## New answers tagged rt.representation-theory

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 ...

4
votes

### 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 ...

2
votes

### 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 $1$, ...

4
votes

### 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.
...

4
votes

### 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 ...

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 $n \geq 2$ ...

4
votes

### 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 ...

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6
votes

### $G$-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 ...

11
votes

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, ...

3
votes

### What are the Schur indices of irreducible representations of $\operatorname{SL}(2,p)$?

This is not a complete answer, but shows how this can be calculated from the character table (at least in this case) without using the usual Frobenius-Schur formula $\nu(\chi) = \frac{1}{|G|} \left( \...

9
votes

### What are the Schur indices of irreducible representations of $\operatorname{SL}(2,p)$?

If $p \equiv 1$ mod $4$ then all faithful irreducibles have Schur index two, and Schur indicator $-1$. If $p \equiv 3$ mod $4$, then most do, but there are two irreducibles of dimension $(p+1)/2$ with ...

7
votes

### Is any representation of a finite group defined over the algebraic integers?

I just stumbled across this ancient question, and I want to point out my notes here that prove the result in question. The proof is basically the same as moonface's accepted answer, but with two ...

6
votes

Accepted

### Does hereditary and connected imply that the underlying ring $k$ of a $k$-algebra is a field?

$R$ doesn't need to be connected, so long as $k$ is (and if $R$ is connected then $k$ is, since a nontrivial idempotent of $k$ would be a nontrivial central idempotent of $R$). Also, $R$ doesn't need ...

2
votes

### Iwahori action on the $p$-ordinary line of a principal series representation

As discussed in the comments, the answer to (1) is "yes", at least if the residue field of $F$ is not $\mathbb F_2$, because $\mathrm{Iw}_\alpha$ is the product of its derived subgroup, on ...

12
votes

Accepted

### Distinct characters with the same character values, outer automorphisms and Galois conjugation

Take $G = S_3 \times S_4$ and consider the unique two-dimensional irreducible representation of $S_3$ and the unique two-dimensional irreducible representation of $S_4$. These have the same character ...

6
votes

### How to see that Eisenstein series are eigenfunctions of the laplacian?

Here's an answer to Question 1 that reduces the result to checking that
$D$ is invariant under hyperbolic isometries $z \mapsto (az+b)/(cz+d)$
plus a couple of elementary identities. I'll let others ...

0
votes

Accepted

### Chevalley restriction theorem

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\trace{trace}$I can write the details of Humphreys' argument: For each integer $k\geq 1$ it suffices to show that the map $\theta:\Sym^k({\mathfrak g}^...

3
votes

Accepted

### Orbital integrals of $\operatorname{SL}_2$ and the fundamental lemma

$\gamma$ and $\gamma^{-1}$ are rationally conjugate, by $\begin{pmatrix} i \\ & -i \end{pmatrix}$ if $q \equiv 1 \pmod4$ and $\begin{pmatrix} & \sqrt{-\epsilon^{-1}} \\ \sqrt{-\epsilon} \end{...

10
votes

Accepted

### Coefficients when rewriting the Hook-Content polynomials in terms of binomial polynomials

These coefficients can be understood using the theory of $(P,\omega)$-partitions, as discussed for example in Section 3.15 of Stanley's "Enumerative Combinatorics," Volume 1, 2nd edition.
...

1
vote

### Is this concept of a left-abelian category studied?

Some notion of this kind exists, see:
Breitsprecher. Lokal endlich präsentierbar Grothendieck-Kategorien. Mitt. Math. Sem. Giessen Heft, 85:1–25, 1970.
Rump. Locally finitely presented categories of ...

0
votes

### Maximal submodule of Verma

I believe I figure out a proof; here is how it goes.
Clearly we have an inclusion
$$\sum_{\alpha\in I}M(s_\alpha\cdot\lambda)\subset N_I(\lambda)+\sum_{\alpha\in I}M(s_\alpha\cdot\lambda)\subset N(\...

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