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Single-SEM Schubert Polynomials

Dora Woodruff

TL;DR

The paper addresses when Schubert polynomials simplify to a single standard elementary monomial or a single complete homogeneous monomial by translating these simplifications into pattern-avoidance criteria for the permutation $w$. It develops a framework based on Lehmer codes, pipe dreams, and divided-difference operators, proving that $S_w$ is a single SEM iff $w$ avoids $\{312,1432\}$ and $S_w$ is a single CHM iff $w$ avoids $\{321,231\}$, with the proofs leveraging bottom pipe dreams, ladder moves, and Monk's rule. Counting results show the number of such $w\in S_n$ equals $F_{2n}$ for SEMs and $2^{n-1}$ for CHMs (and $C_n$ for single monomials), linking these algebraic objects to classical combinatorial sequences. The work provides computationally advantageous cases for quantum Schubert polynomials and reveals structural insights via pipe-dream decompositions and pattern-avoidance.

Abstract

We give a pattern-avoidance characterization of $w \in S_n$ such that the Schubert polynomial $\mathfrak{S}_w$ is a standard elementary monomial. This characterization tells us which quantum Schubert polynomials are easiest to compute. We solve a similar problem for complete homogeneous monomials.

Single-SEM Schubert Polynomials

TL;DR

The paper addresses when Schubert polynomials simplify to a single standard elementary monomial or a single complete homogeneous monomial by translating these simplifications into pattern-avoidance criteria for the permutation . It develops a framework based on Lehmer codes, pipe dreams, and divided-difference operators, proving that is a single SEM iff avoids and is a single CHM iff avoids , with the proofs leveraging bottom pipe dreams, ladder moves, and Monk's rule. Counting results show the number of such equals for SEMs and for CHMs (and for single monomials), linking these algebraic objects to classical combinatorial sequences. The work provides computationally advantageous cases for quantum Schubert polynomials and reveals structural insights via pipe-dream decompositions and pattern-avoidance.

Abstract

We give a pattern-avoidance characterization of such that the Schubert polynomial is a standard elementary monomial. This characterization tells us which quantum Schubert polynomials are easiest to compute. We solve a similar problem for complete homogeneous monomials.

Paper Structure

This paper contains 9 sections, 17 theorems, 15 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Background
  4. Standard Elementary Monomials and Divided Difference Operators
  5. Permutations and Pipe Dreams
  6. Proofs of Theorems \ref{['patternavoidanceSEMs']} and \ref{['patternavoidanceCHMs']}
  7. Proof of Theorem \ref{['patternavoidanceSEMs']}
  8. Proof of Theorem \ref{['patternavoidanceCHMs']}
  9. Further Directions

Key Result

Theorem 1.1

The Schubert polynomial $\mathfrak{S}_w$ is a single monomial $x_1^{l_1}x_2^{l_2}\dots x_{n-1}^{l_{n-1}}$ if and only if the following equivalent conditions hold:

Figures (5)

  • Figure 1: The two (reduced) pipe dreams for $4132$. The pipe dream on the left is the bottom pipe dream of $4132$ because its crossings are left-adjusted, and indeed, $L(4132) = (3, 0, 1, 0)$.
  • Figure 2: A ladder move of order $4$
  • Figure 3: Above is the bottom pipe dream $P$ for $35427861$. (Cross signs denote squares with crossings; empty squares denote squares with elbows). We can construct $P$ by starting with the bottom pipe dream $P'$ of $34526781$ (in black) and adding the outer crossings (in red) to the outermost edge of $P'$. There are two outer columns of length 1 and 2.
  • Figure 4: The grey arrows represent the possible places where we can slide each red outer crossing. In this example, the outer columns contribute factors of $e^2_1$ and $e^6_2$ each. Then, the dominant bottom pipe dream $P'$ contributes $e^3_3e^7_7$, so we get $\mathfrak{S}_{35427861} = e^2_1e^3_3e^6_2e_7^7$.
  • Figure 5: Above is the bottom pipe dream for $1427356$ (cross signs denote crossings, and empty boxes denote non-crossings). If we draw a diagonal line out from the first crossing in row 3, then this diagonal line never intersects or is directly to the right of another crossing. However, the diagonal line emitting from the first box in row 5 enters the square directly to the right of the last crossing of row 3. Indeed, the subsequence $473$ is a $231$ pattern.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3: pipedreams
  • Definition 2.4
  • Theorem 2.5: pipedreams
  • Theorem 2.6
  • ...and 18 more