_k$ is proved in The Jucys-Murphy elements commute pairwise.