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Special values of Grothendieck polynomials in terms of hypergeometric functions

Taikei Fujii, Takahiko Nobukawa, Tatsushi Shimazaki

Abstract

We give some special values of Grothendieck polynomials and an explicit formula for the number of set-valued tableaux. For Young diagrams consisting of a single row or a single column, both the value and number are written by the Gauss' hypergeometric function ${}_2F_1$. For general Young diagrams, the Holman hypergeometric function $F^{(n)}$ is used to represent both the value and count. As an application, we derive a summation formula for $F^{(n)}$.

Special values of Grothendieck polynomials in terms of hypergeometric functions

Abstract

We give some special values of Grothendieck polynomials and an explicit formula for the number of set-valued tableaux. For Young diagrams consisting of a single row or a single column, both the value and number are written by the Gauss' hypergeometric function . For general Young diagrams, the Holman hypergeometric function is used to represent both the value and count. As an application, we derive a summation formula for .
Paper Structure (9 sections, 10 theorems, 48 equations)

This paper contains 9 sections, 10 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partitions and Young diagrams
  4. Semi-standard tableaux and Schur polynomials $s_\lambda$
  5. Set-valued tableaux and Grothendieck polynomials $G_\lambda$
  6. Special values of Grothendieck polynomials in terms of hypergeometric functions
  7. Special cases of Young diagrams
  8. General cases of Young diagrams
  9. Conclusions

Key Result

Proposition 3.1

For any $\lambda=(k)\ (k\in \mathbb{Z}_{>0})$, we have

Theorems & Definitions (20)

  • Definition 2.1
  • Example 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.1
  • Corollary 3.2
  • Theorem 3.1
  • proof
  • ...and 10 more