Entropies associated with orbits of finite groups
Ryan Leal, Jingtong Sun, Juan Pablo Vigneaux
TL;DR
This work builds a bridge between symmetry and information theory by studying orbit counts of flag-stabilizers in finite groups and extending entropy-like growth notions to broader group families. It introduces two new entropic functionals, reflective entropy $H_R(P)$ and symplectic entropy $H_{Sp}(P)$, derived from asymptotic quotients of finite reflection groups and the symplectic group, respectively, with explicit formulas and chain-rule analogues. The authors develop a two-tier framework—combinatorial (via Poincaré polynomial factorizations) and probabilistic (via coarse-graining and conditional structure)—to extend entropy concepts beyond the classical Shannon and Tsallis cases to Dynkin-type series $A,B,C,D$ and to isotropic subspace counts. The results unify algebraic counting with information-theoretic behavior, offering new functionals and chain rules with potential implications for combinatorics, representation theory, and entropy theory in discrete symmetry contexts.
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.