Let P denote the set of partitions, let widetildeG_lambda be the modified stable Grothendieck symmetric function, and let leqslant_operatornameg denote Grothendieck positivity. Thomas–Yong–Mihalcea's Grothendieck positivity conjecture. For all partitions lambda,mu∈ P, one has widetildeG_lambda,widetildeG_muleqslant_operatornamegwidetildeG_lambda veemu,widetildeG_lambda wedgemu. The corresponding Schur-positivity inequality is proved in the paper, whereas the stronger Grothendieck-positivity assertion remains open.

Thomas–Yong–Mihalcea Grothendieck positivity inequality

Less than 1 year old · traced to 2026

Let $\mathcal P$ denote the set of partitions, let $\widetilde{G}_{\lambda}$ be the modified stable Grothendieck symmetric function, and let $\leqslant_{\operatorname{g}}$ denote Grothendieck positivity. Thomas–Yong–Mihalcea's Grothendieck positivity conjecture. For all partitions $\lambda,\mu\in\mathcal P$ , one has

\widetilde{G}_{\lambda}\\,\widetilde{G}_{\mu}\leqslant_{\operatorname{g}}\widetilde{G}_{\lambda\vee\mu}\\,\widetilde{G}_{\lambda\wedge\mu}.

The corresponding Schur-positivity inequality is proved in the paper, whereas the stronger Grothendieck-positivity assertion remains open.

References

Primary source

Swee Hong Chan, Hong Chen, Igor Pak and Daniel Soskin, “Correlation inequalities for Schur positivity”, arXiv:2606.06688 (2026).

Posted 1 month ago

Progress summary

Refreshed 4 weeks ago

Open

The stronger version of a known positivity inequality remains open; recent papers prove only a weaker version and report no proof or counterexample.

Thomas–Yong formulated the Grothendieck-positivity conjecture, also associated in the literature with Mihalcea's 2024 work. It asserts that the product indexed by two partitions is bounded, in Grothendieck positivity, by the products indexed by their join and meet.

Known results

Theorem 1.8 proves the corresponding Schur-positivity inequality $\widetilde{G}_{\lambda}\widetilde{G}_{\mu}\leqslant_{\mathrm{s}}\widetilde{G}_{\lambda\vee\mu}\widetilde{G}_{\lambda\wedge\mu}$ for all partitions $\lambda,\mu$ .
The same source explicitly treats the stronger Grothendieck-positivity assertion as open and presents the Schur result only as evidence for it.

Current status (as of September 2026): The Schur-positivity inequality is proved, but the Grothendieck-positivity conjecture remains open, with no publicly reported proof, counterexample, or claimed resolution.

Sources

No solutions have been posted yet.