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Symmetric Entropy Regions of Degrees Six and Seven

Zihan Li, Shaocheng Liu, Qi Chen

TL;DR

This work classifies when symmetry constraints render the G-symmetric almost entropic region exactly describable by Shannon-type inequalities for six and seven random variables. It develops an orbit-structure and fix_G framework to define $G$-symmetric entropy regions $\Psi_G^*$ and their outer bounds $\Psi_G$, and uses $H$- and $V$-representations along with computational tools to verify Shannon-tightness across representative subgroups. The main contributions are explicit lists of subgroups $G$ for which $\overline{\Psi_G^*}=\Psi_G$ and for which $\overline{\Psi_G^*}\subsetneq\Psi_G$ in degrees six and seven, complemented by Hasse diagrams and appendix details of the symmetric polymatroidal regions. These results illuminate when symmetry simplifies entropy characterizations, with potential implications for network information theory problems exhibiting permutation symmetry.

Abstract

In this paper, we classify all G-symmetric almost entropic regions according to their Shannon-tightness, that is, whether they can be fully characterized by Shannon-type inequalities, where G is a permutation group of degree 6 or 7.

Symmetric Entropy Regions of Degrees Six and Seven

TL;DR

This work classifies when symmetry constraints render the G-symmetric almost entropic region exactly describable by Shannon-type inequalities for six and seven random variables. It develops an orbit-structure and fix_G framework to define -symmetric entropy regions and their outer bounds , and uses - and -representations along with computational tools to verify Shannon-tightness across representative subgroups. The main contributions are explicit lists of subgroups for which and for which in degrees six and seven, complemented by Hasse diagrams and appendix details of the symmetric polymatroidal regions. These results illuminate when symmetry simplifies entropy characterizations, with potential implications for network information theory problems exhibiting permutation symmetry.

Abstract

In this paper, we classify all G-symmetric almost entropic regions according to their Shannon-tightness, that is, whether they can be fully characterized by Shannon-type inequalities, where G is a permutation group of degree 6 or 7.
Paper Structure (9 sections, 6 theorems, 21 equations, 11 figures)

This paper contains 9 sections, 6 theorems, 21 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Orbit structure
  4. Symmetries in the entropy space
  5. Symmetric entropy functions of degree six and seven
  6. Symmetric entropy functions of $6$ random variables
  7. Symmetric entropy functions of $7$ random variables
  8. More details about the symmetric polymatroidal regions
  9. The Hasse diagram of $P_6$ and $P_7$

Key Result

Theorem 1

partition Let $p=\{N_1, N_2, \cdots , N_t\}$ be a partition of $N$, that is, ${N_i}$, $i=1,2,\cdots ,t$ are disjoint and $\bigcup\limits_{i=1}^{t} N_i=N$. Let $G_p=S_{N_1}\times S_{N_2}\times \cdots \times S_{N_t}$, where $S_{N_i}$ are symmetric groups on ${N_i}$. For $\lvert N \rvert \geq 4$, if and only if $p=\{N\}$ or $\{\{i\},N\textbackslash\{i\}\}$ for some $i \in N$.

Figures (11)

  • Figure 1: Orbit structures
  • Figure 1: $\textfrak{O}_{\mathrm{PSL}_2(5)}$
  • Figure 2: The Hasse diagram of critical equivalence classes of $P_6$. The "green" nodes indicate $\overline{\Psi_G^*} = \Psi_G$, and "red" nodes indicate $\overline{\Psi_G^*}\subsetneq \Psi_G$.
  • Figure 2: $\textfrak{O}_{S_3\mathrm{wr}_2C_2}$
  • Figure 3: The Hasse diagram of critical part of $P_7$
  • ...and 6 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Example 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Lemma 1
  • proof
  • ...and 2 more