Questions tagged [weights]
The weights tag has no usage guidance.
66
questions
22
votes
0answers
556 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
1answer
87 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 ...
9
votes
1answer
438 views
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
0answers
117 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
43 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
213 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
0answers
66 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
235 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
220 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
93 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
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
1answer
654 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
81 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
60 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
88 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
74 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
82 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
78 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
218 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
185 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
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
2answers
201 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
194 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
344 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
141 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
160 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
76 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
603 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
112 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
147 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
465 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
230 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
143 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
651 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 ...
15
votes
1answer
520 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, ...
5
votes
0answers
289 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
364 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 ...
2
votes
1answer
305 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
584 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
170 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
191 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 ...
6
votes
1answer
902 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
243 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
131 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 ...
