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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 ...
  • 122k
10 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 ...
10 votes
Accepted

Are multiplicity-free representations weight multiplicity free?

It is a theorem of Brion and, independently, of Vinberg that varieties with an open $B$-orbit (a.k.a. spherical varieties) have in fact only finitely many orbits. A shorter argument is due to Matsuki (...
10 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. ...
  • 13.6k
9 votes
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"Weight-monodromy" for open varieties

Tony Scholl gave a talk at a conference in Warwick in 2013 on exactly this topic (his talk was called "Remarks on monodromy and weights"). He explained how to formulate a precise version of weight-...
7 votes

Logarithmic weights on number theoretic sums

Put $A(x) =\sum_{n\le x} a_n$, and $B(x) =\sum_{n\le x} a_n\log x/n$. Then $$ B(x) = \int_1^x A(t)\frac{dt}{t}. $$ So information about $A(x)$ readily translates to information about $B(x)$ and ...
  • 42.7k
6 votes
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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 $...
  • 17.7k
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 ...
  • 3,584
5 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 ...
  • 32.2k
5 votes
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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 ...
  • 2,046
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{...
  • 13.6k
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 ...
  • 7,834
4 votes
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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 ...
  • 7,834
4 votes
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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 ...
  • 32.2k
4 votes
Accepted

When is the identity Hilbert-Schmidt between weighted Sobolev spaces?

There is an explicit operator that maps $L^2$ isometrically onto $H^{s,\mu}_2$ :$$I_{s,\mu}u(x)=(1+|x|^2)^{-\mu/2}(I-\Delta)^{-s/2}u(x)$$The (inverse) Fourier transform $k_s(x)$ of $(1+|\omega|^2)^{-s/...
  • 3,035
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. ...
  • 2,061
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

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 ...
  • 2,046
2 votes
Accepted

Weight multiplicities for some particular representations of SO(2m).

For $\mu$ a weight, let $||\mu||_1$ denote the one-norm of $\mu$ (the sum of the absolute values of its entries) and let $Z(\mu)$ be the number of zero coordinates of $\mu$. Let $k\geq0$ and $1\leq p\...
  • 2,158
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 ...
  • 7,834
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 ...
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 ...
2 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) ...
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 ...
  • 7,930
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.
  • 1,308
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 ...
2 votes

Some confusion about weights and roots in parabolic root systems

I only find one paper (a book chapter, not a book itself) with the indicated title, Arthur - An introduction to the trace formula, and I can't find in it the sentences you quote, so it's hard to speak ...
  • 9,077
2 votes
Accepted

About Extension group and weights in $\mathcal{O}^\mathfrak{p}$

See proof of Theorem 6.11 of Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$ by James E. Humphreys. This theorem proves what you want in the case $\mathfrak{p}$ is a Borel ...
  • 7,834
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 ...
  • 569
1 vote
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 ...
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