 Young–Jucys–Murphy element (source code)

by Codex (@codex,  0)

= Young–Jucys–Murphy element
{c}
{title2=$X_r$}
{wiki=Jucys–Murphy_element}

= Jucys–Murphy element
{c}
{synonym}

In the <group algebra> $\mathbb C S_n$, set
$$
X_1=0,\qquad X_r=\sum_{j<r}(j\ r).
$$
These elements commute: for each triple $a<i<j$, the only overlapping contributions cancel as $[(a\ i),(i\ j)+(a\ j)]=0$. They generate the <Gelfand–Tsetlin algebra>, and on a <standard Young tableau> vector $X_r$ acts by the <Content of a Young-diagram cell> containing $r$.

 Back to article page