Asymptotic essential-dimension ratio of the symmetric groups

Upper bound, last-listed row

| $1$ | [BR1997] | Follows from $ed_{\mathbb C}(S_n)\le n-3$ for all $n\ge 5$, hence $C_{79}\le \limsup_{n\to\infty}(n-3)/n = 1$. [BR1997-sym-bounds] |

exact pinned row

| $1$ | <a href="#BR1997">[BR1997]</a> | Follows from $ed_{\mathbb C}(S_n)\le n-3$ for all $n\ge 5$, hence $C_{79}\le \limsup_{n\to\infty}(n-3)/n = 1$. <a href="#BR1997-sym-bounds">[BR1997-sym-bounds]</a> |

tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $1/2$ | [BR1997], [ER2024] | In characteristic $0$, Buhler–Reichstein proved $ed_k(S_n)\ge \lfloor n/2 \rfloor$, and Duncan improved this to $ed_k(S_n)\ge \lfloor (n+1)/2 \rfloor$ for $n\ge 7$; ER2024 notes the $\lfloor n/2 \rfloor$ argument also works for $\mathrm{char}(k)\ne 2$. In particular, over $\mathbb C$ either estimate yields $C_{79}\ge 1/2$. [ER2024-intro-bounds-open] [BR1997-sym-bounds] |

exact pinned row

| $1/2$ | <a href="#BR1997">[BR1997]</a>, <a href="#ER2024">[ER2024]</a> | In characteristic $0$, Buhler–Reichstein proved $ed_k(S_n)\ge \lfloor n/2 \rfloor$, and Duncan improved this to $ed_k(S_n)\ge \lfloor (n+1)/2 \rfloor$ for $n\ge 7$; ER2024 notes the $\lfloor n/2 \rfloor$ argument also works for $\mathrm{char}(k)\ne 2$. In particular, over $\mathbb C$ either estimate yields $C_{79}\ge 1/2$. <a href="#ER2024-intro-bounds-open">[ER2024-intro-bounds-open]</a> <a href="#BR1997-sym-bounds">[BR1997-sym-bounds]</a> |

tracked since 2026-07-24 — no movement seen yet