All Questions
Tagged with rt.representation-theory ag.algebraic-geometry
782 questions
3
votes
0
answers
163
views
A naive question about representations of group stacks
For an algebraic group $G$ over a field $k$, the abelian category $ Rep_k(G)$ of $k$-linear locally finite representations of $G$ can be identified $QCoh(BG_{lis-et})$. Suppose now I replace $G$ with ...
- 595
3
votes
0
answers
504
views
On Local Langlands correspondences
Both over global function fields and $p$-adic fields, we have a series of conjectures under the name of “geometric Langlands conjectures”.
Over global function fields of char $p$, they are due to ...
user125985
3
votes
0
answers
180
views
Endomorphism sheaves of vector bundles
Let $X$ be a hyperelliptic curve, $\pi: X \to \mathbb{P}^{1}$ denote the ramified covering and $W$ the set of Weirstrass points. Let $F,G$ be two involution invariant (with respect to the ...
- 1,981
3
votes
0
answers
157
views
semistability of parabolic bundles
Let $X$ be a rational curve and $E$ a stable parabolic vector bundle on $X$. Is the sheaf of parabolic endomorphisms (i.e the endomorphisms preserving the flag) of the sheaf $E$ also stable (or ...
- 1,981
3
votes
0
answers
175
views
Geometric interpretation of homological quantities in Artinian local Gorenstein algebras
By corollary 3.5. of http://www.ams.org/journals/tran/2012-364-09/S0002-9947-2012-05430-4/S0002-9947-2012-05430-4.pdf the classification of local artinian Gorenstein algebras (all algebras here are ...
- 26.5k
3
votes
0
answers
136
views
Representations of the intersections of two algebraic subgroups
Let $G$ be an algebraic group over a field $k$ (say of characteristic $0$) and let $H,H'$ be two closed subgroups. I would like to understand the category $Rep_k(H \cap H')$ of finite dimensional ...
- 123
3
votes
0
answers
70
views
Geometric properties of fine moduli space of representations of a bound quiver
We know that for a quiver without relations, if one has a fine moduli space $M$ of representations wrt a certain dimension vector and character, then $M$ is smooth and projective (a standard ref is ...
- 815
3
votes
0
answers
144
views
Computation of nearby cycles, monodromy action and action of $sl_{2}$ on $\operatorname{gr}(Ψ)$ for Picard-Lefshetz family
Let $f: \mathbb{A}^{2} \rightarrow \mathbb{A}^{1}$ be a map that sends $(x,y)$ to $xy$. Let $U \hookrightarrow \mathbb{A}^{2}$ be the preimage $f^{-1}(\mathbb{A}^{2} \setminus \{0\})$ and $X:=f^{−1}(0)...
- 103
3
votes
0
answers
139
views
Cartan decomposition for $G[z]$
Let $G$ be a reductive group over complex numbers. Fix some maximal torus $T$. Let $\Lambda^{+}$ be the monoid of dominant coweights. It is known that one has a Cartan decomposition $$G((z))=\coprod\...
- 31
3
votes
0
answers
150
views
Integral Homology of GIT Quotients
Is there any example of GIT quotients of linear actions on a Euclidean space ${\mathbb C}^n$ that satisfy the following conditions?
The quotient is compact and smooth.
The homology of the quotient ...
- 1,207
3
votes
0
answers
188
views
Bott-type vanishing results for the weighted Grassmannian wGr(2,5)
If $G=Gr(k,n)$ denotes the Grassmannian of k-dimensional subspaces in $V= \mathbb C^n$, representation theory gives us a Bott-type result for the cohomology groups $H^q(G, \Omega^p(k))$ of the twisted ...
- 776
3
votes
0
answers
236
views
Deligne-Simpson problem for classical groups
Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n =...
- 181
3
votes
0
answers
286
views
line bundle on affine grassmannian and central extension
Let $G$ be a connected reductive group over $\mathbb{C}$, let $Gr$ be the affine grassmannian of $G$. On $Gr$, we know that there is a canonical line bundle $L$ (the generator of $Pic(Gr)$).
Now $G(\...
- 3,472
3
votes
0
answers
168
views
Invariant Theory over finite adeles
Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...
- 2,384
3
votes
0
answers
125
views
Non-linearly isomorphic non-equivalent $G-$representations?
Let $G$ be an algebraic group (or a group scheme) over a field $\Bbbk$, and let $V$ be an algebraic $G-$representation (I mean, corresponding to a homomorphism of $\Bbbk-$group schemes $G\rightarrow \...
- 23.3k
3
votes
0
answers
240
views
Is the formula for plethysm $S^n(S^3)$ known explicitely?
Is the formula for plethysm (in this the decomposition into irreducible GL representations of the composition of symmetric powers) $S^n(S^3)$ known explicitely? I know $S^n(S^2)$ e.g. in Macdonald's ...
user27328
3
votes
0
answers
236
views
Equivariant Cohomology of Non-Compact Spaces via Fixed Points
Let $T$ be a complex torus and $X$ a smooth quasi-projective $T$-variety with finitely many fixed points. Denote by $\varphi:H_{T}(X)\rightarrow H_{T}(X^T)$ the map on equivariant cohomology induced ...
