# Tag Info

### 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

### 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
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
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
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

### 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
Accepted

• 3,035
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

### 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 ...

### 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