# Questions tagged [weights]

The weights tag has no usage guidance.

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### 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 = \...

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### 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(\...

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### 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 ...

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### 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_{[\...

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59 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 $\langle \lambda + \rho, \alpha^\lor\rangle \not\in \mathbb{Z}^{>0}$ ...

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144 views

### 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 ...

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49 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}^\...

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139 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 $\mathfrak{g}$ ...

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70 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 $m\neq n\in \omega$ ...

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### 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 ...

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### 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 $...

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175 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 ...

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145 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 $...

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### 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 $W\cdot\lambda$?

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### 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 ...

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### 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 ...

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139 views

### 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 ...

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70 views

### 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 ...

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### Maximum size of a freely rankable family

This is a follow-up question to an older one.
Let $\mathbb{N}$ denote the set of the positive integers and let $n\in\mathbb{N}$. If $w:\{1,\ldots,n\}\to \mathbb{N}$ is a function (the letter $w$ ...

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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 $...

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352 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 ...

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### Distribution of maximum-weight matching in a random graph

Consider the $k$-th power of a non-directed cycle graph with $n$ vertices $C_n^k$. In other words vertex $i \in \{1, ..., n\}$ is connected to vertices $i-k$ to $i+k$ (modulo $n$). Here is an Example ...

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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 ...

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140 views

### 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 ...

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318 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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147 views

### 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 $\{ \...

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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 \...

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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 ...

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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 ...

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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, ...

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### 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}...

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265 views

### Logarithmic weights on number theoretic sums

Suppose we are interested in the sum
$\sum _{n\leq x}a_n.$
The study of the sum
$\sum _{n\leq x}a_n\log (x/n)$
may be easier.
What can one say about the first sum from knowing the behaviour of ...

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190 views

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

Set $w(x) = (1 + |x|^2)^{1/2}$ with $|\cdot|$ the Euclidian norm on $\mathbb{R}^n$. For $s,\mu \in \mathbb{R}$, we define the Sobolev space
$$H_2^{s}(\mathbb{R}^n) = \left\{f : \lVert f \rVert_{s} := \...

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### Intrinsic definition of the weight filtration

Let $X$ be a smooth quasiprojective complex variety. Then Deligne (Theorie de Hodge II) defined a weight filtration on the Betti cohomology of $X$. The general philosophy is quite simple: express the ...

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### What sorts of weights for perverse sheaves were or can be computed?

I am studying certain weights for (triangulated categories of relative) motives. Those are interesting; yet one can hardly say that they are very much explicit or effectively computable. So, I would ...

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### Convergence of weighted double sum of random variables

I'm looking for convergence results of particular weighted sum:
$$S_n=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}X_i X_j.$$
when random variables $X_i$ ar i.i.d. Are there any investigation ...

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462 views

### “Weight-monodromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...

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225 views

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

I am looking for explicit formulas for the weight multiplicities of some particular irreducible representations of $SO(2m)$.
It is possible that they have been already computed; in this case I will ...

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### Is semistability of smooth Weil sheaf preserved under tensor product?

Let $X_0$ be a smooth, geometrically connected scheme over $\mathbb{F}_q$. As usual, let $\tau : \bar{\mathbb{Q}}_{\ell} \simeq \mathbb{C}$ be a fixed isomorphism. Let $\mathcal{C}$ be the category of ...

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### Hodge structure versus Weight structure

This is a naive question.
One is told that, somehow, Hodge theory for varieties over complex numbers, is an analog of weight theory for varities over finite fields. In weight theory, one considers ...

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### Tiling a rectangle with weighted cells (min-max problem)

I have been struggling with a research problem. The problem can be formalized as follows:
Given a $n\times m$ matrix $A$ containing cells with non-negative integer values, partition it in $J$ ...

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### A_infinity structure on cohomology and the weight filtration

Let $X$ be a complex algebraic variety. The rational cohomology of $X(\mathbb{C})$ carries a canonical filtration called the weight filtration. It also carries a canonical equivalence class of $A_\...

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### Weights of restricted modules of some Cartan type Lie algebras

Let $L$ be a simple Lie algebra of Cartan type of absolute toral rank 2 over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq 5$.
Denote by $L_{[p]} $ the minimal $p$-envelope of $L$...

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### Universal Correlation measure — ranking correlations

I have time series data of experimental observations for two related processes. I want to measure correlation for use in further analysis.
Correlation of the series changes over time and across ...

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### weighted to centre mean

I'm not even sure if what I want to do is a good idea but I figure I'll experiment and see.
I have two predicted ratings in the range of 1-5 based on two different algorithms for predicting movie ...

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458 views

### Which (reducible) projective varieties could be presented as 'relatively smooth' hyperplane sections of irreducible (normal) ones?

Are there any restrictions known on a (complex reducible) projective variety $Y$ that can be presented as $Z\cap H$, where $Z$ is a normal (or just irreducible) closed subvariety of a smooth ...

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### A Weyl invariance constructed from Clebsch-Gordan Coefficients.

Let $V$ and $\tilde{V}$ be irreducible representations of SU(N) with tensor decomposition:
\begin{equation}
V \otimes \tilde{V} = \bigoplus_i U_i
\end{equation}
\noindent were $U_i$ are also irreps ...

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### Does Artin's vanishing hold for '$E_2$-weight pieces' for (torsion) cohomology of affine varieties?

Let $X$ be an affine variety of dimension $n$ (say, over complex numbers). Then Artin's vanishing theorem yields: both singular and etale cohomology $H^i(X):=H^i(X,\mathbb{Z}/l\mathbb{Z})=0$ for $i>...

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### Does intersection pairing on `$IH^*(X)$` agree with cup-product on `$H^*(X)$`?

Let $X$ be a proper singular variety over $k=\overline{\mathbb F}_p,$ irreducible of dimension $d.$ Let $H^*(X)$ and $IH^*(X)$ be the $l$-adic cohomology groups and $l$-adic intersection cohomology ...

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474 views

### Smooth in codimension-k and the weight filtration

Let $X$ be an algebraic variety. Then $H_{et}^k(X)$ has a filtration whose associated graded pieces are labeled by "weights", certain integers between $0$ and $2k$. If $X$ is smooth, then the weights ...