All Questions
Tagged with rt.representation-theory ag.algebraic-geometry
782 questions
2
votes
2
answers
342
views
semisimple category with finite number of simple objects
Since I am not an expert in algebraic groups; Is there a description of algebraic groups with semisimple category of finite dimensional representations such that they have only finitely many ...
- 165
2
votes
1
answer
281
views
resolution of strata of the affine grassmanian
Let G a semisimple simply connected group over an algebraically closed field.
Let $Gr:= G(k((t))/G(k[[t]])$ be the affine grassmanian. It admits a stratification indexed by the dominant cocaracter
$...
- 3,472
2
votes
2
answers
353
views
springer resolution over $\wedge^3 \mathbb{C}^6$
The action of $GL_6$ on $P(\wedge^3 \mathbb{C}^6)=P^{19}$ has 4 orbits (of dim 19, 18, 14, 9). Can you describe how the springer resolution applies to each of these orbits? It should have positive ...
- 3,779
2
votes
1
answer
938
views
S. Agnihotri, "Quantum cohomology and the Verlinde algebra"
I am looking for the Oxford PhD thesis of S. Agnihotri, "Quantum cohomology and the Verlinde algebra". I can't seem to find it online. Does anyone know how / where I can find this? Thank you!
- 21k
2
votes
2
answers
527
views
Density and irreducibility
Let G be an linear algebraic group, and let H be a zariski dense subgroup of G. Then does H have to be a irreducible subgroup of G? Here H being irreducible means that H has no nontrivial invariant ...
- 159
2
votes
1
answer
198
views
Orbital integral in Cluckers and Denef
My questions concerns Definition 1.2 of an orbital integral in the paper Orbital integrals on General Linear Groups by Cluckers and Denef. I will recall the definition below, but my question is: how ...
- 3,799
2
votes
1
answer
84
views
reduction of torsion modules
Let $G$ be a profinite group.
Let $K(G,\mathbb Z_\ell)$ be the Grothendieck group of the derived category of finitely generated $\mathbb Z_\ell$-modules with continuous $G$-action.
Let $K(G,\mathbb ...
- 135
2
votes
1
answer
196
views
Dimension of spaces of invariants/tableaux functions
The Hook lenght formula gives the number of standard Young tableaux on a given diagram.
A variant gives the number of semistandard tableuax.
Does there exist a formula for counting "weighted ...
- 3,779
2
votes
1
answer
177
views
Action of $O(3,\mathbb{R})$ on the conic $\{x^2+y^2+z^2=0\}$
The action of the orthogonal group $O(3,\mathbb{R})$ on the conic
$C= \{ x^2+y^2+z^2=0 \}$ in $\mathbb{P}^2$ must be well-understood, but I could not find any reference.
Is it doubly transitive?
- 14k
2
votes
2
answers
270
views
Zariski closure of the image of an induced representation
Let $G$ be a finitely generated discrete group, $H\le G$ a subgroup of finite index $d$, and let $\rho : H\rightarrow \operatorname{GL}(n,\mathbb{C})$ be a representation.
Let $\tilde{\rho} := \...
- 7,527
2
votes
1
answer
186
views
Orbits in the open set given by Rosenlicht's result
Let $G$ be a linearly reductive algebraic group, and let $X$ be an irreducible affine variety, over an algebraically closed field $\mathbb{K}$, with a regular action of $G$. By Rosenlicht's result, we ...
- 839
2
votes
1
answer
249
views
Is there an $SL_n$-invariant functional on the space of rational functions on the projective space $\mathbb P^{n-1}$?
Let the group $SL_n$ act on the projective space $\mathbb P^{n-1}$ in the standard way (both defined over $\mathbb C$).
Is there an $SL_n$-invariant (linear) functional on the space of rational ...
- 2,639
2
votes
1
answer
98
views
A weaker version of strongly graded algebras
Let $A = \oplus_{i \in \mathbb{Z}} A_i$ be a graded algebra. We say that it is strongly graded if $A_i.A_j = A_{i+j}$, for all $i,j \in \mathbb{Z}$. Can there be existing a graded algebra such that
$$...
2
votes
1
answer
261
views
Property of representations of reductive group schemes over characteristic 0 field
I originally posted this on Maths SE, but then I thought it MO might be more fitting.
Let $k$ be a characteristic $0$ field and let $G$ be a linear algebraic group scheme over $k$. Then is it true ...
