Questions tagged [weights]
The weights tag has no usage guidance.
1
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0answers
44 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
1answer
39 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
1answer
50 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 ...
1
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0answers
53 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
0answers
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}$ ...
0
votes
2answers
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 ...
2
votes
1answer
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}^\...
1
vote
1answer
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}$ ...
0
votes
1answer
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$ ...
2
votes
0answers
28 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
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0answers
59 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
2answers
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 ...
1
vote
1answer
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 $...
0
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0answers
41 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$?
2
votes
2answers
210 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
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0answers
73 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
2answers
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 ...
4
votes
0answers
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 ...
0
votes
0answers
44 views
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$ ...
1
vote
2answers
45 views
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
3answers
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 ...
0
votes
0answers
16 views
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 ...
2
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0answers
101 views
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
1answer
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 ...
4
votes
1answer
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 ...
1
vote
0answers
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 $\{ \...
1
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0answers
73 views
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
1answer
442 views
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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0answers
281 views
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
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0answers
54 views
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, ...
3
votes
0answers
187 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}...
6
votes
1answer
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 ...
0
votes
1answer
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} := \...
10
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0answers
463 views
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 ...
3
votes
0answers
154 views
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 ...
2
votes
1answer
162 views
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 ...
5
votes
1answer
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 ...
3
votes
2answers
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 ...
1
vote
0answers
119 views
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 ...
9
votes
2answers
713 views
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 ...
3
votes
0answers
353 views
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 $J$ ...
12
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2answers
601 views
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_\...
4
votes
2answers
369 views
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 $L$...
1
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0answers
173 views
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 ...
1
vote
1answer
88 views
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 ...
2
votes
1answer
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 ...
1
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0answers
135 views
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 ...
3
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0answers
305 views
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 $i>...
7
votes
1answer
513 views
Does intersection pairing on `$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 ...
5
votes
4answers
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 ...
