Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Newton polytopes of dual $k$-Schur polynomials

Bo Wang, Candice X. T. Zhang, Zhong-Xue Zhang

Abstract

Rado's theorem about permutahedra and dominance order on partitions reveals that each Schur polynomial is M-convex, or equivalently, it has a saturated Newton polytope and this polytope is a generalized permutahedron as well. In this paper we show that the support of each dual $k$-Schur polynomial indexed by a $k$-bounded partition coincides with that of the Schur polynomial indexed by the same partition, and hence the two polynomials share the same saturated Newton polytope. The main result is based on our recursive algorithm to generate a semistandard $k$-tableau for a given shape and $k$-weight. As consequences, we obtain the M-convexity of dual $k$-Schur polynomials, affine Stanley symmetric polynomials and cylindric skew Schur polynomials.

Newton polytopes of dual $k$-Schur polynomials

Abstract

Rado's theorem about permutahedra and dominance order on partitions reveals that each Schur polynomial is M-convex, or equivalently, it has a saturated Newton polytope and this polytope is a generalized permutahedron as well. In this paper we show that the support of each dual -Schur polynomial indexed by a -bounded partition coincides with that of the Schur polynomial indexed by the same partition, and hence the two polynomials share the same saturated Newton polytope. The main result is based on our recursive algorithm to generate a semistandard -tableau for a given shape and -weight. As consequences, we obtain the M-convexity of dual -Schur polynomials, affine Stanley symmetric polynomials and cylindric skew Schur polynomials.
Paper Structure (7 sections, 19 theorems, 28 equations, 11 figures)

This paper contains 7 sections, 19 theorems, 28 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $k+1$-cores and $k$-bounded partitions
  4. Dual $k$-Schur functions
  5. M-convexity of dual $k$-Schur polynomials
  6. M-convexity of affine Stanley symmetric polynomials
  7. Future directions

Key Result

Theorem 1.1

Let $d$ be a positive integer, and $\lambda, \, \mu$ be two partitions of $d$. Then $\mathcal{P}_{\mu}\subseteq \mathcal{P}_{\lambda}$ if and only if $\mu \trianglelefteq \lambda$ (meaning that $\mu$ is less than or equal to $\lambda$ in dominance order).

Figures (11)

  • Figure 2.1: The Young diagram of $\lambda=(4,4,4,2,1)$.
  • Figure 2.3: Two $3$-SSYTs.
  • Figure 3.1: The sets $A,\,B,\,C,\, D$ and cells $b_i,\,c_i,\,d_i$ for $i=1,\,2$ of shape $\mathfrak{c}(\lambda)$.
  • Figure 3.2: The construction of the SSYT $T$.
  • Figure 3.3: The filling operation $\mathfrak{c}(\lambda)\leftarrow \{5\}$.
  • ...and 6 more figures

Theorems & Definitions (35)

  • Theorem 1.1: Rado
  • Proposition 2.1: LM2005
  • Example 2.2
  • Proposition 2.3: LM2005
  • Proposition 2.4: LM2005
  • Proposition 2.5: StaEC2
  • Proposition 2.6: LM2007
  • Proposition 2.7: LM2008
  • Proposition 2.8: LM2005
  • Proposition 3.1
  • ...and 25 more