Entropies associated with orbits of finite groups

TL;DR

This paper explores the asymptotic entropies associated with orbits of finite reflection and classical groups, extending classical information-theoretic formulas to new algebraic structures and entropic functionals.

Contribution

It introduces a novel information-theoretic perspective on finite reflection groups and classical groups over finite fields, linking orbit cardinalities to new entropic functionals.

Findings

Asymptotic orbit cardinalities relate to known entropies like Shannon and Tsallis.

Extends entropy concepts to new group classes beyond symmetric and linear groups.

Identifies connections between group orbit structures and novel entropic measures.

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.