# Questions tagged [weights]

The weights tag has no usage guidance.

74
questions

4
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### Takesaki lemma 1.16 (volume II, chapter VII)

I am trying to understand the proof of the implication $(i)\implies (ii)$ in Takesaki's book "Theory of operator algebras II", chapter VII, which says the following:
The relevant setting ...

0
votes

0
answers

29
views

### How to recalculate the weights for an event that happens multiple times and requires all outcumes to be unique?

I think it's easiest to explain with an example.
I have a weighted probability list
A : 0.15
B : 0.15
C : 0.15
D : 0.1
E : 0.1
F : 0.1
G : 0.1
H : 0.075
I : 0.075
...

4
votes

2
answers

239
views

### Takesaki volume II chapter VII lemma 1.15

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\...

3
votes

0
answers

91
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### Weighted limit calculus

In coend calculus by Fosco Loregian, it is mentioned that Lawvere conjectured that co/ends constitute a categorification of logical calculus. In his own words:
The somewhat far-fetched conjecture ...

2
votes

2
answers

106
views

### Decomposition of an element as a difference of positive elements in the definition subalgebra of a weight (Takesaki)

I originally asked this on MSE, but did not get an answer there.
Let $M$ be a von Neumann algebra. Let $\varphi: M_+ \to [0, \infty]$ be a weight on $M$. Consider
\begin{align*}&\mathfrak{p}_\...

3
votes

0
answers

117
views

### When the Leray spectral sequences for nice compactifications give the Deligne's weight ones?

Assume that $X$ is a proper smooth variety over an algebraically closed field $k$, $U=X\setminus (\cup D_i))$ where $D_i$ are closed subvarieties such that the set-theoretic intersections of all sets ...

3
votes

1
answer

245
views

### Comparison of weight filtration on cohomology of complex manifold

Let $X$ be a smooth scheme of finite type over $\mathbb{Z}$ (or let's say a finitely generated $\mathbb{Z}$ algebra). To each prime $p \in \mathbb{Z}$ we can consider the $\mathbb{F}_p$ variety $$X_{\...

1
vote

1
answer

54
views

### inequivalent vertex weights on finite poset

Let $m\geq1$ and $P$ be an arbitrary poset with vertex set $V=\{v_1,\dots,v_n\}$, edge set $E,$ and set $O$ of orbits under $\text{Aut}(P).$ Can we efficiently generate all inequivalent nonnegative ...

26
votes

0
answers

678
views

### Nearby cycles without a function

Suppose that:
$X$ is a smooth complex algebraic variety,
$f : X \to D$ is a proper map to a small disc, smooth away from 0,
$Z_\epsilon = f^{-1}(\epsilon)$, and $Z = Z_0$.
Then there is a procedure (...

3
votes

1
answer

93
views

### Functoriality of weighted limits

Let $C$ be a complete category, let $I$ be a small category, let $F,G:I\to C$ be functors, and let $W,U:C\to\mathrm{Set}$ be also functors, which we view as "weights".
The weighted limits ...

10
votes

1
answer

932
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### Effective weight-monodromy conjecture

$\DeclareMathOperator\Gr{Gr}$Let $G$ be the absolute Galois group of a finite extension of $\mathbb{Q}_p$ with inertia subgroup $I$, and let $V$ be an $\ell$-adic representation of $G$. Grothendieck's ...

3
votes

0
answers

120
views

### Where general mixed Galois representations are defined?

I am interested in etale cohomology of varieties, and respectively, in mixed $\mathbb Q_{\ell}$-adic Galois representations over finitely generated fields. What is the canonical reference for this ...

1
vote

0
answers

47
views

### Is a $\tau$-mixed Weil sheaf always a direct summand of a $\tau$-real sheaf?

Fix an isomorphism $\tau: \overline{\mathbf{Q}_\ell} \cong \mathbf{C}$ and consider Weil sheaves on a scheme $X$ over $\mathbf{F_q}$.
It is a theorem that a $\tau$-real sheaf $\mathscr{F}$ is $\tau$-...

3
votes

0
answers

247
views

### What is an example of a lisse Weil sheaf which is not mixed?

Fix a base field $\mathbf{F}_q$. What is a simple example of a lisse Weil sheaf which is not $\tau$-mixed for any identification $\tau: \overline{\mathbf{Q}_\ell} \cong \mathbf{C}$?
Addendum: it ...

2
votes

0
answers

75
views

### Weights of restriction to a Levi subgroup

Let us consider the group $G = \mathrm{GSp_{2g}/\mathbf{Q}}$ with respect to the symplectic pairing given by the matrix $\begin{pmatrix} & s \\ -s & \end{pmatrix}$, where $s$ is the $g\times ...

2
votes

0
answers

241
views

### Which fields and schemes "have enough finite residue fields"?

I am looking for assumptions on the spectrum $S$ of a field $K$ that ensure the following: there exists an excellent noetherian finite dimensional (integral) scheme $S'$ such that $S$ is its generic ...

