All Questions
Tagged with rt.representation-theory ag.algebraic-geometry
782 questions
4
votes
2
answers
335
views
Deformation of "Hecke modification"
Let $X$ be a smooth curve over $\mathbb{C}$. I wish to compute the deformation of the following data $(E,F,x)$. $E$ and $F$ are locally free sheaves over $X$ and $x$ is a point on $X$. They satisfy:
...
- 125
4
votes
1
answer
196
views
Number of boundary divisors and colors of a Spherical variety
Let $X$ be a Spherical variety for a reductive group $G$ with a Borel subgroup $B$. A boundary divisor of $X$ is a $G$-invariant divisor and a color of $X$ is a $B$-invariant divisor which is no $G$-...
user75933
4
votes
2
answers
956
views
Borel--Bott--Weil for the Grassmannians
The Borel--Bott--Weil Theorem is usually stated for the complete flag manifold of $SU(N)$. Does an analogue hold for the other flags, for example the Grassmannians?
More precisely, suppose $G(\mathbf ...
- 449
4
votes
2
answers
254
views
Invariant planes of a nilpotent matrix with two Jordan blocks of size two
Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map
$$
N = \begin{pmatrix}
0 & 1 & & \\
0 & 0 & & \\
& & 0 &...
- 1,123
4
votes
1
answer
549
views
are quiver varieties local complete intersections?
Is it known when a Nakajima quiver variety happens to be a local complete intersection?
[For simplicity consider an affine quiver variety, i.e. the categorical quotient of the zero set of the moment ...
- 1,526
4
votes
3
answers
1k
views
Fractional Quantum Hall Effect - Mathematics
Just to include something that starts to answer my own question Topological Quantum Computation Lecture notes covers a lot of the Mathematics of the Fractional Quantum Hall effect, or topological ...
- 379
4
votes
2
answers
675
views
Smoothness of orbit of group scheme
Let $G$ be a smooth affine group scheme over a base $S$. $G$ acts on a scheme $X$ over $S$. Let $x$ be an $S$-point in $X$. Then we have an orbit map $G\to X$. I wonder when the image (set-...
- 1,457
4
votes
1
answer
210
views
A map on Grassmannian
Let $G=SL_{2n}$ and let $\sigma:G \to G$ be defined by $\sigma (A)= E(A^t)^{-1}E^{-1}$, where $E=antidiag(1,1, ... ,1,-1,-1,...,-1)$. Then the maximal parabolic associated to the simple root $\...
- 121
4
votes
1
answer
2k
views
Reference request for an introduction to deformation theory in algebraic geometry
I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...
4
votes
1
answer
334
views
"Eigenvalue characters"
This question is an addition to my question on simultaneous diagonalization from yesterday and it is probably also obvious but I just don't know this: Let $G$ be a commutative affine algebraic group ...
- 5,243
4
votes
1
answer
392
views
References on Namikawa-Weyl group
What are the most reasonable references on the definition of the Namikawa-Weyl groups and the first results about them?
In particular, are there more recent (or more educational) texts than the ...
- 153
4
votes
2
answers
869
views
Research topics in representation theory of algebras [closed]
I was wondering what are some of the hot topics in quiver representation or representation theory of algebras that can lead to good mathematics and is important to many mathematicians and top ...
4
votes
1
answer
464
views
Describing the affine Grassmanian via $G$-bundles on $\mathbb{P}^1$
Let $G$ be a simple algebraic group, $\mathcal{O}=\mathbb{C}[[t]], \mathcal{K}=\mathbb{C}((t))$ and let $\text{Gr}_G=G(\mathcal{K})/G(\mathcal{O})$ be the affine Grassmanian. My main question:
Why ...
- 2,216
4
votes
1
answer
482
views
Linear reductivity of $SL_n$ in char $0$: proof in Mukai's book
I'm reading through Mukai's excellent book "Introduction to Invariants and Moduli", and am stuck on a proof in Chapter 4. He's proving that $G = SL_n$ over a field $k$ of characteristic $0$ is ...
- 3,035
4
votes
1
answer
215
views
Proper morphism
Maybe this could be a silly question, but I am considering the following problem.
Let $G$ be a connected reductive group and $X$ be an affine $G$-variety, i.e., $G$ acts on $X$. (You can assume the ...
- 147
4
votes
1
answer
344
views
Tannakian fundamental group of automorphic representations
Let $\mathcal{C}_{\mathrm{aut}}(G, F)$ be the category of automorphic representations of a connected reductive group $G$ over a number field $F$.
If this is a Tannakian category, it has an associated ...
- 87
4
votes
1
answer
553
views
Do all orbits have the same dimension?
Well, I've already asked this question at math.SE — but no-one's answered or commented. So now I'm posting it here (it's about the research paper — I think that it isn't an off-topic for this forum) — ...
- 41
4
votes
1
answer
343
views
Subgroups of algebraic groups containing regular unipotent elements
Let G be a simple algebraic group. Let H be a reductive subgroup of G which contains a regular unipotent element of G. Such subgroups were classified by Saxl and Seitz in all good characteristics. I'...
