Counting conjugacy classes of elements of finite order in p-compact groups

TL;DR

The paper expresses the sets $[B{\mathbb Z}/p^n,BX]$ for connected $p$-compact groups in terms of the Weyl group action on the maximal torus and derives exact counts in both non-modular and modular (exotic) cases. It develops a uniform framework using the lattice $L$, the Weyl group $W$, Burnside counting, and torsion data in cokernels to compute the cardinalities, with explicit closed-form formulas for several modular exotic groups and for generalized Grassmannians in family 2a. The results illuminate the structure of representations of cyclic $p$-groups into exotic $p$-compact groups, providing concrete polynomial formulas in $p^n$ for five modular exceptional cases and a general counting strategy adaptable to GAP and related computations. These findings connect the homotopy-theoretic study of $p$-compact groups with finite reflection groups and yield precise enumerative data relevant to the homotopical classification, with potential implications for understanding vacua in related quantum field theories.

Abstract

We express the set of representations from a cyclic $p$-group to a connected $p$-compact group in terms of the associated reflection group and compute its cardinality for each exotic $p$-compact group.

Theorems & Definitions (62)

• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Proposition 1.3
• proof
• Corollary 1.4
• proof
• Lemma 1.5
• proof