0
votes

1
answer

475
views

### Significance of the Eigenvalues of the adjacency matrix of a weighted di-graph

I'm currently running a simulation on a bunch of randomly generated points, each with two randomly selected 'partners' from the set of points. In the simulation the points try to move such that they ...

4
votes

0
answers

95
views

### About the geometry of the set of weights that is strongly linked to $\lambda$

Define $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_\alpha\cdot\lambda<\lambda$ for some $\alpha\in\Phi^+$. More generally, $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_{\alpha_1}s_{\...

0
votes

2
answers

148
views

### About weights in $\mathfrak{h}^*$

On p.24 of the book "Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$".
I quote:
"For example, the fixed point $-\rho$ under the dot action lies in a linkage
class by ...

2
votes

1
answer

810
views

### Highest weight of a representation of a Lie Algebra [closed]

Given a Lie Algebra $\mathfrak{g}$, its Cartan Matrix $A$ and a finite representation $R$, is there a way of determining its highest weight $\Lambda$ in a simple way?
In my course, we consider $\...

2
votes

1
answer

83
views

### Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight

Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight.
It is an exercise of Bröcker's book on Representations of Compact Lie ...

1
vote

0
answers

77
views

### Barycentric weighting of a point in a mesh

I know there is a way to get a barycentric weighting of a point $p$ inside a convex polygon; for weighting $\omega_n$ of component $q_n$ with adjacent angles $\gamma_n$ and $\delta_n$:
$$\omega_n = \...

0
votes

1
answer

144
views

### About regular infinitesimal character

Let $Z(\mathfrak{g})$ be the centre of $U(\mathfrak{g})$ and let $\chi_\lambda$ be an algebra homomorphism
$Z(\mathfrak{g}) \to \mathbb{C}$ such that
$
z \cdot v = \chi_\lambda(z)v
$
for all $z \in Z(\...

1
vote

1
answer

83
views

### About finite direct sum of full subcategory of category $\mathcal{O}^\mathfrak{p}$

As is shown in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, every nonzero module $M \in \mathcal{O}^\mathfrak{p}$ has a finite filtration with nonzero quotients, each ...

0
votes

0
answers

87
views

### Associated subgroup of Weyl group

Let $\Phi$ be a root system.
For a weight $\lambda\in\mathfrak{h}^*$,
start by defining
$
\Phi_{[\lambda]}:=\{\alpha\in \Phi \ | \ \langle \lambda,\alpha^{\lor}\rangle\in\mathbb{Z} \}
$
and
$
W_{[\...

0
votes

0
answers

79
views

### Simplicity Criterion for Verma module

In Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$,
$\lambda\in\mathfrak{h}^*$ is
antidominant if $\langle \lambda + \rho, \alpha^\lor\rangle \not\in \mathbb{Z}^{>0}$ ...

0
votes

2
answers

234
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### Some confusion about weights and roots in parabolic root systems

I was reading James Arthur's book An Introduction to the Trace Formula and had a couple of questions. Here $A_0$ is a maximal split torus of a reductive group $G$, $P_0 \supset A_0$ is a minimal ...

2
votes

1
answer

55
views

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

Denote $M_I(\lambda)$ be the generalized Verma module with highest weight $\lambda$ and $L(\mu)$ is the simple highest weight module with highest weight $\mu$.
Suppose $\text{Ext}_{\mathcal{O}^\...

1
vote

1
answer

195
views

### About Hom and weight space of nilpotent Lie algebra cohomology

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Denote by $\Phi$
the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by $\mathfrak{g}_\alpha$ the root subspace of $\mathfrak{g}$ ...

0
votes

1
answer

74
views

### Minimizing the set of "wrong" edges in $K_\omega$ with $\{0,1\}$-weights

For any set $X$, let $[X]^2 = \big\{\{x,y\}:x\neq y\in X\big\}$.
Let $f:[\omega]^2\to\{0,1\}$ be a function. The principal goal is to find a partition of $\omega$ such that if $m\neq n\in \omega$ ...

2
votes

0
answers

33
views

### About Extension group in Category $\mathcal{O}^\mathfrak{p}$

Let $M_I(\lambda)$ be the generalized Verma module with highest weight $\lambda$, $L(\lambda)$ be the simple highest weight module with highest weight $\lambda$.
Suppose $\mu\le \lambda\le \nu$, does ...

1
vote

0
answers

94
views

### about weight decomposition of U(sl3) [closed]

the universal enveloping algebra is $\mathbb{Z}^2$-graded algebra. $U(sl_3)= \bigoplus_{(i_1,i_2) \in \mathbb{Z}^2 } U_{i_1\alpha_1+i_2\alpha_2}$ $($the weight decomposition$)$ where $\alpha_1$ and $...