- 4,920
3
votes
0
answers
247
views
Koszul duality, and coherent sheaves on $pt/G \times_{\mathfrak{g}/G} pt/G$
My questions are the following (from this paper of Arinkin-Gaitsgory):
Q1 Let $P \subset G$ be algebraic groups (in my case, $P$ being a parabolic subgroup of a reductive group $G$, but the following ...
- 2,216
3
votes
0
answers
334
views
Local counterpart of the NON-Hitchin Hecke eigen-sheaves ?
Insight of Beilinson and Drinfeld at early 90-ies - that Hitchin's D-modules are Hecke eigen-D-modules. However they are NOT all Hecke-eigensmodules and actually they are only the half-dimensional ...
- 24.9k
3
votes
0
answers
516
views
Schubert varieties of flag variety , perverse sheaves
The set of Schubert varieties in a flag variety is in one-to-one correspondence with elements of the Weyl group via left cells. There is also some relation between products of Schubert varieties and ...
- 51
2
votes
2
answers
635
views
Are there other ways to show Pic(G)is trivial when G is a simple-connected semisimple algebraic groups over C?
Indeed,I'm reading the book《representation theory and complex geometry》,there is a proof of the fact that Pic(G)is trivial when G is a simple-connected semisimple algebraic group over C,but the proof ...
- 73
2
votes
3
answers
1k
views
A question on a unipotent element in reductive algebraic groups
Let G be a connected reductive group over complex numbers whose derived subgroup is simply connected. Let u be a unipotent element of G. The centralizer of u in G is denoted by Z_(u). Let F_(u) be a ...
- 21
2
votes
2
answers
533
views
elements in the weyl group
Let W be the Weyl a group of a semisimple simply connected group over C.
Let I={1,...,r} the set of simple roots.
For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...
- 3,472
2
votes
2
answers
756
views
Du Val Singularities and Dynkin diagrams references
May I ask whether there are good references for computing blowups of the Du Val Singularities? Also, how are these singularities related to the Dynkin diagrams?
- 1,719
2
votes
2
answers
87
views
Computation of ideal of functions, given by explicit quadratic equations, vanishing on $G/P$ for the exceptional Lie group $G_2.$
In Section 10.6.6 of Procesi's "Lie Groups" he writes that a theorem due to Kostant tells us that for an algebraic group $G$ and a parabolic subgroup group $P,$ the ideal of functions ...
- 201
2
votes
2
answers
1k
views
Poincaré Polynomial and Counting Rational Points
I am currently reading a paper from Sankaran and Vanchinathan where they compute certain Kazhdan-Lusztig polynomials.
Sankaran, P.; Vanchinathan, P.: Small resolutions of Schubert varieties and ...
- 783
2
votes
3
answers
299
views
Can the 'linkages' between equivalent extensions of modules of an algebraic group be taken to have bounded length?
I have a conjecture concerning how "tightly" two equivalent $n$-fold extensions of modules over an algebraic group over a field might
be "linked". I suspect that the
question has been already ...
- 511
2
votes
1
answer
429
views
Representation ring of the general linear group
The ring of representations of the symmetric group is isomorphic to the ring of symmetric functions. The Schur-Weyl duality relates the irreducible representations of the symmetric group and that of ...
- 673
2
votes
1
answer
275
views
Irreducible representations of $SL_n \mathbb Z$
I understand that via the Borel density theorem given a finite dimensional (polynomial) representation of the simple non-compact Lie groups $SL_n \mathbb R$ or $Sp_n \mathbb R$, I get an irreducible ...
- 236
2
votes
2
answers
483
views
Coherent cohomology of G/U, G = reductive group, B = TU Borel subgroup
Hi,
Let $G$ be an algebraic reductive group over an algebraically closed field $k$, $T$ a maximal torus and $B = TU$ a Borel subgroup containing it. I'm interested in computing $H^*(G/U,\mathcal O_{G/...
- 2,842
2
votes
3
answers
181
views
Stabilizers of the action of Levi on abelianization of nilpotent radical
$\DeclareMathOperator\Lie{Lie}$Let $G$ be a simple connected reductive group over $\mathbb C$. Consider a parabolic subgroup $P=MU$ of $G$, where $M$ is a Levi of $P$ and $U$ is the unipotent radical ...
- 6,622
2
votes
1
answer
361
views
Computation of $H^*_c(\mathcal{M}_{1,[2]},\mathbb{V}_1)$
As a term of a Serre spectral sequence, I would like to compute the cohomology group with compact support $H^*_c(\mathcal{M}_{1,[2]},\mathbb{V}_1)$ of the moduli space of genus $1$ curves with $2$ ...
- 33
2
votes
3
answers
294
views
Space of representations of surface group into Lie groups
In the context of Goldman's paper The symplectic nature of fundamental groups of surfaces:
Consider a closed oriented surface $S$ with fundamental group $\pi$, and let $G$ be a connected Lie group. ...
