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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 $v=1$?
Nov
20
comment “Nyldon words”: understanding a class of words factorizing the free monoid increasingly
@TheMaskedAvenger This is definitely not what is going on. The modifications you are proposing would yield Lyndon words for the associated ordering. Nyldon words behave very differently. For example, the Nyldon words $101$ and $1011$ defy this kind of description.