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A pipe dream perspective on totally symmetric self-complementary plane partitions

Daoji Huang, Jessica Striker

Abstract

We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams. Using this characterization and the recent bijection of [Gao-Huang] between reduced pipe dreams and reduced bumpless pipe dreams, we give a bijection between alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We also give a different bijection in the 1432- and 2143-avoiding case that preserves natural poset structures on the associated pipe dreams and bumpless pipe dreams.

A pipe dream perspective on totally symmetric self-complementary plane partitions

Abstract

We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams. Using this characterization and the recent bijection of [Gao-Huang] between reduced pipe dreams and reduced bumpless pipe dreams, we give a bijection between alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We also give a different bijection in the 1432- and 2143-avoiding case that preserves natural poset structures on the associated pipe dreams and bumpless pipe dreams.
Paper Structure (14 sections, 22 theorems, 13 equations, 9 figures, 1 table)

This paper contains 14 sections, 22 theorems, 13 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Alternating sign matrices
  4. Totally symmetric self-complementary plane partitions
  5. TSSCPP boolean triangles and a permutation case bijection
  6. Bumpless pipe dreams
  7. Pipe dreams
  8. Reduced BPD-PD bijection
  9. Characterizing TSSCPP as pseudo-Yamanouchi pipe dreams
  10. Mapping TSSCPP into pipe dreams
  11. A Yamanouchi-like condition on bounded compatible sequences
  12. A bijection between TSSCPP and ASM in the reduced, 1432-avoiding case
  13. A poset-preserving bijection in the $2143$- and $1432$-avoiding case
  14. Concluding remarks

Key Result

Theorem 1.1

Let $\pi\in S_n$. There is an explicit weight-preserving injection $\varphi$ from $\mathop{\mathrm{TSSCPP}}\nolimits^{red}(\pi)$ to $\mathop{\mathrm{ASM}}\nolimits^{red}(\pi)$. If $\pi$ avoids $1432$, then $\varphi$ is a bijection.

Figures (9)

  • Figure 1: An example of the bijection of this paper. From left to right the objects are: TSSCPP, pipe dream, bumpless pipe dream, ASM. The pipe dream and bumpless pipe dream both have permutation $135264$, which avoids $1432$. Note the black rhombi in column $k$ (from the left) of the TSSCPP fundamental domain correspond to the cross tiles in row $k$ (from the top) of the pipe dream. This equals the number of blank tiles in row $k$ of the bumpless pipe dream, which is the number of positive inversions of row $k$ of the ASM.
  • Figure 2: An alternating sign matrix and its corresponding square ice configuration, six-vertex configuration, and bumpless pipe dream.
  • Figure 3: An example of the permutation case bijection of PermTSSCPP
  • Figure 4: $\mathop{\mathrm{PD}}\nolimits^{red}(1432)$
  • Figure 5: An example of transforming a TSSCPP to a pipe dream. Note the weight of this TSSCPP is $x_2^2x_3x_4$.
  • ...and 4 more figures

Theorems & Definitions (55)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: Def 2.12 of PermTSSCPP
  • Proposition 2.7: Prop 2.13 of PermTSSCPP
  • Definition 2.8: Def 3.1 of PermTSSCPP
  • Theorem 2.9: Theorem 3.5 of PermTSSCPP
  • ...and 45 more