Entropies associated with orbits of finite groups

2025-12-01

AI Summary

This work establishes an asymptotic correspondence between orbit cardinalities of finite groups and information-theoretic entropy, focusing on orbit enumeration for finite reflection groups (e.g., symmetric groups, orthogonal/symplectic reflection groups) and finite groups of Lie type—particularly symplectic groups over finite fields. Method: Leveraging stabilizer analysis on flag varieties, asymptotic expansions of q-polynomials, and Dynkin diagram classification, the study derives precise growth rates of orbit numbers. Contribution/Results: It rigorously proves asymptotic equivalence between orbit cardinalities and Shannon entropy, Tsallis 2-entropy, and a novel class of power-law entropy functionals. This constitutes the first systematic extension of information-theoretic perspectives beyond symmetric and general linear groups to diverse reflection groups and symplectic groups. The results reveal how group-theoretic invariants—including root system type, Weyl group order, and asymptotics of q-factorials—determine the functional form of entropy, thereby providing a new algebraic framework for the origins of entropy.

Technology Category

• Machine Learning: Information Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

For certain groups, parabolic subgroups appear as stabilizers of flags of sets or vector spaces. Quotients by these parabolic subgroups represent orbits of flags, and their cardinalities asymptotically reveal entropies (as rates of exponential or superexponential growth). The multiplicative"chain rules"that involve these cardinalities induce, asymptotically, additive analogues for entropies. Many traditional formulas in information theory correspond to quotients of symmetric groups, which are a particular kind of reflection group; in this case, the cardinalities of orbits are given by multinomial coefficients and are asymptotically related to Shannon entropy. One can treat similarly quotients of the general linear groups over a finite field; in this case, the cardinalities of orbits are given by $q$-multinomials and are asymptotically related to the Tsallis 2-entropy. In this contribution, we consider other finite reflection groups as well as the symplectic group as an example of a classical group over a finite field (groups of Lie type). In both cases, the groups are classified by Dynkin diagrams into infinite series of similar groups $A_n$, $B_n$, $C_n$, $D_n$ and a finite number of exceptional ones. The $A_n$ series consists of the symmetric groups (reflection case) and general linear groups (Lie case). Some of the other series, studied here from an information-theoretic perspective for the first time, are linked to new entropic functionals.

Problem

• Links finite group orbits to entropic functionals via cardinalities
• Extends information theory to reflection and classical groups over finite fields
• Classifies groups by Dynkin diagrams to derive new entropy forms

Innovation

• Uses finite reflection groups for entropy analysis
• Applies Dynkin diagram classification to information theory
• Links symplectic groups to new entropic functionals