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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 ...
Dave Benson's user avatar
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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 ...
Will Sawin's user avatar
  • 133k
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 $1$, ...
Will Sawin's user avatar
  • 133k
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. ...
David E Speyer's user avatar
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 ...
Sebastien Palcoux's user avatar
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 $n \geq 2$ ...
Damien's user avatar
  • 146
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 ...
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 ...
Benjamin Steinberg's user avatar
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, ...
Andy Putman's user avatar
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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( \...
Geoff Robinson's user avatar
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 ...
Dave Benson's user avatar
  • 11.2k
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 ...
Andy Putman's user avatar
  • 43.2k
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 ...
Jeremy Rickard's user avatar
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 ...
LSpice's user avatar
  • 11.1k
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 ...
Will Sawin's user avatar
  • 133k
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 ...
Noam D. Elkies's user avatar
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}^...
F Zaldivar's user avatar
  • 1,385
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{...
LSpice's user avatar
  • 11.1k
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. ...
Sam Hopkins's user avatar
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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 ...
Ivan Di Liberti's user avatar
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(\...
Estwald's user avatar
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