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higher group character

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

$${\chi }^{(k)}:G^k\to ℂ$$\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:

$$[g_1,\cdots ,g_k]:=\{(g{\prime }_1,\cdots ,g{\prime }_k)\in G^k\mid \exists (h_1,\cdots h_k)\in G^k:g{\prime }_1=h_1{g}_1{h}_2^{-1}\vee g{\prime }_2=h_2{g}_2{h}_3^{-1}\vee \cdots g{\prime }_k=h_k{g}_2{h}_1^{-1}\}.$$[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

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

Functorial reformulation

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