All Questions
Tagged with rt.representation-theory ag.algebraic-geometry
782 questions
2
votes
0
answers
127
views
Is the image of the exponential map of a complex semisimple group Zariski open?
Let $G$ be a semisimple complex algebraic group. Is the image of the exponential map
$$\exp : \mathfrak{g} \to G$$
Zariski open in $G$?
- 443
2
votes
0
answers
187
views
Counting fixed points on flag variety and Deligne-Lusztig functors
Let $G=GL_n(q)$ be the general linear group over $\mathbb{F}_q$ and $T$ be the torus of diagonal matrices. We also pick a Levi subgroup of the form $L=GL_{n_1}(q)\times GL_{n_2}(q) \times \cdots \...
- 1,061
2
votes
0
answers
255
views
For a nilpotent matrix A, are the cardinalities of sets: 1) B: commute with A, 2) B: anticommute with A, 3) B: q-commute with A — the same?
Let us work over finite fields $F_{p^k}$. Simulations seems to indicate:
Question 1: Consider a nilpotent matrix $A$, consider the set of all matrices $B$, such that $AB-qBA=0$, then cardinality of ...
- 24.9k
2
votes
0
answers
102
views
Deforming the affine flag variety into the product of affine Grassmannian and the usual flag variety
In Gaitsgory - Construction of central elements in the affine Hecke algebra via nearby cycles the idea of deforming the affine flag variety into the product of the affine Grassmannian and the usual ...
- 31
2
votes
0
answers
140
views
Integral functional on algebraic varieties
Suppose that $X$ is a smooth complex algebraic variety and $X_{\mathbb{R}}$ is a real form of $X$. If $X_{\mathbb{R}}$ is compact and oriented as a real manifold, then it will admit a natural ...
- 1,077
2
votes
0
answers
84
views
Weights of finite abelian group actions on submanifolds/subvarieties
(cross-posted from https://math.stackexchange.com/questions/4125529/weights-of-finite-abelian-group-actions-on-submanifolds-subvarieties)
How do weights associated to actions of finite subgroups of $\...
- 521
2
votes
0
answers
270
views
Road map: beyond Artin-Wedderburn theorem
For a noncommutative semisimple ring $R$, its structure and its category of representations can be largely understood using Artin-Wedderburn theorem. Such structure theory is useful, for example, in ...
- 5,230
2
votes
0
answers
74
views
Berenstein-Fomin-Zelevinsky's Ininital seeds and initial seeds from Postnikov diagrams
In Cluster algebra III by Berenstein-Fomin-Zelevinsky, Theorem 2.10, for any pair of reduced words $(u,v)$, they constructed an initial seed for the cluster algebra $\mathbb{C}[B^{u,v}]$, where $B^{u,...
- 6,201
2
votes
0
answers
107
views
Paramodular forms with level and Iwahori subgroups?
Given an integer $N>0$, not necessarily prime, we have the paramodular group $K(N) \subset \text{Sp}_{4}(\mathbb{Q})$, which consists of matrices of the form
$$\begin{bmatrix} * & *N & * &...
- 1,701
2
votes
0
answers
640
views
Areas of algebraic geometry useful for geometric representation theory
What topics/areas of algebraic geometry (aside from perverse sheaves/D-modules, etale cohomology, and possibly derived algebraic geometry) is it useful to learn/master if one is interested in doing ...
- 2,964
2
votes
0
answers
119
views
relative rank two group: structure of parabolic subgroup-- high-level Jacobson--Morozov sl_2 triple
Given a parabolic subgroup $P=MN$ of a connected reductive group $G$ defined over a local field $F$, let $W_M$ be the relative Weyl group of $M$ in $G$, assume that the reduced roots relative to $M$ ...
- 301
2
votes
0
answers
94
views
Anything similar to cone product formula (for convex polytopes)?
The convex polytope flag vector ring $\mathcal{R}$ satisfies the cone product formula
$$
C(U) C(V) = C(J(U, V)) + DUV
$$
where
$$
J(U, V) = U C(V) + C(U) V - e_1 UV
$$
is the join formula.
Note: ...
- 459
2
votes
0
answers
223
views
Deligne's internal characterisation of Tannakian categories - glueing of algebras
I am currently reading the proof of Deligne's internal characterization of Tannakian categories (Thm. 7.1) in Deligne: Catégories tannakiennes.
My question is similar to this one.
Given a ...
- 51
2
votes
0
answers
157
views
Transformations of the cubic forms [closed]
Is there a way to understand whether there exist linear transformation that brings one cubic form of n variables to another form? In particular one of the examples I am interested in are two cubic ...
2
votes
0
answers
175
views
Minimal Embedding for flags varieties
I would like to understand how to construct a parametization of a flag
variety $F(V,n_1,\ldots,n_r)\subseteq \mathbb{P}^N$ in its minimal embedding.
First, I would like to know if there is a closed ...
- 21
2
votes
0
answers
137
views
Coordinate ring of an equivariant embedding of a homogeneous projective variety
Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $\lambda$ be a dominant integral ...