- 2,751
4
votes
1
answer
180
views
points with small U stabilizer on a spherical variety
Let $(G,H)$ be a spherical pair (i.e. $G$ is a reductive group, $H$ is a closed subgroup and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H$). Let $U$ be the unipotent radical of $...
- 2,639
4
votes
1
answer
303
views
The Gysin Sequence for an Associated Bundle over a Partial Flag Variety
Let $G$ be a connected, simply-connected complex semisimple Lie group, and let $P\subseteq G$ be a parabolic subgroup. Suppose that $V$ is a $1$-dimensional complex $P$-representation and consider the ...
- 4,920
4
votes
2
answers
459
views
Orbits on the affine Grassmanian, and closure ordering
Let $\mathcal{K} = \mathbb{C}((t)), \mathcal{O}=\mathbb{C}[[t]]$, $G=SL_2$ (or any semisimple group), and $\text{Gr}_G=G(\mathcal{K})/G(\mathcal{O})$; there is a left action of $G(\mathcal{O})$ on $\...
- 2,216
4
votes
1
answer
283
views
Minimal relative Schubert modules
I am trying to better understand the definition of certain objects called minimal relative Schubert modules. My primary reference is Chapters 1 and 2 of Wilberd van der Kallen's Lectures on Frobenius ...
- 2,160
4
votes
1
answer
252
views
Symplectic structure of Higgs branch
I've been reading Kamnitzer's survey Symplectic resolutions, symplectic duality, and Coulomb branches. Here the Higgs branch is defined as a projective GIT quotient, but I couldn't figure out how the ...
- 323
4
votes
1
answer
296
views
Computation of multiplicity of irreducible representation in some representation via geometric Satake correspondence
Let $G$ be a reductive algebraic group over $\mathbb{C}$. Let $\operatorname{Gr}_{G}$ be the corresponding affine Grassmannian ($\operatorname{Gr}_{G}(\mathbb{C})=G(\mathbb{C}((z)))/G(\mathbb{C}[[z]])$...
- 103
4
votes
1
answer
319
views
How to write down the connection morphism in the long exact sequence in Čech cohomology explicitly in this specific case?
Fix an integer $k$. Let $X=G/P$ be a complex rational homogeneous variety. I assume here $G$ is a simply connected semi simple complex Lie group and $P=P_k$ is a maximal parabolic subgroup defined by ...
- 279
4
votes
1
answer
192
views
Quiver invariants as polynomials/algebraic curves
I'm interested in algebraic curves one can associate to gauge or string theories. Examples involve Seiberg-Witten curves or family of A-polynomials which define holomorphic Lagrangian submanifolds for ...
- 243
4
votes
0
answers
81
views
Classification of nilpotent orbits over local fields (for type ABCD via partitions )
Let $\mathfrak g$ be a simple Lie algebra over a char $0$ local field $F$ (e.g. $F=\mathbb R$ or $F=\mathbb Q_p$) with its adjoint group $G$. Let $\mathcal N \subseteq \mathfrak g$ be its nilpotent ...
- 6,622
4
votes
0
answers
129
views
How does one compute the group action of the automorphism group on integral cohomology?
Suppose I have a curve $X$ (for concreteness, we can take $X$ to be a smooth, projective curve over a finite field $\mathbb F_q$, and even more concretely consider the family of curves described by ...
- 7,746
4
votes
0
answers
186
views
Relation between exotic sheaves in Achar's notes and in Bezrukavnikov-Mirkovic
I am trying to calculate explicitly a certain simple exotic sheaf (a simple object of the heart of the exotic t-structure on the Springer resolution, which is defined in Theorem 1.5.1 of Bezrukavnikov-...
- 2,964
4
votes
0
answers
129
views
Identify the universal centralizer of $G$ as moduli space of 'flat connections' on lagrangians in cotangent bundle of $G^{\vee}$
Note that $\mathbb{C}^*$ can be interpreted as the space of flat $\mathbb{C}^*$-connections on the dual of $\mathbb{C}^*$. Our goal is to find a similar construction for $\mathbb{C}$, particularly in ...
- 303
4
votes
0
answers
219
views
Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$
EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect.
Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
- 7,300
4
votes
0
answers
105
views
Matrix description for automorphisms of genus $2$ curve split into two copies of an elliptic curve
Consider a genus $2$ curve $C$ and its Jacobian $J$ (for simplicity, over the field $\overline{\mathbb{Q}}$ or $\mathbb{C}$). Assume that $J$ is $(2,2)$-isogenous to the direct square $E^2$ of an ...
- 1,833
4
votes
0
answers
106
views
Confusion about D-affineness and jet sheaves on projective line
I have encountered what feels like a very basic confusion relating to the Beilinson Bersnstein localisation theorem.
This theorem in particular states that if $F$ is a $D_{\mathbb{P}^{1}}$ module, ...
user108998
4
votes
0
answers
177
views
Reference for Iwahori-Hecke algebras
I recently came across the notion of an Iwahori-Hecke algebra. I would like to learn the basics about this type of algebras (mainly to get an intuition about them, as they seem to be related to some ...
