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Special values of $K$-theoretic Schur $P$- and $Q$-functions

Takahiko Nobukawa, Tatsushi Shimazaki

TL;DR

The paper determines exact special values for skew and set-valued shifted $K$-theoretic Schur functions, proving that $GP_{oldsymbol{ u}/oldsymbol{ au}}(eta, frac{}{}eta, frac{}{}eta, frac{}{}eta, frac{}{}eta igm| -eta^{-1})$ and $GQ_{oldsymbol{ u}/oldsymbol{ au}}(eta, frac{}{}eta, frac{}{}eta, frac{}{}eta, frac{}{}eta igm| -eta^{-1})$ equal the monomial $eta^{|oldsymbol{ u}/oldsymbol{ au}|}$, via sign-reversing involutions on shifted set-valued skew tableaux. This yields that the number of such tableaux is odd when nonzero and extends the evaluation to a second skew type, while also showing that a related double-skew family vanishes under the same specialization. The work builds on and extends prior results for Grothendieck and shifted functions, linking combinatorial tableaux with $K$-theoretic geometry and providing a robust involutive cancellation framework. Potential directions include seeking bi-alternant formulas and exploring connections to hypergeometric functions for explicit counting, as well as further combinatorial bijections induced by the involutions.

Abstract

We provide the special values of the skew version of the $K$-theoretic Schur $P$- and $Q$-functions. Using these special values, we show an oddness property of the number of shifted set-valued skew tableaux. Additionally, we generalize these special values to another skew case. Based on these special values, we give pairs among certain shifted set-valued skew tableaux.

Special values of $K$-theoretic Schur $P$- and $Q$-functions

TL;DR

The paper determines exact special values for skew and set-valued shifted -theoretic Schur functions, proving that and equal the monomial , via sign-reversing involutions on shifted set-valued skew tableaux. This yields that the number of such tableaux is odd when nonzero and extends the evaluation to a second skew type, while also showing that a related double-skew family vanishes under the same specialization. The work builds on and extends prior results for Grothendieck and shifted functions, linking combinatorial tableaux with -theoretic geometry and providing a robust involutive cancellation framework. Potential directions include seeking bi-alternant formulas and exploring connections to hypergeometric functions for explicit counting, as well as further combinatorial bijections induced by the involutions.

Abstract

We provide the special values of the skew version of the -theoretic Schur - and -functions. Using these special values, we show an oddness property of the number of shifted set-valued skew tableaux. Additionally, we generalize these special values to another skew case. Based on these special values, we give pairs among certain shifted set-valued skew tableaux.

Paper Structure

This paper contains 9 sections, 8 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strict partitions and shifted Young diagrams
  4. Shifted semistandard tableaux and Schur $P$- and $Q$-functions
  5. Shifted set-valued tableaux and $K$-theoretic Schur $P$- and $Q$-functions
  6. Shifted set-valued skew tableaux and $K$-theoretic skew Schur $P$- and $Q$-functions
  7. Special values of $GP_{\lambda/\mu}$ and $GQ_{\lambda/\mu}$
  8. Special values of $GP_{\lambda/\!\!/\mu}$ and $GQ_{\lambda/\!\!/\mu}$
  9. Conclusions

Key Result

Theorem 3.1

Let $\lambda$ and $\mu$ be partitions such that $\lambda \supset \mu$. We have

Theorems & Definitions (24)

  • Definition 2.1
  • Example 2.1
  • Definition 2.2: IN13
  • Example 2.2
  • Theorem 3.1: FNS23
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • ...and 14 more