K-theoretic Schur P/Q polynomials

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#K-theoretic Schur P/Q polynomials

In [IN13], T. Ikeda and H. Naruse introduce $K$-theoretic analogs of the Schur $P$ and Schur $Q$ functions, as well as factorial versions of these. The factorial versions are denoted $\schurKP_{\lambda}(\xvec|b)$ and $\schurKQ_{\lambda}(\xvec|b),$ where $b = (b_1,b_2,\dotsc)$ is a vector of parameters. When all these are set to $0,$ we obtain $\schurKP_{\lambda}(\xvec)$ and $\schurKQ_{\lambda}(\xvec).$ These are $K$-theoretic analogs of $\schurP_{\lambda}(\xvec)$ and $\schurQ_{\lambda}(\xvec).$ Ikeda and Naruse give several formulas for these functions, including a ratio-of-Pfaffians formula, a shifted set-valued tableau formula, and an excited Young diagram formula.

They give a presentation for equivariant $K$-theory classes of the orthogonal and Lagrangian Grassmannian.

E. Marberg and B. Pawlowski give $K$-theory formulas for orthogonal and symplectic orbit closures [MP20]. Their stable Grothendieck analogues for fixed-point-free involutions expand positively in $K$-theoretic Schur $P$-functions, and related involution families are connected to $K$-theoretic Schur $Q$-functions. S. Iwao gives a boson–fermion correspondence for dual $K$-theoretic $P$- and $Q$-functions [Iwa27]. The construction uses $\beta$-deformed neutral fermions and vertex operators, and recovers generating functions for the dual $K$-theoretic Schur $P$- and $Q$-functions.

J. B. Lewis and E. Marberg prove the Nakagawa–Naruse conjecture that the dual $K$-theoretic Schur $P$- and $Q$-functions are generating functions for shifted plane partitions [LM24]. They also describe the images of these functions under the involution $\omega$ and verify an Ikeda–Naruse basis conjecture for the $GQ$-functions.

Y.-C. Chiu and E. Marberg prove identities expanding $K$-theoretic Schur $Q$-functions in terms of $K$-theoretic Schur $P$-functions [CM24]. Their main formula is a signed finite expansion, and they characterize when its coefficients lie in $\setN[\beta]$; see [Thm. 1.1 and Cor. 1.2, CM24]. Their formulas extend to skew and dual versions, and they also prove a shifted skew Cauchy identity for symmetric Grothendieck polynomials. E. Marberg proves that the $K$-Stanley symmetric functions of classical types expand positively into these $K$-theoretic Schur $P$- and $Q$-functions [Mar25].

Example (A $Q$-to-$P$ expansion).

In the notation of [Thm. 1.1, CM24], the coefficient of $\schurKP_\lambda$ in $\schurKQ_\mu$ is obtained by summing over strict partitions $\lambda \supseteq \mu$ with $\ell(\lambda)=\ell(\mu)$ such that the shifted skew diagram $\lambda/\mu$ has at most one box in each row. The coefficient is \[2^{\ell(\mu)}(-1)^{\operatorname{col}(\lambda/\mu)} \left(-\frac{\beta}{2}\right)^{|\lambda/\mu|},\] where $\operatorname{col}(\lambda/\mu)$ is the number of occupied shifted columns. For $\mu=(3,2),$ the only strict partitions which contribute are $(3,2),$ $(4,2),$ and $(4,3),$ so \[\schurKQ_{(3,2)} =4\schurKP_{(3,2)} +2\beta\schurKP_{(4,2)} -\beta^2\schurKP_{(4,3)}.\] The negative term occurs because the two boxes of the shifted skew diagram $(4,3)/(3,2)$ lie in the same shifted column. In contrast, every part of $(4,2)$ differs from the next by at least two, and the same rule gives only nonnegative $\beta$-coefficients: \[\schurKQ_{(4,2)} =4\schurKP_{(4,2)} +2\beta\schurKP_{(4,3)} +2\beta\schurKP_{(5,2)} +\beta^2\schurKP_{(5,3)}.\]

#Properties

The fundamental quasisymmetric expansion of $\schurKP_{\lambda}(\xvec)$ and $\schurKQ_{\lambda}(\xvec)$ can be found in [HKPW+17].

T. Nobukawa and T. Shimazaki compute special values of skew $K$-theoretic Schur $P$- and $Q$-functions [NS24]. For strict partitions $\lambda \supset \mu,$ their main evaluation gives $\schurKP_{\lambda/\mu}(\beta,\dotsc,\beta\mid -\beta^{-1}) = \schurKQ_{\lambda/\mu}(\beta,\dotsc,\beta\mid -\beta^{-1}) = \beta^{|\lambda/\mu|}$ in their notation. The proof uses sign-reversing involutions on shifted set-valued tableaux and implies oddness results for the corresponding tableau counts.

#Littlewood–Richardson rule

The Littlewood–Richardson rule for $\schurKP_{\lambda}(\xvec)$ is due to Clifford–Thomas–Yong [CTY14]. Their result is based on a Pieri rule due to A. Buch and V. Ravikumar, [BR12].

Problem (Littlewood–Richardson rule).

Find a Littlewood–Richardson rule for $\schurKQ_{\lambda}(\xvec).$

Bibliography

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[CTY14]Edward Clifford, Hugh Thomas and Alexander Yong. K-theoretic Schubert calculus for ${OG}(n, 2n+1)$ and jeu de taquin for shifted increasing tableaux. Journal für die reine und angewandte Mathematik (Crelles Journal), 2014(690), January 2014.

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