Identification of set-valued decomposition tableau polynomials

Prove that, for every strict partition $\lambda$, the weight-generating polynomial of semistandard set-valued decomposition tableaux of shape $\lambda$ coincides with the $K$-theoretic Schur $P$-function associated with $\lambda$.

Background

For a strict partition λ\lambda, the paper defines a polynomial by summing the monomial weights of semistandard set-valued decomposition tableaux of shape λ\lambda. Prior work cited in the paper establishes only a unitriangular relation between this polynomial and the corresponding KK-theoretic Schur PP-function.

The paper proves that the decomposition-tableau polynomial is Grothendieck-positive and verifies the conjectured equality for one-row strict partitions, but it does not establish the equality for arbitrary strict partitions.

References

Cho and Ikeda have conjectured Conj.~3.2 that this polynomial coincides with the {$K$-theoretic Schur $P$-function} $-0.2mm_\lambda(x_1,x_2,\dots,x_n)$ introduced in , which is the weight-generating function for a different family of {(semistandard) set-valued marked shifted tableaux}.

— Grothendieck positivity for normal square root crystals  (2501.16640 - Marberg et al., 28 Jan 2025) in Section 1, subsection “Generating functions of set-valued decomposition tableaux”