nLab
higher group character

A kk-character, kk \in \mathbb{N}, of a group GG is a certain function

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

which descends to a function of kk-conjugacy classes.

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

[g 1,,g k]:={(g 1,,g k)G k|(h 1,h k)G k:g 1=h 1g 1h 2 1g 2=h 2g 2h 3 1g k=h kg 2h 1 1}. [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 kG^k to their kk-conjugacy class is called (at least for k=2k=2) the weak Cayley table of GG.

Remarks

  • Evidently, on any kk-conjugacy class of GG we canonically obtain kk different commuting actions of GG.

Reference

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

Functorial reformulation

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

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