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Results tagged with gr.group-theory
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user 2530
Questions about the branch of algebra that deals with groups.
5
votes
Accepted
Center of a Symmetric Group on an Infinite Set
I think this is easy. We shall prove that $f=\mathrm{id}$ for every $f\in Z\left(G\right)$. In fact, let $x\in X$ be such that $f\left(x\right)\neq x$ (if no such $x$ exists, we are done anyway). Now …
- 33.8k
3
votes
Accepted
Mean on the natural numbers which is invariant with respect to the power
Posting it as an answer, since OP confirmed I did understand the question right.
Such a mean cannot exist. In fact, if it would, we could denote it by $M$ and compute:
$M\left(S\right)=M\left(2^S\ri …
- 33.8k
29
votes
Accepted
How big can the irreps of a finite group be (over an arbitrary field)?
EDIT: Part 4 added. EDIT2: Second proof of Part 4 added.
1. The answer is no (as long as we are working over a field - of any characteristic, algebraically closed or not). If $k$ is a field and $G$ is …
- 33.8k
4
votes
Accepted
vanishing of certain product in group algebra
I think the answer is "if and only if the group $G$ is not cyclic". Why?
1) An element of $\mathbb C\left[G\right]$ is zero if and only if it acts as zero on each irreducible representation of $G$ (s …
- 33.8k
7
votes
Accepted
Occurence of trivial representation in a tensor square.
The trivial representation appears in $\wedge^2 V$ if and only if the representation $V^{\ast}$ has a $G$-invariant alternating bilinear form (because $\wedge^2 V\cong\wedge^2\left(\left(V^{\ast}\righ …
- 33.8k
12
votes
An easy proof that $S(n)$ does not embed into $A(n+1)$?
I think this is solved on http://www.artofproblemsolving.com/Forum/viewtopic.php?f=61&t=333049 .
- 33.8k
11
votes
Faithful representations and tensor powers
See problem 3.26 in Etingof's "Introduction to representation theory". If you have troubles with understanding the hint, feel free to ask me. (The first sentence uses the fact that if a vector space o …
- 33.8k
2
votes
Reference request: from a reduced expression of an element in a Coxeter group to another red...
George Lusztig, Hecke algebras with unequal parameters, arXiv:math/0208154v2, Theorem 1.9.
7
votes
Results from abstract algebra which look wrong (but are true)
Quite a few things in the Hopf algebra world are surprising:
Takeuchi's theorem: Every connected graded bialgebra is a Hopf algebra. (No finiteness assumptions!) Takeuchi was actually more general: I …
4
votes
On permanents and determinants of finite groups
For the sake of completeness, here is the answer to Question 1, part of which
is missing from the other answers:
Proposition 1. Let $G$ be a finite group. Consider the representations of
$G$ over $\m …
- 33.8k
4
votes
3
answers
496
views
Molien for modular representations?
Let $G$ be a finite group, and let $k$ be a field whose characteristic divides $\left|G\right|$. Let $\rho:G\to \mathrm{End} V$ be a (finite-dimensional) representation of $G$ over $k$. Prove or dispr …
- 33.8k
12
votes
Conjectures in the representation theory of the symmetric group
Just a few things that come to my mind; most are about the base ring $\mathbb{Z}$, but a good answer would tell us something interesting about $\mathbb{Q}$ as well.
Specht modules can be defined not …
- 33.8k