Questions tagged [weights]
The weights tag has no usage guidance.
63
questions
3
votes
0answers
108 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
0answers
40 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
0answers
183 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 ...
1
vote
0answers
60 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
0answers
228 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
1answer
167 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
0answers
91 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
2answers
139 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 ...
0
votes
1answer
441 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
1answer
72 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
0answers
52 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
61 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
67 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
0answers
71 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
71 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
209 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
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
1answer
175 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
72 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
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
0answers
87 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
196 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
181 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
0answers
45 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
2answers
265 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
0answers
127 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
153 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
75 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 ...
1
vote
2answers
48 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
503 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
0answers
109 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
144 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 ...
5
votes
1answer
413 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
200 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
vote
0answers
107 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
574 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 ...
10
votes
0answers
333 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
vote
0answers
58 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, ...
6
votes
0answers
284 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
317 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
262 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
votes
0answers
536 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
164 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
175 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
620 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
237 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
129 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
858 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
370 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
votes
2answers
671 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_\...
