2022/09/13 by Theo Douvropoulos, Douvropoulos, Theo
Computer Science · Mathematics · #05E99 #20F55 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2209.06201
openalex publication_date 2022/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a finite Coxeter group W and with two given conjugacy classes of parabolic subgroups [X] and [Y], we count those parabolic subgroups of W in [Y] that are full support, while simultaneously being simple extensions (i.e., extensions by a single reflection) of some standard parabolic subgroup of W in [X]. The enumeration is given by a product formula that depends only on the two parabolic types. Our derivation is case-free and combines a geometric interpretation of the "full support" property with a double counting argument involving Crapo's beta invariant. As a corollary, this approach gives the first case-free proof of Chapoton's formula for the number of reflections of full support in a real reflection group W.