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.

翻译：对于特定群，抛物子群表现为集合或向量空间旗的稳定子。这些抛物子群的商表示旗的轨道，其基数渐近地揭示了熵（作为指数或超指数增长的速率）。涉及这些基数的乘法“链式法则”在渐近意义上诱导了熵的加法类比。信息论中许多传统公式对应于对称群的商，对称群是一类特殊的反射群；此时，轨道基数由多项式系数给出，并与香农熵渐近相关。类似地可处理有限域上一般线性群的商；此时，轨道基数由$q$-多项式给出，并与Tsallis 2-熵渐近相关。本文中，我们考虑其他有限反射群以及作为有限域上典型群（李型群）例子的辛群。两种情形下，群均按Dynkin图分类为无限系列相似群$A_n$、$B_n$、$C_n$、$D_n$及有限个例外群。$A_n$系列包含对称群（反射情形）和一般线性群（李情形）。其他系列中部分群系首次从信息论视角进行研究，它们关联于新的熵泛函。