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## Hot answers tagged weights

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 ...
• 32.7k
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 (...
• 13.9k
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
Accepted

### "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-...
• 32.7k
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
Accepted

• 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

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\})$but ... • 51.8k 2 votes Accepted ### Achieving every possible ranking by rearranging weights The following should give an example where you can't achieve every ranking. Let$n=4$and let$\mathcal{A} = \{ (1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\} = [4]^{(2)}$. I claim that we can't achieve any ... • 462 2 votes ### About block of category$\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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