# nLab higher group character

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

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

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

# Remarks

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

# Reference

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

# Functorial reformulation

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)