- 1,516
2
votes
1
answer
202
views
Core components of quiver varieties as fiber bundles of flag varieties
Is there an example of Nakajima quiver variety of type A which has all core components smooth, such that at least one of them is NOT an iterated fibre bundle of flag manifolds (i.e. a space obtained ...
- 1,677
2
votes
1
answer
165
views
Representation of symmetric group as Cremona transformations
Question from me and a colleague:
Given a matrix
\begin{equation}
U =
\begin{bmatrix}
U_{11} & U_{12} \\
U_{21} & U_{22}
\end{bmatrix}
\quad \text{with } U_{22} \neq 0,
\end{equation}
...
- 533
2
votes
1
answer
273
views
Multiplication of section of pushforward structure sheaf via finite flat morphism
Let $f: X \rightarrow Y$ be a finite, flat morphism of curves of degree $n$. The direct image of the structure sheaf $f_* O_X$ is a locally free $O_Y$-module.
Given a local section $s$ of $f_* O_X$ ...
- 479
2
votes
1
answer
306
views
connectedness of fibers of torus-equivariant moment maps
Given a possibly singular, connected, symplectic algebraic variety with a torus action, every fiber of the moment map admits a torus action. Is each fiber of this moment map connected? Any examples or ...
- 1,719
2
votes
2
answers
877
views
Tangent space of moduli of stable principal $G$-bundles on a compact Riemann surface
This is probably a dumb question.
Let $G$ be a connected complex reductive group and $X$ a compact Riemann surface. Consider a stable principal $G$-bundle $P$ on $X$. I am interested in how one ...
- 59
2
votes
1
answer
372
views
Ext groups in the equivariant derived category
I apologize in advance that this question is probably too basic for MO, but I reckoned I would not get an answer on Math.Stackexchange.
I am starting to learn about perverse sheaves, the ...
2
votes
1
answer
1k
views
What is the cohomology of the tangent bundle of a flag variety?
Let $G$ be the general linear group $\operatorname{GL}(n,\mathbb{C})$ and $P$ a parabolic subgroup with Lie algebra $\mathfrak{p}$. Consider the vector bundles
$$
\mathcal{P} = G\times_P \mathfrak{...
2
votes
1
answer
282
views
Why are Stab(M) and Aut_R(M) isomorphic as schemes?, and why is Aut_R(M) smooth over F?
I have been trying to understand a sketch of a proof from P. Gabriel's article "Finite representation type is open".
Let $F$ be an algebraically closed field, and let $R$ be a finite dimensional $F$-...
2
votes
1
answer
593
views
semisimple restricted representation
As a consequence of THEOREM 2.6 in [ROBERT J. BLATTNER, SUSAN MONTGOMERY, CROSSED PRODUCTS AND GALOIS EXTENSIONS OF HOPF ALGEBRAS, PACIFIC JOURNAL OF MATHEMATICS
Vol. 137, No. 1, 1989, 37-54], we know ...
- 491
2
votes
1
answer
677
views
About localization theorem for affine Lie algebra?
Here is my question: how to define global section functor from D-module on affine flag variety to representation of affine Lie algebra?
Let's me explain the difficulty: it seems there doesn't exist ...
- 1,457
2
votes
1
answer
287
views
On the definition of the Cherednik algebra of a variety with a finite group action
Let $X$ be a connected complex smooth affine variety, acted on by a finite group $G$. We define a reflection hypersurface $(Y,g)$ as a smooth codimension one subvariety $Y\subset X$ which is fixed by $...
- 541
2
votes
3
answers
651
views
Simultaneous similarity of pair of matrices
Let $k$ be an arbitrary field, and $A,B,A',B'\in M_n(k)$. Do we have any algorithm with polynomial complexity to determine the simultaneous similarity of the pair $(A,B)$ with $(A',B')$?
I found the ...
- 129
2
votes
1
answer
381
views
Counting cosets in the Quotient of Weyl groups
Let $\Sigma=G/P$ be a flag variety of type $A$,i.e. $G=SL_{n+1}$ and is a stabilizer of the partial flag $0 \subset V_{d_1}\subset \cdots\subset V_{d_r}\subset V_{n+1}$ of length $r$. Let $W$ be its ...
- 83
2
votes
1
answer
311
views
No cuspidal character sheaves on GL(n)
We need a reference for the fact that there are no cuspidal character sheaves on $GL_n$ unless $n=1$.
See page 11 of http://www.kurims.kyoto-u.ac.jp/~arakawa/Henderson_mgsctalk2.pdf.
