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Opposite Schubert localization formula in equivariant K-theory

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Consider the Grassmannian \limfuncGrn,N\limfunc{Gr}_{n,N} and the ring

KT(\limfuncGrn,N)=qhn∗/⟨q,\Greekmath010C+1,tj−1+e\Greekmath0122N+1−i⟩.K_{\mathbb{T}}(\limfunc{Gr}_{n,N})=qh_n^{\ast}/\langle q,{\Greekmath 010C}+1,t_j-1+e^{{\Greekmath 0122}_{N+1-i}}\rangle.

For partitions \Greekmath0115{\Greekmath 0115} and \Greekmath0116{\Greekmath 0116} indexing opposite Schubert varieties and fixed points, let G\Greekmath0115∨G_{{\Greekmath 0115}^{\vee}} and G\Greekmath0115∗G_{{\Greekmath 0115}^{\ast}} denote the corresponding factorial Grothendieck polynomials, and let t′t', ⊖t\ominus t, and ⊖t\Greekmath0116∗\ominus t_{{\Greekmath 0116}^{\ast}} use the formal inverse operation of the generalized cohomology theory. Opposite Schubert localization conjecture. The localized class of the sheaf O\Greekmath0115\mathcal{O}^{{\Greekmath 0115}} of the opposite Schubert variety is

[O\Greekmath0115]\Greekmath0116=Π(t∅)Π(t\Greekmath0115)G\Greekmath0115∨(t\Greekmath0116∣⊖t′)=Π(t\Greekmath0115∗)Π(t(kn))G\Greekmath0115∗(⊖t\Greekmath0116∗∣t).[\mathcal{O}^{{\Greekmath 0115}}]_{{\Greekmath 0116}}=\frac{\Pi(t_{\emptyset})}{\Pi(t_{{\Greekmath 0115}})}G_{{\Greekmath 0115}^{\vee}}(t_{{\Greekmath 0116}}\mid\ominus t')=\frac{\Pi(t_{{\Greekmath 0115}^{\ast}})}{\Pi(t_{(k^n)})}G_{{\Greekmath 0115}^{\ast}}(\ominus t_{{\Greekmath 0116}^{\ast}}\mid t).

This conjecture proposes the equivariant K-theoretic analogue of the known non-equivariant opposite Schubert relations and is motivated by the preceding dual-basis and product identities; the source does not state that it has been proved.

References

Primary source

Vassily Gorbounov and Christian Korff, “Quantum Integrability and Generalised Quantum Schubert Calculus”, arXiv:1408.4718 (2017).

equivariant-k-theorygrassmanniansschubert-calculus
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Progress summary

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No public proof or counterexample has been found, although related localization formulas exist and the exact conjecture remains open.

The conjecture proposes an equivariant KK-theoretic formula for restrictions of opposite Schubert classes on a Grassmannian, expressed through factorial Grothendieck polynomials. No source retrieved attributes its formulation to a particular mathematician or states that it has been proved.

Known results

  • Kreiman, 2005: obtained equivariant KK-theory restrictions of opposite Schubert varieties using double Grothendieck polynomials.
  • Buch, Kresch, and Tamvakis, 2015: proved combinatorial rules for equivariant KK-theory structure constants, including a special localization case.
  • Hudson, Ikeda, Matsumura, and Naruse, 2021: established factorial Grothendieck-polynomial representatives for Grassmannian Schubert classes and discussed fixed-point localizations.

Current status (as of August 2026): The exact factorial-Grothendieck localization identity remains unproved and unrefuted; related restriction and localization formulas do not establish its equivalence to the conjecture.

Sources

Solutions 0

No solutions have been posted yet.