Open AccessComplexity and randomness in the Heisenberg groups (and beyond)
Author(s) -
Persi Diaconis,
M. Malliaris
Publication year - 2021
Publication title -
new zealand journal of mathematics
Language(s) - English
Resource type - Journals
eISSN - 1179-4984
pISSN - 1171-6096
DOI - 10.53733/134
Subject(s) - randomness , conjugacy class , heisenberg group , limiting , mathematics , limit (mathematics) , combinatorics , character (mathematics) , discrete mathematics , pure mathematics , mathematical analysis , statistics , mechanical engineering , geometry , engineering
By studying the commuting graphs of conjugacy classes of the sequence of Heisenberg groups $H_{2n+1}(p)$ and their limit $H_\infty(p)$ we find pseudo-random behavior (and the random graph in the limiting case). This makes a nice case study for transfer of information between finite and infinite objects. Some of this behavior transfers to the problem of understanding what makes understanding the character theory of the uni-upper-triangular group (mod p) “wild.” Our investigations in this paper may be seen as a meditation on the question: is randomness simple or is it complicated?