0
votes

2
answers

208
views

### About block $\mathcal{O}_\lambda$ of Category $\mathcal{O}$

The blocks of $\mathcal{O}$ are precisely the subcategories consisting of modules whose composition factors all have highest weights linked by $W_{[\lambda]}$ to an
antidominant weight $\lambda$.
I ...

1
vote

1
answer

213
views

### About subcategory of parabolic Category $\mathcal{O}^\mathfrak{p}$

It follows from Exercise 1.13 in Humphreys' Category $\mathcal{O}$ book that $M\in\mathcal{O}^\mathfrak{p}_{\chi_\lambda}$ has a direct sum decomposition $M=\oplus M_i$ such that all weights of each $...

0
votes

0
answers

46
views

### About weight of $M\in\mathcal{O}_{\chi_\lambda}$ in the Category $\mathcal{O}$

Given $M\in\mathcal{O}_{\chi_\lambda}$, how does the weights of $M$ relate to $\lambda$? Is it something about $W\cdot\lambda$?

3
votes

2
answers

412
views

### About block of category $\mathcal{O}$

In the book "Representation of Semisimple Lie Algebra in the BGG Category $\mathcal{O}$".
Exercise 1.13. Suppose $\lambda\not\in\Lambda$, so the linkage class $W\cdot\lambda$ is the disjoint union of ...

1
vote

0
answers

174
views

### Highest weight category and weight structures

In various branches of representation theory, there is a notion of highest weight category.
On the other hand there is a notion of weight structure on a triangulated category $C$ (introduced by ...

1
vote

2
answers

164
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### About Morita equivalent and regular block of category $\mathcal{O}^\mathfrak{p}$

In the following paper: Representation type of the blocks of category $\mathcal{O}_S$
https://www.sciencedirect.com/science/article/pii/S0001870804002853
On p.196, it states that "When $\mu$ is ...

4
votes

0
answers

78
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### Good range and fair range

Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...

1
vote

2
answers

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### Achieving every possible ranking by rearranging weights

This problem actually arose as a question in the real world (see the paragraph "Origin of the problem" below).
Let $\mathbb{N}$ denote the set of the positive integers and let $n\in\mathbb{N}$. If $...

7
votes

3
answers

706
views

### Decomposition of tensor power of symmetric square

Studying some representation theory I came up with the following problem.
We work over a field of characteristic $0$. Let $V$ be the standard representation of $\mathrm{GL}_n$ and let $W$ be the ...

2
votes

0
answers

115
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### Stability of mixed complexes under open embeddings

In Weil II, Deligne proves that the six functors preserve the category of mixed $l$-adic complexes. Most difficulties are encountered when proving that if $f\colon X\to Y$ is a finite type separated ...

1
vote

1
answer

149
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### How to compute the index of a given weight?

I am learning the Borel–Weil–Bott theorem in rational homogeneous varieties. Working on them, I need to know two questions:
How to judge if a weight is singular?
How to compute the index of a given ...

5
votes

1
answer

550
views

### Comparing Frobenius weights with Mixed Hodge theory

For a variety over a finite field Deligne define weights by looking at the eigenvalues of the Frobenius. On the other hand, if we take a variety over $\mathbb{Q}$, at least for its constant sheaf we ...

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vote

0
answers

259
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### Element of the Weyl group acting on co-roots

Let $W$ be the Weyl group of a semisimple algebraic group defined over say $\mathbb C$. We have the root space $\mathfrak a^*$ spanned by the roots $\{\beta \in \Delta\}$ and also by the weights $\{ \...

1
vote

0
answers

184
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### Interior and boundary vertices of weighted graphs

Xu He's article Rigidity of Infinite Disk Patterns and I have a problem with a statement he makes on page 7.
He considers weighted embedded planar graphs $G=(V, E)$ with weight function $\Theta: E \...

7
votes

1
answer

770
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### Are multiplicity-free representations weight multiplicity free?

A rational representation $(G,V)$ of a complex reductive linear algebraic group is called multiplicity-free if the decomposition of $\mathbb C[V]$ into irreducible $G$-modules contains each ...

15
votes

1
answer

547
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### Integrability property of polynomials in several variables

This might be very trivial, or not.
Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there ...

1
vote

0
answers

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### Robust weighted estimator of location

Let $X = (x_1, \ldots, x_n)$ be a sample of i.i.d values. There are several robust estimators of sample location, most notably sample median and Hodges-Lehmann estimator.
Now let $W = (w_1, \ldots, ...

6
votes

0
answers

298
views

### Embeddings between weighted Besov spaces

Consider the Besov spaces $B_{p,q}^s(\mathbb{R}^d)$ for parameters $0<p,q\leq \infty$ and $s\in \mathbb{R}$. The weighted Besov space $B_{p,q}^s(\mathbb{R}^d;\mu)$ is defined for $\mu \in \mathbb{R}...