12
votes
Accepted
What's the intuition for weighted limits?
In enriched category theory, weighted limits may be strictly more general than conical limits, in the sense that an enriched category with all conical limits may fail to have all weighted limits.
...
- 15.9k
11
votes
Decomposition of tensor power of symmetric square
By Pieri's formula, a partition with $2n$ elements in $n$ rows, corresponding to a representation of $GL_n$, occurs in this representation with multiplicity equal to the number of ways of obtaining ...
- 148k
11
votes
Effective weight-monodromy conjecture
You can bound the filtration length (assuming the WM conj) using the weight spectral sequence of Rapoport--Zink. This is a sp seq converging to $H^*(X_{\overline{k}})$; and if the WM conj holds, then ...
- 37k
7
votes
Accepted
Comparison of weight filtration on cohomology of complex manifold
Yes, the $\ell$-adic weight filtration is compatible with the weight filtration in mixed Hodge theory under the comparison isomorphism. These facts go back to Deligne, and are described in his ...
- 35.2k
6
votes
Accepted
Takesaki lemma 1.16 (volume II, chapter VII)
$\newcommand{\con}{\operatorname{conv}}$[Edit: This has turned out to be more involved than I thought, and my argument is now somewhat different from the book.] Firstly, some facts about $A_+$. Let $...
- 18.7k
5
votes
Decomposition of tensor power of symmetric square
The answer is yes.
Let $e_1,\ldots, e_n$ be the standard basis of $V$. Consider the morphism
$$
f \colon \det(V) \to V^{\otimes n}
$$
given by
$$
f(e_1 \wedge \cdots \wedge e_n) = \sum_{\sigma \in ...
- 3,704
5
votes
Accepted
Takesaki volume II chapter VII lemma 1.15
Given the comment of @Andromeda to my first answer, let me provide the following alternative proof, which only uses results that are proven earlier in the Takesaki books.
Let $a_n$ be a bounded ...
- 4,351
5
votes
Accepted
Every locally compact group gives rise to a locally compact quantum group
In the general, not necessarily $\sigma$-compact setting, one has to interpret $L^\infty(G,\lambda)$ as the von Neumann algebra of locally measurable functions that are bounded outside a locally null ...
- 4,351
5
votes
Integral weight filtration on cohomology with no compact support
The results of the paper you mention (https://arxiv.org/pdf/1403.6805.pdf) are for analytic spaces, but the same proofs (actually simplified) give a weight filtration on the cohomology with ...
- 96
4
votes
Accepted
Comparing Frobenius weights with Mixed Hodge theory
For constant coefficients, the comparison statement, along with a sketch, appears in Deligne's ICM talk, Poids dans la cohomologie....
For things to work the way you seem to want in your second ...
- 35.2k
4
votes
Accepted
About weights in $\mathfrak{h}^*$
Any element of $\mathfrak{h}^*$ is called weight. I don't know why in this case the author used definite article. As a non-native speaker I would use indefinite one, i.e. "a weight is called regular ...
- 8,597
4
votes
Accepted
Highest weight of a representation of a Lie Algebra
To obtain the highest weight of a semisimple Lie algebra $\mathfrak{g}$ you first have to choose Cartan subalgebra $\mathfrak{h} \leq \mathfrak{g}$ and then set of positive roots (or alternatively ...
- 8,597
4
votes
Accepted
Functoriality of weighted limits
Let $X$ be an arbitrary object in $\mathcal{C}$.
I write $\{ W, F \}$ for the limit of $F$ weighted by $W$.
By definition,
$$\mathcal{C} (X, \{ W, F \}) \cong [\mathcal{I}, \textbf{Set}] (W, \mathcal{...
- 15.9k
4
votes
Accepted
Decomposition of an element as a difference of positive elements in the definition subalgebra of a weight (Takesaki)
I don't know what Takesaki had in mind for the proof, but what you're asking is incorrect. Here is a counterexample where $\phi$ in the counterexample is a normal faithful semifinite (n.f.s.) weight.
...
- 2,471
4
votes
Tensor product of operator values weights (in the theory of locally compact quantum groups)
Each normal weight is a supremum of normal positive functionals (second volume of Takesaki, theorem 1.11 in Chapter VII). For a normal positive functional $\phi$ you can just define $(\iota \otimes \...
- 5,005
4
votes
Accepted
Weights in the proof of Chen's theorem in Nathanson's "Additive Number Theory The Classical Bases"
Choosing optimal weights in sieve theory is a very difficult problem that is often done by trial and error. In Nathanson's book it seems as if he was attempting to produce the simplest and shortest ...
