Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Expanding the unicellular LLT polynomials of two-headed melting lollipops into ribbon Schurs

Victor Wang

TL;DR

The paper addresses expanding unicellular $LLT$ polynomials associated to a broader class of unit interval graphs, namely two-headed melting lollipops, into ribbon Schur functions. It derives a simple formula $LLT_{\mathbf a}(\mathbf x;q)=\sum_{\alpha \vDash n+m} q^{\mathbf b(\text{set}(\alpha))} r_\alpha$ for these graphs, with a modified area sequence $\mathbf b$, proven via double and triple inductions; it further translates this into an explicit Schur expansion using standard Young tableaux and their descent sets. This extends the earlier melting lollipop result of Huh, Nam, and Yoo by providing nonnegative ribbon Schur expansions in a larger graph class and gives a combinatorial handle on Schur-positivity for unicellular LLT polynomials. The findings strengthen the connections between LLT polynomials, ribbon Schur positivity, and unit interval graph combinatorics, offering tools for explicit Schur expansions and potential further generalizations.

Abstract

We prove a simple formula expanding the unicellular LLT polynomials of a class of graphs we call two-headed melting lollipops into ribbon Schur functions. Our work extends the Schur expansion originally found for melting lollipop graphs by Huh, Nam, and Yoo.

Expanding the unicellular LLT polynomials of two-headed melting lollipops into ribbon Schurs

TL;DR

The paper addresses expanding unicellular polynomials associated to a broader class of unit interval graphs, namely two-headed melting lollipops, into ribbon Schur functions. It derives a simple formula for these graphs, with a modified area sequence , proven via double and triple inductions; it further translates this into an explicit Schur expansion using standard Young tableaux and their descent sets. This extends the earlier melting lollipop result of Huh, Nam, and Yoo by providing nonnegative ribbon Schur expansions in a larger graph class and gives a combinatorial handle on Schur-positivity for unicellular LLT polynomials. The findings strengthen the connections between LLT polynomials, ribbon Schur positivity, and unit interval graph combinatorics, offering tools for explicit Schur expansions and potential further generalizations.

Abstract

We prove a simple formula expanding the unicellular LLT polynomials of a class of graphs we call two-headed melting lollipops into ribbon Schur functions. Our work extends the Schur expansion originally found for melting lollipop graphs by Huh, Nam, and Yoo.

Paper Structure

This paper contains 5 sections, 8 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Melting lollipop graphs
  4. Two-headed melting lollipop graphs
  5. Acknowledgments

Key Result

Theorem 2.5

HuhNamYoo Let area sequence $\mathbf a=(a_1,\dots,a_{n-1})$ and let $i$ be such that $a_{i-1}+1\le a_i$ (letting $a_0$ be $0$). Suppose area sequences $\mathbf a'$ and $\mathbf a"$ differ from $\mathbf a$ only in position $i$ with $a_i=a_i'+1=a_i"+2$. If $a_{i+a_i-1}=a_{i+a_i}+1$, then

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Proposition 2.7
  • Example 2.8
  • Definition 3.1
  • Definition 3.2
  • ...and 16 more