- 187
2
votes
0
answers
369
views
Koszul duality and coherent sheaves on projective space
There are different descriptions of the category $Coh(\mathbf{P}^{n})$. One can either describe it as modules over Beilinson's quiver algebra (Let us denote it by $A$) using the exceptional collection ...
- 377
2
votes
0
answers
176
views
Quotient of a finite morphism by an action of a reductive group is still finite?
Let $X, Y$ be two quasi-affine schemes over $\mathbb{C}$. Let $G$ be a reductive algebraic group. Suppose that we are given an action of $G$ on $X,Y$ and a $G$-equivariant
finite morphism $X \...
- 143
2
votes
0
answers
203
views
Mistake in Discriminants, Resultants, and Multidimensional Determinants?
I'm studying this proof in the book cited in the title.
The author say that any multiplicative function $\chi\colon\mathsf{GL}(k)\longrightarrow\mathbb{C}^\times$ is a power of the determinant. But ...
- 1,252
2
votes
0
answers
103
views
A question on Lusztig's Intersection cohomology complexes on a reductive group
This is a proposition in Lusztig's Intersection cohomology complexes on a reductive group (Invent. Math. 75, 205-272).
Proposition 6.3. The triple $(L,C_1, \mathcal{E}^{\cdot}_1)$ above is unqiuely ...
- 21
2
votes
0
answers
232
views
Didactic (counter-)examples in algebraic groups and groups schemes
Algebraic groups are very rich objects. As such, a large bag of examples against which one can test his intuition can be very helpful in learning the general theory.
What are some good didactic (...
- 7,789
2
votes
0
answers
208
views
Representations, ADE singularities
This sounds very classical, but it would be great if someone can point out a reference to me.
Let $G \leq \mathrm{SL}(2, \mathbb{C})$ be a finite group with its natural action on $\mathbb{C}[x, y]$, ...
- 143
2
votes
0
answers
97
views
Can we write an element in a super Grassmannian as a pair of matrices?
Super Grassmannians are introduced by Manin, see for example.
Elements in a grassmannian can be written as matrices, see for example.
Can we write an element in a super Grassmannian as a pair of ...
- 6,201
2
votes
0
answers
118
views
extending local systems on a neighbourhood
Let $Y$ an affine finite type scheme over an algebraically closed field $k$.
Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$.
If we have a $\mathbb{Z}_{\ell}$ local ...
- 3,472
2
votes
0
answers
91
views
Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane
In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...
- 1,509
2
votes
0
answers
157
views
An equality of discriminant and resultant divisors
Let $\Phi$ be the root system of a split group $G$ over a field $k$. The differentials $d\alpha$ of the roots define a polynomial called the discriminant
$$\prod_{\alpha\in\Phi}d\alpha$$
on $\mathfrak ...
- 3,799
2
votes
0
answers
156
views
Extension of the Hilbert-Mumford Criterion
Let $X$ be a smooth variety, $L$ a line bundle on $X$ and $G$ a reductive group actin on $X$ with a linearization of the action to $L$. Say we are over the complex numbers.
Both the concept of GIT ...
- 2,384
2
votes
0
answers
94
views
Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?
Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...
- 853
2
votes
0
answers
659
views
Constant group scheme and torsors
Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism:
$$g_Y \colon Y \times_X Y \cong Y \...
- 781
2
votes
0
answers
316
views
Dimension of affine Springer fiber and its functor of points as an ind-scheme
Let $k$ be a finite field and let $F = k( (t))$ with ring of integers $\mathfrak{o} = k[ [t]]$. Let $G$ be a connected linear algebraic $k$-group with Lie algebra $\mathfrak{g}$. Suppose that $\gamma\...
user1437
2
votes
0
answers
149
views
Invariant generalized sections of dual vector bundles
Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My ...
- 335
2
votes
0
answers
151
views
About the reduceness of the commuting scheme associated with a symmetric pair
my question is the following one:
Let $G$ be a connected reductive algebraic group over the field of complex numbers, and let $V$ be a linear representation of $G$ obtained as the isotropy ...
- 21
2
votes
0
answers
311
views
Generators of the algebra of invariant polynomials on a Lie algebra and the root-space decomposition
Let $G$ be a connected, simply-connected complex semisimple group with Lie algebra $\mathfrak{g}$. Fix a pair $T\subseteq B\subseteq G$ of a maximal torus and Borel subgroup, and let $\mathfrak{t}$ ...
- 4,920
2
votes
0
answers
357
views
$G$-equivariant coherent sheaves on Bott$-$Samelson resolutions
Let $G$ be a Lie group and $B$ a Borel subgroup. $G/B$ is the corresponding flag variety.
Let $w$ be an element of the Weyl group $W$ with a reduced expression
$w = s_1 \cdots s_n$. Let $X_w$ be ...
- 1,719
2
votes
0
answers
163
views
A hypersurface in the Grassmannian of endomorphisms.
Let $V$ be complex vector space of dimension $n$, let ${\frak s\frak l }(V)$ denote the traceless endomorphisms, and let $G(k,W)$ denote the Grassmannian.