- 465
3
votes
Takesaki volume II chapter VII lemma 1.15
Yes, the metric $d$ in Takesaki's proof metrizes the $\sigma$-strong topology on $\mathscr{M}p \cap \mathscr{S}$. Here $\mathscr{S}$ denotes the unit ball of $\mathscr{M}$ and it thus suffices to ...
- 4,351
3
votes
About block of category $\mathcal{O}$
I think you're overcomplicating things. You have showed that: (a) As a vector space we have the following decomposition:
$$
M = \bigoplus_{[\nu]\in\mathfrak{h}^*/\Lambda_r} M^{[\nu]}.
$$
(b) ...
- 116
3
votes
Decomposition of tensor power of symmetric square
Just an addendum to Ricky's answer: the multiplicity is indeed 1 which can be proved as follows.
An occurrence of ${\rm det}(V)^{\otimes 2}$ inside $({\rm Sym}^2(V))^{\otimes n}$ is the same thing as ...
3
votes
Accepted
Takesaki II "Connes cocycle derivative"
The operator $S$ is defined as the closure of an operator $S_0$ that is described by (10) and (6*). So, by definition, the graph $K$ of $S_0$ (viewed as a linear subspace of $\mathfrak{H}_\rho \oplus \...
- 4,351
3
votes
Accepted
Takesaki: question about lemma in section "Left Hilbert algebras and weights"
In order for the answer to be more self-contained, note that we are in the following general setting. We are given a von Neumann algebra $M$ (namely $\mathcal{M}'$) and a left ideal $B \subset M$ (...
- 4,351
3
votes
Accepted
Tensor product of operator values weights (in the theory of locally compact quantum groups)
The approach I had in mind is the following. For a reference, see Section 4, Chapter IX of Takesaki 2. The extended positive part $\widehat{M_+}$ is by definition the space of positive-homogeneous, ...
- 18.7k
2
votes
Achieving every possible ranking by rearranging weights
Consider $\mathcal A = \{\{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}\}$.
Then $sc(\{1,3\}) - sc(\{1,4\}) = sc(\{2,3\}) - sc(\{2,4\})$. In particular, it's impossible to have $sc(\{1,3\}) > sc(\{1,4\})$ but ...
- 54.2k
2
votes
Accepted
Achieving every possible ranking by rearranging weights
The following should give an example where you can't achieve every ranking.
Let $n=4$ and let $\mathcal{A} = \{ (1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\} = [4]^{(2)}$.
I claim that we can't achieve any ...
- 472
2
votes
How to compute the index of a given weight?
Just evaluate $\langle \lambda, \alpha \rangle$ for all roots $\alpha$. It is sufficient to test the equality for positive roots only. Also, the weight $\lambda$ is regular if and only if it has ...
- 8,597
2
votes
Accepted
Minimizing the set of "wrong" edges in $K_\omega$ with $\{0,1\}$-weights
I believe not: let $f$ be any colouring and take a maximal equivalence relation $\sim$ on $\omega$ with the property that $m\sim n$ implies $f(\{m,n\})=0$. Note that $\sim$ can be extreme: the ...
- 11.4k
2
votes
About block $\mathcal{O}_\lambda$ of Category $\mathcal{O}$
Submodules/quotients have the same (edit: generalized) infinitesimal character. Now use Harish-Chandra theorem.
EDIT: Definitely an overkill. Easier solution is in the comments of the question.
- 1,368
2
votes
Decomposition of an element as a difference of positive elements in the definition subalgebra of a weight (Takesaki)
This is a reply to the comments. I didn't have enough space there.
Assume $M$ acts on the Hilbert space $H$.
Let $ \mathfrak{p},\mathfrak{m}, \mathfrak{n}$ as in lemma 1.2, chapter VII of Takesaki. We ...
- 175
2
votes
Accepted
Significance of the Eigenvalues of the adjacency matrix of a weighted di-graph
Your idea is pretty much spot on; this is the area of spectral graph theory. Often the graph Laplacian is used rather than its adjacency matrix -- the Laplacian is defined as $L = D - A$ where $A$ is ...
- 4,529
2
votes
About weights in $\mathfrak{h}^*$
In Humphreys, weights are defined in section 0.7 "Representations". He notes that in the finite case all weights are integral, but that this is not true for infinite dimensional representations.
Note ...
- 116
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