- 541
4
votes
0
answers
204
views
Explicit description of wonderful compactification for PGL_3
Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry ...
- 163
4
votes
0
answers
194
views
Reference request: ranks of quadrics containing Segre variety
Let $U_{n}$ be a vector space of dimension $n$.
From plethysm we obtain an isomorphism
$$\mathrm{Ker}(S^2(U_4\otimes U_3)^{\vee}\stackrel{p}{\rightarrow} S^2U_4^{\vee}\otimes S^2 U_3^{\vee})\simeq \...
- 51
4
votes
0
answers
204
views
$\DeclareMathOperator\sym{sym}$Does $L(s, \sym^2 f \times \sym^2 g)$ have a pole at $s=1$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I encountered a question on the poles of $\GL_3\times \GL_3$ $L$-function, which needs the knowledge of the experts ...
- 563
4
votes
0
answers
163
views
External tensor product Calabi-Yau DG categories
Let $\mathcal{C}$ be a smooth proper DG-category such that the shift $[p]$ is a Serre functor for $D^{perf}(\mathcal{C})$ (we say that $D^{perf}(\mathcal{C})$ is $p$-Calabi-Yau). I am looking for a ...
- 7,300
4
votes
0
answers
78
views
Minimal set generators ideal submaximal minors
Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as:
$$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...
- 7,300
4
votes
0
answers
115
views
Tensors of minimal rank in Schur modules $S_{\lambda}V \subset V^{\otimes |\lambda|}$
It is well known that for a vector space $V$ with $\dim(V)=n+1$ the $GL(V)-$module $V^{\otimes d}$ splits as a sum of irreducible representations (with suitable multiplicities) $S_{\lambda}V$, where $\...
- 1,343
4
votes
0
answers
264
views
Singular del Pezzo surfaces and degeneration of root systems
Let $S$ be a smooth del Pezzo surface of degree $d$ and $K_S^*$ the anticanonical class. It is well known that the set of classes
$$R(S)=\{\alpha\in H^2(S,\mathbb Z)|\alpha^2=-2,\alpha\cdot K_S^*=0\},$...
- 1,803
4
votes
0
answers
82
views
Support of the fundamental class of irreducible components of partial Springer fibres corresponding to flag signatures
The (partial) Springer resolution is defined as a map $\mu: T^*\mathcal{F} \to \mathcal{N}$, where $\mathcal{F}$ is the partial flag variety consisting of $n$-step partial flags of $\mathbb{C}^d$, and ...
- 159
4
votes
0
answers
140
views
Quotient Jordan property
The Jordan property for finite subgroups of ${\rm GL}_n(\mathbb{C})$ says that there exists a constant $c(n)$ so that for any finite subgroup $G$ of ${\rm GL}_n(\mathbb{C})$ there is a normal abelian ...
- 405
4
votes
0
answers
141
views
Singular schemes with a torus action and embedded points
I've got a couple rather geometric questions about the following setup.
Let $X$ be a scheme over an algebraically closed field ($\mathbb{C}$, say) with the action of a torus $T$, such that the action ...
- 841
4
votes
0
answers
156
views
Generalized minors and Pfaffians
When $P$ is the maximal parabolic corresponding to the spin node in type $B_n$, the homogeneous coordinate ring of $G/P$ is generated by the generalized minors associated to the weights in the Weyl ...
- 81
4
votes
0
answers
215
views
Local structure of non-normal toric varieties---possible mistake in "Discriminants, Resultants and Multidimensional Determinants"
I believe I may have a counterexample to Theorem 5.3.1 on page 179 from the book book Discriminants, Resultants and Multidimensional Determinants by Gel'fand, Kapranov, and Zelevinsky. To summarize ...
- 3,079
4
votes
0
answers
486
views
Plucker coordinates of flag varieties
I am interested in understanding Lemma A.2 in the paper "Moduli spaces of principal F-bundles" by varshavsky which you can find here. It uses so called "Plücker" coordinates of the flag variety for ...
- 41
4
votes
0
answers
241
views
Semisimple vs ordinary trace of Frobenius on nearby cycles of affine flag variety
Let $G \rightarrow X$ be a parahoric group scheme over a curve, with parahoric level structure at $x_0$. Gaitsgory essentially showed that the nearby cycles functor $R\Psi$ takes perverse sheaves on ...
- 2,809
4
votes
0
answers
233
views
"Lifting" of Jacobi forms of weight zero vs. index one?
In this question I'll try to avoid using the words "Borcherd's Lift" only because I'm not sure in what setting it applies properly. What I will be asking about is sometimes called "second quantized ...
- 1,701
4
votes
0
answers
76
views
Comparing parametrizations of unipotent radical
Let ${G}$ be a simple algebraic group over $\mathbb{C}$ with maximal torus $T$ and set of simple roots $\{\alpha_i\}_{i\in \Delta}$. We then have a Borel supgroup $B=TU$ with unipotent radical $U$. ...
- 2,516