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18h
awarded  Yearling
21h
comment The algebra of natural transformations of the n-th power tensor functor
@anton: the proof of Schur-Weyl duality in the blog post I link to above doesn't assume this, only that the characteristic is $0$.
22h
comment The algebra of natural transformations of the n-th power tensor functor
@anton: look at the action of $\phi$ on $V$ such that the action of $k[S_n]$ on $V^{\otimes n}$ is faithful, and compare it to the action of elements of $k[S_n]$. Not very satisfying, I admit.
22h
revised Extending a homotopy equivalence
edited tags
22h
comment Are logarithms of prime numbers algebraically independent?
As far as I know this is an open problem, but like many open problems of this form it follows from Schanuel's conjecture (en.wikipedia.org/wiki/Schanuel%27s_conjecture).
22h
comment Lifting a principal G-bundle to a principal bundle with structure group a covering of G
@Student: yes, that's starting all the way from $H = O(n)$. I started from $H = SO(n)$, which corresponds to picking an orientation, so $w_1$ already needs to vanish. I'd guess that there's a proof in Milnor-Stasheff, but not sure.
22h
answered How to recover the cohomology of a torus from its description of a quotient
23h
answered Lifting a principal G-bundle to a principal bundle with structure group a covering of G
23h
answered The algebra of natural transformations of the n-th power tensor functor
2d
comment The Relationship between Separable Functors and Faithful Functors
That isn't the statement of Rafael's theorem on the nLab: ncatlab.org/nlab/show/separable+functor
Jul
25
awarded  Revival
Jul
25
comment Why Composition and Dihedral Group have reverse order of operation?
The order you choose for composition is a convention; as long as you pick a consistent convention, it doesn't matter. Any group $G$ is naturally isomorphic to the same group $G^{op}$ with reversed composition, with the isomorphism given by $g \mapsto g^{-1}$.
Jul
24
comment Would this be a homology theory?
@Michael: yes, it's quite different, which is why I wrote "looks like."
Jul
24
comment what is exactly the fine-scale geometry of Euclidean space?
I don't understand the question. Can you be more specific?
Jul
24
answered Would this be a homology theory?
Jul
24
comment Is there a systematic way to determine the irreducible representations of a finite group?
"Systematic" and "efficient" don't necessarily mean the same thing. A thing you might want from "systematic" is a method you can apply to understand a large class of related groups at once. Such methods are known for families of groups such as the symmetric groups $S_n$ and the general linear groups $GL_n(\mathbb{F}_q)$, but the answers get pretty complicated and it's quite nontrivial to describe them.
Jul
23
awarded  Nice Question
Jul
21
comment What are the most important results in graph theory?
@M.M: that's Euler's formula for convex polyhedra. The difference between this and Euler's formula for planar graphs comes down to whether you think the "face at infinity" of a planar graph (the complement of the interior) counts as a face. Said another way, it depends on whether you think a planar graph encodes a CW decomposition of a disk or a sphere.
Jul
20
comment Terminological conventions regarding group actions
Almost certainly it means the former. What reason do you have for thinking that it means the latter?
Jul
20
comment How to see that SL(2,C) is simply connected?
The deformation retract to $SU(2)$ strategy works. The homotopy is given by performing the Gram-Schmidt process smoothly.