higher group character

A $k$-character, $k\in \mathbb{N}$, of a group $G$ is a certain function

$${\chi}^{(k)}:{G}^{k}\to \u2102$$

which descends to a function of *$k$-conjugacy classes*.

A $k$-conjugacy class in ${G}^{k}$ is a minimal subset of ${G}^{k}$ which is closed under $k$-fold conjugation in the following sense:

$$[{g}_{1},\cdots ,{g}_{k}]:=\{(g{\prime}_{1},\cdots ,g{\prime}_{k})\in {G}^{k}\mid \exists ({h}_{1},\cdots {h}_{k})\in {G}^{k}:g{\prime}_{1}={h}_{1}{g}_{1}{h}_{2}^{-1}\vee g{\prime}_{2}={h}_{2}{g}_{2}{h}_{3}^{-1}\vee \cdots g{\prime}_{k}={h}_{k}{g}_{2}{h}_{1}^{-1}\}\phantom{\rule{thinmathspace}{0ex}}.$$

The quotient map which sends elements in ${G}^{k}$ to their $k$-conjugacy class is called (at least for $k=2$) the weak Cayley table of $G$.

- Evidently, on any $k$-conjugacy class of $G$ we canonically obtain $k$ different commuting actions of $G$.

For a brief review and a collection of many relevant references see

- Kenneth W. Johnson, Sandro Mattarei and Surinder K. Sehgal,
*Weak Cayley tables*, Journal of the London Mathematical Society 2000 61(2):395-411 (pdf)

There is a succinct functorial reformulation of $k$-conjugacy classes. This is described at higher group characters.

Revised on June 25, 2009 01:00:30
by Toby Bartels
(71.104.230.172)