All Questions
Tagged with qa.quantum-algebra finite-groups
12 questions
20
votes
1
answer
586
views
$q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$?
The following formula of astonishing beauty and power (imho):
$$ \sum_{n \ge 0} \frac{| \mathrm{Hom}(G,S_n) | }{n! } z^n = \exp\left( \sum_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^...
- 24.9k
15
votes
1
answer
657
views
Is every finite quantum group a quantum symmetry group?
This post is basically a quantum extension of Is every finite group a group of “symmetries”?
Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra.
Frucht's theorem ...
14
votes
0
answers
347
views
What is the mathematical name for the anomaly for an action of a group on a lattice conformal field theory?
Suppose $V$ is a (bosonic) chiral conformal field theory which is "holomorphic" in the sense that its category of vertex modules is trivial. (The definition of "chiral conformal field theory" might be ...
- 54.6k
10
votes
1
answer
1k
views
Are there workable algebraic geometry approaches for the pentagon equation?
A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring.
A fusion ring is given by a finite set of integer ...
8
votes
3
answers
528
views
Classification of $\operatorname{Rep} D(G)$
Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However,...
- 5,230
7
votes
3
answers
650
views
Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$
Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\...
- 5,230
7
votes
0
answers
140
views
Triviality of Semisimple Hopf Algebras of Cyclic Dimension
A cyclic number is a natural number $n$ such that any group of order $n$ is cyclic. A003277
Theorem (T. Szele, 1947): A number $n$ is cyclic if and only if it is coprime to its Euler totient $\varphi(...
5
votes
0
answers
172
views
Are the symmetric groups integrable as Hopf algebras?
Let $G$ be a group. For $g,h \in G$, let $[g,h]=g^{-1}h^{-1}gh$ be a commutator. The normal subgroup $G' = \langle [g,h] \ | \ g,h \in G \rangle$ is called the commutator subgroup or derived subgroup. ...
4
votes
0
answers
326
views
Are there infinitely many simple integral fusion rings of rank $4$?
$\DeclareMathOperator\ch{ch}$$\DeclareMathOperator\FPdim{FPdim}$We refer to [EGNO15, Chapter 3] for the notion of fusion ring and basic results. The type of a fusion ring $R$ is the list $(\FPdim(b_i)...
4
votes
0
answers
190
views
A group-theoretical analogous of Temperley-Lieb-Jones subfactor planar algebras
The Temperley-Lieb-Jones subfactor planar algebra $\mathcal{TLJ}_{\delta}$ admits the following properties:
maximal,
it exists for every possible index, i.e. $\delta^2 \in \{4cos^2(\pi/n) \ | \ n \...
3
votes
0
answers
271
views
Are there workable numerical approaches for the pentagon equation?
Warning: this post is the "numerical" analog of
Are there workable algebraic geometry approaches for the pentagon equation?
I've replaced "algebraic geometry" by "numerical" in its content,
...
2
votes
0
answers
63
views
Existence of a unitary fusion category with this relation ruled out on finite groups
In this answer, Geoff ruled out the existence of a finite group $G$ such that the fusion category $\mathrm{Rep}(G)$ has simple objects $5_1$ and $7_1$ of FPdim $5$ and $7$ resp., with (for some object ...