MathOverflow

Let $S_N^+$ be the quantum permutation group on the $N$-point set.
Theorem. A group dual $\widehat{\Gamma}\subset S_N^+$ is always obtained from the quotient map$$\mathbb{Z}_{n_1}*\dots*\mathbb{Z}_{n_r}\longrightarrow\Gamma. $$
Let me elaborate more. Such a quotient group homomorphism induces the dual embedding$$\widehat{\Gamma}\subset(\mathbb{Z}_{n_1}*\dots*\mathbb{Z}_{n_r})^{\wedge}\subset S_N^+, $$where $N=n_1+\dots+n_r$. The claim is that every quantum group embedding of...

I have worked on this problem for a few hours but I completely can't solve it. Prove that for real numbers $a,b,c>0$ that:$$\frac 1 a + \frac 1 b + \frac 1 c + \frac 4 {a+b} + \frac 4 {b+c} + \frac 4 {c+a} \geq \frac {12} {3a+b} + \frac {12} {3b...

Let $ S_n $ denote the symmetric group on $n$ letters, and $ \mathbb{C}[S_n] $ its group algebra.
Let $X_n$ be the $n$-th Jucys–Murphy element $X_n = \sum_{k=1}^{n-1} (k\ n)$.
Denote by $Z_n = Z(\mathbb{C}[S_n])$ the center of the group algebra, and consider the centralizer $Z_{(n-1,1)} := Z_{\mathbb{C}[S_n]}(\mathbb{C}[S_{n-1}])$ of the subalgebra $\mathbb C[S_{n-1}]$ inside $\mathbb{C}[S_n]$....

I recently came across a recent paper presenting a proof of Zorn’s Lemma that seems very short and elementary. I found it interesting because the proof does not use transfinite induction or any set theory concepts.
It seems like it could be useful for teaching, for instance in linear algebra, to show students a simple proof of Zorn’s Lemma without requiring them to learn transfinite induction....