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irrep, $V_x\otimes \lambda$, and we evaluate $\lambda(x)$ to get our pairing:$$\langle \langle x,y\rangle\rangle_V\in \mathbb{C}^*$$ This form is alternating, bilinear, and controls the behaviour of characters …

invariant functions on $G$ under convolution, and surprisingly, this induced map is also an algebra automorphism, as can be seen by passing to $\mathbb{C}$ and noting that this is the Galois action on characters …

Since it is an adjoint, this map preserves characters/virtual representations. … On characters, this map is given by:$$\chi_{\nu^k V}(g)=\sum_{h^k=g}\chi_V(h).$$ My question is then, does a natural lift of $\nu^k$ to the Burnside ring exist? …

One interesting way to describe the ordinary (over $\mathbb{C}$) character theory of finite groups is to view the categories $Rep(G)$ together in a $2$ category with bimodules as morphisms. This $2$ c …