# Tag Info

1 vote

### Infinite radical ideal cubed equals zero for tame hereditary Artin algebras

I think that for tame hereditary algebras, the maps in the infinite radical go preprojective $\to$ regular $\to$ preinjective, so composing three of these gives zero. A possible route to confirming ...
• 11.2k

### Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?

The Hurwitz representation's existence implies the inequality $(s-2) n - \sum_i c_i + 2\geq 0$ in general - there is no need to restrict to the case of $G$ acting on itself. If $G$ acts transitively ...
• 133k

### Factorizations of an $n$-cycle in $S_n$ into a product $xy$ where $|x| = 2, |y| = 3$

Always (assuming $x$ and $y$ are allowed to have order $1$ in small $n$ cases). For an example take a tree with $n/2$ edges, with half-edges allowed, where each vertex has degree either $3$ or $1$, ...
• 133k

### Factorizations of an $n$-cycle in $S_n$ into a product $xy$ where $|x| = 2, |y| = 3$

Looks to me like it is always possible (for $n \geq 3$). If $n = 3m$, then take $\sigma =(123)(456) \cdots (3m-2\ 3m-2 \ 3m)$, $\tau = (34)(67) \cdots (3m-3\ 3m-2)$ and $\tau \sigma$ is an $n$-cycle. ...
• 150k

### Some fusion rings/categories I don't recognize

These fusion categories are all weakly integral, each with an FPdim less than 84, and therefore, they are all weakly group-theoretical by this paper. Consequently, they can all be described using ...
• 25.7k
1 vote

### $G$-module structure of the relation module for a presentation of a finite group $G$

I think we may be able to extend the result to fields if the characteristic of the field $F$ does not divide the order of the group $G$. Proposed Theorem: Let $F_n$ be a free group of rank $n \geq 2$ ...
• 146

### The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group

It turns out that my original question does indeed have a positive answer. In fact, one can show that if $G$ has an irreducible character of degree $\geq 3$ then ${\rm AD}(G) \geq 2+ |G'|^{-1}$. The ...

### $G$-module structure of the relation module for a presentation of a finite group $G$

The question has been answered over fields of characteristic $0$ but not over $\mathbb Z$. as originally asked. It turns out that the statement is never true for $G$ noncyclic. It is proved in Lemma ...
• 37.2k
Accepted

### $G$-module structure of the relation module for a presentation of a finite group $G$

Your memory is correct, at least if you replace $\mathbb{Z}$ with a field $k$ of characteristic $0$. This is a theorem of Gaschütz. See W. Gaschütz, Über modulare Darstellungen endlicher Gruppen, ...
• 43.2k

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