Finite vertical-strip expansion formula for shifted stable Grothendieck polynomials

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Let μ\mu be a strict partition. For strict partitions λ⊇μ\lambda\supseteq\mu with ℓ(λ)=ℓ(μ)\ell(\lambda)=\ell(\mu), let SDλ/μ\textsf{SD}_{\lambda/\mu} be the shifted skew diagram. A vertical strip is a subset of Z>0×Z>0\mathbb{Z}_{>0}\times\mathbb{Z}_{>0} containing at most one position in each row, and let c(λ/μ)c(\lambda/\mu) be the number of distinct columns occupied by positions in SDλ/μ\textsf{SD}_{\lambda/\mu}. Vertical-strip expansion conjecture. Then

GQμ(β)=2ℓ(μ)⋅∑λ(−1)c(λ/μ)⋅(−β/2)∣λ/μ∣⋅GPλ(β),G\hspace{-0.2mm}Q^{(\beta)}_\mu=2^{\ell(\mu)}\cdot\sum_{\lambda}(-1)^{c(\lambda/\mu)}\cdot(-\beta/2)^{|\lambda/\mu|}\cdot G\hspace{-0.2mm}P^{(\beta)}_\lambda,

where the sum is over the finite set of such strict partitions λ\lambda for which SDλ/μ\textsf{SD}_{\lambda/\mu} is a vertical strip. The source reports that computer calculations suggest this finite formula and gives it as the ensuing conjectural description of the expansion.

References

Primary source

Joel Brewster Lewis and Eric Marberg, “Enriched set-valued P-partitions and shifted stable Grothendieck polynomials”, arXiv:1907.10691 (2020).

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