{"id":"hexagon/2609.00135","pdf":"https://hexagonmath.org/pdf/2609.00135v1","url":"https://hexagonmath.org/abs/2609.00135v1","tags":["math.RT","msc:20C30","msc:05E99","Lie algebra","symmetric group algebra","symmetric group representations"],"score":0,"title":"A counterexample to the Burman--Kulishov conjecture on Lie elements","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Darij Grinberg"],"reviews":0,"subject":"math.CO","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-09-29","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}