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Puzzle Ideals for Grassmannians

Chenqi Mou, Weifeng Shang

Abstract

Puzzles are a versatile combinatorial tool to interpret the Littlewood-Richardson coefficients for Grassmannians. In this paper, we propose the concept of puzzle ideals whose varieties one-one correspond to the tilings of puzzles and present an algebraic framework to construct the puzzle ideals which works with the Knutson-Tao-Woodward puzzle and its $T$-equivariant and $K$-theoretic variants for Grassmannians. For puzzles for which one side is free, we propose the side-free puzzle ideals whose varieties one-one correspond to the tilings of side-free puzzles, and the elimination ideals of the side-free puzzle ideals contain all the information of the structure constants for Grassmannians with respect to the free side. Besides the underlying algebraic importance of the introduction of these puzzle ideals is the computational feasibility to find all the tilings of the puzzles for Grassmannians by solving the defining polynomial systems, demonstrated with illustrative puzzles via computation of Gröbner bases.

Puzzle Ideals for Grassmannians

Abstract

Puzzles are a versatile combinatorial tool to interpret the Littlewood-Richardson coefficients for Grassmannians. In this paper, we propose the concept of puzzle ideals whose varieties one-one correspond to the tilings of puzzles and present an algebraic framework to construct the puzzle ideals which works with the Knutson-Tao-Woodward puzzle and its -equivariant and -theoretic variants for Grassmannians. For puzzles for which one side is free, we propose the side-free puzzle ideals whose varieties one-one correspond to the tilings of side-free puzzles, and the elimination ideals of the side-free puzzle ideals contain all the information of the structure constants for Grassmannians with respect to the free side. Besides the underlying algebraic importance of the introduction of these puzzle ideals is the computational feasibility to find all the tilings of the puzzles for Grassmannians by solving the defining polynomial systems, demonstrated with illustrative puzzles via computation of Gröbner bases.
Paper Structure (15 sections, 17 theorems, 19 equations, 21 figures)

This paper contains 15 sections, 17 theorems, 19 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schur polynomial and Littlewood-Richardson coefficient
  4. Knutson-Tao-Woodward puzzle for the Littlewood-Richardson rule
  5. $T$-equivariant puzzle for equivariant cohomology of Grassmannians
  6. Puzzles for $K$-theory of Grassmannians
  7. Atomic puzzle ideal for Knutson-Tao-Woodward puzzle
  8. Forbidding puzzle ideal for $T$-equivariant puzzle
  9. Implying puzzle ideals for $K$-theoretic puzzles
  10. Puzzle ideal
  11. Atomic refinement
  12. Atomic puzzle ideal
  13. Forbidden and implicit puzzle pieces
  14. Puzzle ideal
  15. Side-free puzzle ideal

Key Result

Theorem 2.4

Let $\lambda$, $\mu$, and $\nu$ be partitions in $\genfrac{\{}{\}}{0pt}{}{n}{k}$. Then $\#\mathop{\mathrm{S}}\nolimits(P^{\nu, \Omega_0}_{\lambda \mu}) = c_{\lambda \mu}^{\nu}$.

Figures (21)

  • Figure 1: All the semi-standard Young tableaux of shape $(3, 1)$
  • Figure 2: The binary sequence for $(3, 1)$
  • Figure 3: An illustrative $\bigtriangleup^{\nu}_{\lambda \mu}$
  • Figure 4: The puzzle pieces for Knutson-Tao-Woodward puzzles
  • Figure 5: One tiling of Knutson-Tao-Woodward puzzle $P^{\nu, \Omega_0}_{\lambda \mu}$
  • ...and 16 more figures

Theorems & Definitions (41)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4: K04h
  • Theorem 2.5: K03p
  • Theorem 2.6: V06gW19l
  • Theorem 2.7: P20p
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • ...and 31 more