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Aug
27
comment Generalized identities of (soluble) groups
I don't understand your definition. If $a_1,\ldots,a_n\in G$, what does $x^{a_1}\cdots x^{a_n}$, $x\in G$ mean?
Jul
14
comment What are the “tensor-closed” object of the BGG category $\mathcal{O}$ of a semisimple Lie algebra $\mathfrak{g}$?
No, I was mistaken. I think one can probably show that the tensor product of a Verma module with an infinite dimensional module is not finitely generated.
Jul
14
comment Decomposition of quadratic polynomials inti irreducible representations of affine group over a finit field
@Thomas $V$ is not closed under addition. It is just a set.
Jul
14
comment What are the “tensor-closed” object of the BGG category $\mathcal{O}$ of a semisimple Lie algebra $\mathfrak{g}$?
Have you looked at sl_2 yet? I think that will give a negative answer to your question.
Jul
13
comment Decomposition of quadratic polynomials inti irreducible representations of affine group over a finit field
That is a good point.
Jul
13
answered Decomposition of quadratic polynomials inti irreducible representations of affine group over a finit field
Jul
8
comment SO(3) transformation that produces a reflection
I believe $H(u)=u-(2u\cdot v^T)v$ is what you meant. This is a reflection. The product of two reflections is a rotation.
Jul
7
comment Irreducible unitary representations of semidirect groups of a discrete abelian group by a discrete group
@YemonChoi Frobenius reciprocity seems to be okay. The proof I know is fine for the situation. The only issue might be the equality between $\dim H_\Gamma$ and the hom space, but given we have an invariant inner product, that seems fine too.
Jul
7
comment Irreducible unitary representations of semidirect groups of a discrete abelian group by a discrete group
@YemonChoi I think it's okay. Isn't Fronenius reciprocity really a statement about adjointness of tensor product and hom. The proof I know is for modules over some rings S<R, though I don't recall the exact hypotheses. I will certainly double check.
Jul
7
revised Irreducible unitary representations of semidirect groups of a discrete abelian group by a discrete group
deleted 6 characters in body
Jul
6
answered Irreducible unitary representations of semidirect groups of a discrete abelian group by a discrete group
Jun
29
comment Can the product of a simple and a non-simple indecomposable representation be semisimple?
As Jim Humphreys suggests, it would be helpful to know what kind of "generic irreducible representations" $\sigma$ you are considering? Do they have any special properties? The fact that you have information about the semi-simplicity of $\rho\otimes\sigma$ leads me to think there is more relevant information available.
Jun
26
comment Permutation covering of a $G$-lattice
Ahhh! I have been using the diagonal action of $C_p$ on $L$ to make computations.
Jun
25
comment Permutation covering of a $G$-lattice
I am having a bit of trouble verifying your answer. In particular, I don't see why $L$ has no invariant subspaces. For example, when $p=3$, I calculated that $L$ has exactly two 1-dimensional submodules. This means there should be a permutation covering of rank $5$. Am I missing something here?
Jun
2
awarded  Citizen Patrol
Apr
8
comment An application of Maschke's theorem
@KConrad $7\cdot 7 = 49 = (5\sqrt{2}-1)(5\sqrt{2}+1)$?
Mar
3
awarded  Yearling
Nov
21
comment “Nyldon words”: understanding a class of words factorizing the free monoid increasingly
Also, can you define you ordering more explicitly. Do you read words left-to-right or right-to-left?
Nov
21
comment “Nyldon words”: understanding a class of words factorizing the free monoid increasingly
I don't think your comment above is helpful. I think it was appropriate to delete it.
Nov
21
comment “Nyldon words”: understanding a class of words factorizing the free monoid increasingly
Isn't (3) already false for $w=101$, $u=10$ and $v=1$?