higher group character

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

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

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:

$[g_1, \cdots, g_k]
:=
\{
(g'_1, \cdots, g'_k) \in G^k |
\exists (h_1, \cdots h_k) \in G^k :
g'_1 = h_1 g_1 h_2^{-1} \vee
g'_2 = h_2 g_2 h_3^{-1} \vee \cdots
g'_k = h_k g_2 h_1^{-1}
\}
\,.$

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)