A counterexample to the Burman--Kulishov conjecture on Lie elements
Burman and Kulishov defined Lie elements in the group algebra $\mathbf k[S_n]$ by comparing, on every exterior power of the reflection representation $V$, the usual action of $\mathbf k[S_n]$ with the infinitesimal action induced by its action on $V$. They conjectured that the Lie algebra $\mathcal L_n$ of all Lie elements is generated by the Kirchhoff differences $1-(ij)$. We disprove this conjecture for $n=4$ by exhibiting an explicit counterexample arising from the $(2,2)$-block of $\mathbf k[S_4]$. More generally, we describe $\mathcal L_n$ in terms of the Artin--Wedderburn decomposition…
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