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Generalized Matrix polynomials of Tree Laplacians indexed by Symmetric functions and the GTS poset

Mukesh Kumar Nagar, Sivaramakrishnan Sivasubramanian

Abstract

Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_T^q$ and Laplacian matrix $L_T$. Let $GTS_n$ be the generalized tree shift poset on the set of unlabelled trees on $n$ vertices. Inequalities are known between coefficients of the immanantal polynomial of $L_T$ (and $L_T^q$) as we go up the poset $GTS_n$. Using the Frobenius characteristic, this can be thought as a result involving the schur symmetric function $s_λ$. In this paper, we use an arbitrary symmetric function to define a {\it generalized matrix function} of an $n \times n$ matrix. When the symmetric function is the monomial and the forgotten symmetric function, we generalize such inequalities among coefficients of the generalized matrix polynomial of $L_T^q$ as we go up the $GTS_n$ poset.

Generalized Matrix polynomials of Tree Laplacians indexed by Symmetric functions and the GTS poset

Abstract

Let be a tree on vertices with -Laplacian and Laplacian matrix . Let be the generalized tree shift poset on the set of unlabelled trees on vertices. Inequalities are known between coefficients of the immanantal polynomial of (and ) as we go up the poset . Using the Frobenius characteristic, this can be thought as a result involving the schur symmetric function . In this paper, we use an arbitrary symmetric function to define a {\it generalized matrix function} of an matrix. When the symmetric function is the monomial and the forgotten symmetric function, we generalize such inequalities among coefficients of the generalized matrix polynomial of as we go up the poset.

Paper Structure

This paper contains 5 sections, 19 theorems, 19 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. GMF of $q$-Laplacians arising from symmetric functions
  4. GMF of tree Laplacians arising from $m_{\lambda}$, the monomial symmetric function
  5. GMF of tree Laplacians arising from $f_{\lambda}$, the forgotten symmetric function

Key Result

Theorem 1

Let $T_1$ and $T_2$ be two trees with $n$ vertices such that $T_2$ covers $T_1$ in $\mathsf{GTS}_n$. Let $\mathcal{ L}_{T_1}^q$ and $\mathcal{ L}_{T_2}^q$ be the $q$-Laplacians of $T_1$ and $T_2$ respectively. For $\lambda \vdash n$, let Then for all $\lambda \vdash n$, we assert that $c_{m_{\lambda},r}^{\mathcal{ L}_{T_1}^q}(q) - c_{m_{\lambda},r}^{\mathcal{ L}_{T_2}^q}(q) \in \mathbb{R}^+[q^2]

Theorems & Definitions (21)

  • Theorem 1
  • Lemma 2
  • Lemma 3: Chan and Lam, chan-lam-binom-coeffs-char
  • Theorem 4: Chan and Lam, hook_immanant_explained-chan_lam
  • Theorem 5: Nagar and Sivasubramanian
  • Lemma 6: Nagar and Sivasubramanian
  • Theorem 7: Nagar and Sivasubramanian
  • Theorem 8
  • Lemma 9
  • Lemma 10
  • ...and 11 more