2
votes
1
answer
177
views
Global homological dimension of reductive groups
Let $k$ be an algebraically closed field. Consider a smooth group scheme $G$ over $k$. It is well known that the category $\textbf{Rep}_{G}$ is semisimple if and only if one of the following ...
- 532
2
votes
1
answer
360
views
Maximal compact subroup is dense in Zariski? [closed]
I need help on this one:
In Chriss & Ginzburg book on representation theory and complex geometry I came across the following statement:
maximal compact (in analytic topology) subgroup G_comp ...
- 1,677
2
votes
1
answer
505
views
Grothendieck group of representations
For a linearly reductive group $G$ over $k$ we consider the bounded derived category of finite dimensional representations $D^b(\mathrm{Repr}(G))$. Is the Grothendick group $K_0(D^b(\mathrm{Repr}(G))$ ...
- 741
2
votes
3
answers
600
views
Differential forms with poles on the diagonal
This question arises because I'm reading Frenkel and Ben Zvi's book "vertex algebras and algebraic curves" at the moment.
Let $X$ be a curve, $\Delta \subset X \times X$ the usual diagonal embedding, ...
- 157
2
votes
1
answer
227
views
Representation-induced relations in the Grothendieck of varieties
Let $X$ be a variety over $k = \mathbb{C}$ with an action of a finite group $G$. According to this paper (Section 4), the induced action of $G$ on the cohomology of $X$ respects the mixed Hodge ...
- 197
2
votes
1
answer
118
views
classifying non-split Cartan subalgebras
Let $G$ be a connected reductive group over $\mathbb{C}$. Let $F=\mathbb{C}((t))$ and $\bar{F}$ its algebraic closure. Let $c$ be a Cartan subalgebra of $g$ and $N$ its normalizer. It is written in ...
- 125
2
votes
2
answers
2k
views
Semistability in GIT
If I understand correctly, in geometric invariant theory, polystable points can be defined as those which have a closed orbit. Is it true that semistable points can be characterized as those whose ...
- 1,347
2
votes
1
answer
493
views
Fibers of the Bott-Samelson Resolution of Schubert Varieties
Is there an explicit (perhaps visual) description of the fibers of the Bott-Samelson Resolutions of Schubert Varieties? Let's fix $G$ to be $GL_n(\mathbb{C})$.
Also, how would the answer to the ...
- 1,719
2
votes
1
answer
376
views
Representations of $GL(n)$ containing $S^kV$
Let $V$ be a vector space of dimension $n$.
Let $S^k V$ be a representation of $GL(n)$.
I would like to know if there exists some characterization of finite dimensional $GL(n)$ modules $V_1,V_2$ such ...
- 2,179
2
votes
2
answers
2k
views
Reference needed for representation theory of direct products of algebraic groups over a field (of arbitrary characteristic)
In my dissertation I proved a certain theorem(s) concerning the representation theory of a direct product G x H of algebraic groups over a field, given those of G and H. But I would wager 100:1 that ...
- 511
2
votes
1
answer
160
views
Is the restriction of the Cartan 3-form on conjugacy classes exact?
Let $G$ be a complex semisimple group and $\mathcal{O} \subset G$ a conjugacy class, i.e. $\mathcal{O} = \{gag^{-1} : g \in G\}$ for some $a \in G$. Let $\Omega$ be the Cartan 3-form on $G$ defined by
...
- 443
2
votes
1
answer
133
views
Do representations of same dimension implies isomorphic closed orbits?
Let us recall this fact. Let $G$ be a semisimple algebraic group over $\mathbb C$ and let $V,V'$ be two irreducible $G$-representations. We denote by $X,X'$ the unique closed $G$-orbits contained in $\...
- 381
2
votes
1
answer
340
views
Exterior powers of $Sym^p T$ over Gr(k,n)
Let G=Gr(k,n) the Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ and denote by $T$ the (rank $k$) tautological bundle over $G$, and by $Sym^p T$ its $p$-th symmetric power. Is there any ...
- 776
2
votes
1
answer
223
views
Cluster algebra structure on the coordinate ring of $Mat_3$
Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.
We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the ...
- 6,201
2
votes
2
answers
663
views
Regular embeddings of reductive groups
A regular embedding of a connected reductive linear algebraic group $G$ defined over $\mathbb{F}_q$ is a morphism $\varphi : G \rightarrow G'$ of algebraic groups which is a closed immersion where $G'$...
- 309
2
votes
1
answer
853
views
Various definitions of the Bruhat decomposition of the affine Grassmannian
Let $G$ be a connected, simply connected, complex, semi-simple group with affine Grassmannian $\mathcal Gr \cong G(\mathbb C[t,t^{-1}])/G(\mathbb C[t])$. Fix a choice of maximal torus $T \subset G$ ...
- 627
2
votes
2
answers
508
views
Example of linearization for GIT
Take a vector space $V$ (finite dimensional, over the complex numbers), let $G=SL(V)$. The group $G$ acts on $\mathbb{P}V$ and we can linearize its action to an action on the line bundle $\mathcal{O}(...
- 2,384