Let $E\in G(2q,{\frak s\frak l} (V))$ and ...
- 835
2
votes
0
answers
180
views
on geometric Satake and functions
Let $G(F)/G(O)$ the affine grassmanian with $F=k((t))$ where $k$ is a finite field.
For $\lambda$ a dominant cocharacter, we have by Cartan decomposition the schubert strata $\overline{Gr^{\lambda}}$....
- 3,472
2
votes
0
answers
239
views
Resolution of singularities of this cubic surface?
Let $A = \mathcal O(Y)^{SL_2(\mathbb C)}$ be the ring of invariant functions on $Y := \mathrm{Hom}(\mathbb Z^2, SL_2(\mathbb C))$. We can identify $A$ with the quotient of $\mathbb C[x,y,z]$ by the ...
- 2,869
2
votes
0
answers
282
views
Non-characteristic is to pullback as (blank) is to pushforward.
Suppose $f:X\to Y$ is a map of smooth complex algebraic varieties. There is a pushforward functor
$f_\ast : D(X) \to D(Y)$
on the derived category of $D$-modules. This certainly does not preserve ...
- 6,829
2
votes
0
answers
219
views
Is there algebraic structure (manifold, stack ...) on the SET of irreducible representation of algebraic group ?
Consider algebraic group G over some field "k".
Consider the SET of all its complex irreducible representations (I think I need unitary also).
Question Is there some way to put algebraic structure (...
- 24.9k
2
votes
0
answers
278
views
What is M / G ?
Consider algebraic manifold M (e.g. vector space) and algebraic group G (e.g. unitriangular matrices UT(n)) acting on it.
Question: Is there some way to put algebraic structure on M/G i.e. on the ...
- 24.9k
2
votes
0
answers
597
views
Transverse slices to orbits in the nilpotent cone and affine Grassmanian in type A
Background My question is about the paper http://arxiv.org/abs/0712.4160; specifically about the isomorphism in Theorem $1.2$ (in the Introduction),
$T_{\lambda} \cap \overline{\mathcal{O}_{\mu}} \...
- 2,216
2
votes
0
answers
408
views
Orbit stratification of semi infinite flag manifold?
Denote semi infinite flag manifold by $Fl_{\infty/2}=G((t))/N_-((t))H[[t]]$, denote $B_-((t))=N_-((t))H[[t]]$
from the book of Frenkel and Benzvi" Vertex algebras and algebraic curves", They take ...
- 5,515
2
votes
0
answers
277
views
localization functor for category of quasi coherent sheaves of $Fl_{\infty/2}$?
From series papers of Frenkel-Feign-Gaitsgory-Braveman and Beilinson Drinfeld and their school we know that the algebraic geometric definition and theory of semi infinite flag manifold were not ...
- 5,515
2
votes
0
answers
254
views
Can I recognize the quotient of a group by a closed subgroup? (for example the standard representation of S3 in GL(2,C))
The first question might be too much in general.
The cases I'd like to understand in practice are quotients (as algebraic varieties) of GL(n,C) (or SL(n,C) if you prefer) by finite subgroups.
Is ...
- 1,889
1
vote
1
answer
292
views
Plus and minus Białynicki-Birula decomposition for normal variety
We work over $\mathbb{C}$. Let $X$ be a normal projective irreducible variety, and let $\mathbb{C}^*$ act nontrivially on $X$. The fixed point locus of $X$, namely $X^{\mathbb{C}^*}$, can be ...
1
vote
1
answer
572
views
moduli problem for flag varieties?
Hi,
Suppose $G$ is a reductive group over an algebraiclly closed field $k$
(suppose $k$ of char zero if you want at first). Let $X$ be its flag variety.
Question: What is the moduli problem that $X$ ...
- 2,842
1
vote
1
answer
184
views
General centralizer of algebraic group
Perhaps there is a simple answer, but I'm very puzzled by the following question:
Question: Does there exist a (smooth, connected) algebraic group $G$ such that the general centralizer (i.e. the ...
- 507
1
vote
1
answer
347
views
Do Auslander-Reiten quivers coincide with the McKay quivers for arbitrary subgroups of GL(2,C)?
It is a theorem of Auslander that if $G< GL(2,\mathbb C)$ is a finite subgroup without pseudo-reflections, then the Auslander-Reiten quiver of $K[x,y]^G$ coincides with the McKay quiver of $G$ with ...
- 13
1
vote
1
answer
212
views
Lie algebras and pulled back group schemes
Suppose I have an extension of fields $L/K$, a group scheme $G_K$ over $\operatorname {Spec} K$. Let $G_L$ denote the pullback of $G_K$ to $\operatorname{Spec} L$. Then, under what conditions on the ...
- 13
1
vote
1
answer
466
views
Resolution of Kleinian Singularities using Hilbert schemes of points
Apologies in advance for the naive and rather speculative question.
In this blog post by John Baez, he paints a (perhaps not original) picture of how one might expect that the minimal resolution of a ...
- 471