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A counterexample to the Burman--Kulishov conjecture on Lie elements

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Contributed by Darij Grinberg ↗

SubmitterDarij Grinberg

Version 1 / Sep 29, 2026 / CC0 1.0

Abstract

Burman and Kulishov defined Lie elements in the group algebra k[Sn]\mathbf k[S_n] by comparing, on every exterior power of the reflection representation VV, the usual action of k[Sn]\mathbf k[S_n] with the infinitesimal action induced by its action on VV. They conjectured that the Lie algebra Ln\mathcal L_n of all Lie elements is generated by the Kirchhoff differences 1−(ij)1-(ij). We disprove this conjecture for n=4n=4 by exhibiting an explicit counterexample arising from the (2,2)(2,2)-block of k[S4]\mathbf k[S_4]. More generally, we describe Ln\mathcal L_n in terms of the Artin--Wedderburn decomposition of k[Sn]\mathbf k[S_n]: its hook blocks are determined by the action on VV, whereas its non-hook blocks are arbitrary. Consequently, the primitive central idempotents of the non-hook blocks yield linearly independent obstructions to the conjecture. We also identify the Lie algebra generated by the Kirchhoff differences in terms of the derived algebra of the Lie algebra generated by transpositions. Along the way, we give an integral, and hence characteristic-free, proof that the exterior powers of VV are the hook-shaped Specht modules.

Provenance statement

Generated by GPT-5.6 Sol (Plus), proofread and edited by myself. Also available on my website at https://www.cip.ifi.lmu.de/~grinberg/algebra/burman-cex-gpt.pdf / https://www.cip.ifi.lmu.de/~grinberg/algebra/burman-cex-gpt.tex

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References

  1. Yurii Burman and Valeriy Kulishov, Lie elements and the matrix-tree theorem , Moscow Math. J. 23 (2023), no. 1, 47--58.
  2. Ivan Marin, L'alg\`ebre de Lie des transpositions , J. Algebra 310 (2007), 742--774; https://arxiv.org/abs/math/0502119 arXiv:math/0502119 .arXiv

Version history

  1. v1Initial depositCurrentSep 29, 2026