Let EuScriptE_m be the Fomin–Kirillov algebra over a field mathbfk associated with the symmetric group S_m. Fomin–Kirillov's conjecture. The algebra EuScriptE_m is infinite dimensional for all m≥ 6. This conjecture asks for the dimension of the Fomin–Kirillov algebra; it is known that EuScriptE_m is finite dimensional for all m≤ 5, while the asserted infinite dimensionality for m≥ 6 remains open.

Fomin–Kirillov's infinite-dimensionality conjecture

Fomin and Kirillov conjectured that $\mathcal{E}_m$ is infinite-dimensional for every $m\geq 6$ . The known literature continues to describe this as unresolved, with no reported proof or counterexample.

Known results

$\mathcal{E}_m$ is finite-dimensional for $m\leq 5$ ; known dimensions include $\dim\mathcal{E}_3=12$ , $\dim\mathcal{E}_4=576$ , and $\dim\mathcal{E}_5=8294400$ .
The Nichols-algebra identification is known for $m\leq 4$ by Milinski and Schneider and for $m=5$ by Graña.
For $m\geq 6$ , both infinite-dimensionality and the corresponding Nichols-algebra equality remain open.
A conditional criterion would imply infinite-dimensionality from some $m_0>5$ onward, but is not an unconditional proof.

Current status (as of September 2026): $\mathcal{E}_m$ is settled finite-dimensional for $m\leq 5$ , while the conjectured infinite-dimensionality for every $m\geq 6$ remains open with no reported public progress.

Fomin–Kirillov's infinite-dimensionality conjecture

Primary source

Progress summary

Solutions 0