Let r,N>0, let S=C^2, and set E=bigoplus_i=1^NO_S langle y_i rangle and V=bigoplus_i=1^rO_S langle v_i rangle. Strong Segre symmetry. The Segre series satisfies S_S(E,V;(-1)^Nq)=S_S(V,E;(-1)^rq). This conjecture strengthens the weak symmetry established earlier in the paper and is motivated by symmetry in higher-degree Segre series. Its resolution is not given in the source.

Strong Segre symmetry for equivariant Quot schemes of the plane

About 3 years old · traced to 2023

Let $r,N>0$ , let $S=\mathbb{C}^2$ , and set $E=\bigoplus_{i=1}^N\mathcal{O}_S\langle y_i\rangle$ and $V=\bigoplus_{i=1}^r\mathcal{O}_S\langle v_i\rangle$ . Strong Segre symmetry. The Segre series satisfies

{\mathcal{S}}_{S}(E,V;(-1)^Nq)={\mathcal{S}}_{S}(V,E;(-1)^rq).

This conjecture strengthens the weak symmetry established earlier in the paper and is motivated by symmetry in higher-degree Segre series. Its resolution is not given in the source.

References

Primary source

Arkadij Bojko and Jiahui Huang, “Equivariant Segre and Verlinde invariants for Quot schemes”, arXiv:2303.14266 (2023).

Posted 1 month ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Open

No public proof or counterexample has been found; the symmetry remains an open conjecture.

Bojko and Huang formulate the conjecture that the two equivariant Segre series agree after the stated sign changes, namely $\mathcal{S}_{S}(E,V;(-1)^Nq)=\mathcal{S}_{S}(V,E;(-1)^rq)$ . Their preprint presents it as a strengthening of an earlier weaker symmetry.

Known results

Bojko and Huang establish the weaker symmetry and extend the equivariant Segre–Verlinde correspondence to all degrees and reduced virtual classes; the strong symmetry itself remains conjectural.

Current status (as of September 2026): The weaker symmetry and broader correspondence are reported, but the strong symmetry remains neither proved nor disproved in the retrieved sources.

Sources

No solutions have been posted yet.