Unless I am being usually dumb, while the notion of a normal quantum subgroup is established in the theory of compact quantum groups, it’s something that rarely comes up. I am not entirely sure why this is.

I wondered if the classical permutation group S_N\subset S_N^+ is normal for N\geq 4. I am not sure why I didn’t just Google this: Shuzhou Wang covered this here. I mean it’s in the abstract yet this was my source for learning about normal quantum subgroups.

Tracial central states on C(S_N^+)

Let k=h_{C(S_N)}\circ \pi_{\text{ab}} be the Haar idempotent associated to S_N\subset S_N^+.

See Proposition 2.1 of Wang to see that if S_N is a normal subgroup then k is central. That k is central is to say that for all irreducible representation \alpha\in\mathrm{Irr}(S_N^+), there exists c_\alpha\in\mathbb{C} such that:

\displaystyle k(u_{ij}^\alpha)=c_\alpha\delta_{i,j}.

If a state \varphi is central in this way, and this holds for a general compact quantum group \mathbb{G}, then it is central in the sense that for all states on C(\mathbb{G}) we have:

\displaystyle \varphi\star \rho=\rho\star \varphi\qquad (\rho\in \mathcal{S}(C(\mathbb{G}))).

This is easy: both (\varphi\star \rho)(u_{ij}^\alpha) and (\rho\star \varphi)(u_{ij}^\alpha)=c_\alpha\,\rho(u_{ij}^{\alpha}), and those matrix elements form a basis for the norm-dense \mathcal{O}(\mathbb{G})\subset C(\mathbb{G}).

Recently, Freslon, Skalski & Wang have showed that the set of tracial central states is given by (Theorem 5.6):

\displaystyle \mathrm{TCS}(S_N^+)=\{t\varepsilon+(1-t)h\colon t\in [0,1]\}\qquad (N\geq 6).

The Haar idempotent k satisfies k\star k=k, and if S_N is normal, then k\in \mathrm{TCS}(S_N^+). If you solve \varphi\star \varphi=\varphi for \varphi\in \mathrm{TCS}(S_N^+), you find t=0,1, that is \varphi=\varepsilon\text{ or }h.

This also reproves that S_N^+ is simple for N\geq 6.

The Haar idempotent: it’s not central

So… there must exist a state \varphi on C(S_N^+) that does not commute with k. Can you find such a state and f\in C(S_N^+) such that:

(k\star \varphi)(f)\neq (\varphi\star k)(f)?