- 2,639
2
votes
1
answer
687
views
Derived Push-Forward of Morphism of Perverse Sheaves and Translation Functors
I hope this question is not too vague.
Let $G$ be a complex reductive group, $B$ a Borel subgroup of $G$, and $P$ a parabolic containing $B$.
Denote by $\pi:G/B\to G/P$ the canonical map. Consider ...
- 2,554
2
votes
1
answer
356
views
Observable Unipotent Algebraic Subgroups of the Unipotent Upper Triangular Groups
It is well known that any unipotent algebraic group (over a field) can be embedded as a closed subgroup of $U_n$ for some $n$, where $U_n$ denotes the set of all $n \times n$ upper triangular matrices ...
- 511
2
votes
1
answer
264
views
Springer fibres for nilpotents of type $(n,n)$; framed tangles
My question is about the paper "Affine tangles and irreducible exotic sheaves" (arxiv.org/abs/0802.1070).
Background: Let $\mathcal{B}, \mathcal{N}$ denote the flag variety and nilpotent cone of $G=...
- 2,216
2
votes
0
answers
92
views
Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
2
votes
0
answers
136
views
Irreducible non-reductive subgroup in GL(n) over a characteristic 0 field
Let $V$ be an $n$-dimensional vector space over a characteristic $0$ field $k$ (or better, let $V=\mathbb{A}^n_k$). I wonder whether the following is true:
Absolutely irreducible subgroups $H$ of $\...
- 455
2
votes
0
answers
118
views
Growth of invariants in mod-$p$ representations of $\mathrm{GL}_n$
Let $G$ be smooth admissible mod-$p$ representations of $\mathrm{GL}_n(\mathbb{Q}_p)$. Also suppose $\pi$ is an irreducible infinite-dimensional smooth admissible representation of $G$ over $\mathbb{F}...
2
votes
0
answers
147
views
Abstract definition of hypertoric varieties
I'm reading Proudfoot's survey on hypertoric varieties. In Section 1.4 he mentioned such a conjecture:
Conjecture 1.4.2 Any connected, symplectic, algebraic variety which is projective over its ...
- 341
2
votes
0
answers
143
views
Comparison of IC sheaves on Schubert varieties on two settings (l-adic vs. complex)
This question is basically about comparison of IC sheaves (or their sheaf cohomologies) for the settings: 1. variety is over $\mathbb{C}$ and sheaf is $\mathbb{C}$-linear, 2. variety is over a finite ...
- 323
2
votes
0
answers
132
views
Canonical basis and perverse coherent sheaves on the nilpotent cone
In the paper of Ostrik, he introduced a canonical basis of $K^{G\times {\mathbb C}^*}(\mathcal N)$, where $\mathcal N$ is the nilpotent cone for the group $G$. Question: does this canonical basis ...
- 2,964
2
votes
0
answers
169
views
The dimension of the representation ring
Let $G$ be a compact Lie group. I am trying to characterize the algebraic properties of the representation ring $R(G)$ of $G$. In the case of the $n$-torus, the representation ring $R(T)$ is ...
2
votes
0
answers
198
views
Picard group of a DM-stack
I wonder if the followings are true. Let $k$ be a field, $G/k$ is a finite constant group scheme acting on an affine $k$-scheme $\mathrm{Spec}(A)$. Then a line bundle on the quotient DM stack $[\...
- 455
2
votes
0
answers
176
views
Smooth pullback of holonomic D-modules is fully faithful
Let $X$ and $Y$ be (not necessarily smooth) algebraic varieties over an algebraically closed field of characteristic $0$, and suppose we have a smooth surjective map $f: X \to Y$ of relative dimension ...
- 3,019
2
votes
0
answers
174
views
Representability of the sheaf $\mathrm{Hom}(G,\mathrm{SL}_2)$
$\DeclareMathOperator\Spm{Spm}\DeclareMathOperator\SL{SL}\DeclareMathOperator\Hom{Hom}$Let $T$ be the topos of $\Spm\mathbb{Q}_p$-rigid analytic spaces, $G$ an abstract group, and $\Hom(G,\SL_2)$ the ...
- 2,907
2
votes
0
answers
152
views
Line bundles on toric varieties associated to Weyl chamber
I am interested in studying toric varieties associated to the fan of Weyl chambers. General information would be best but I am also interested in the specific case of the Weyl chamber of $\mathfrak{sl}...
- 41
2
votes
0
answers
206
views
Springer sheaf and Deligne-Lusztig induction
Let $G=Gl_n$ be the general linear group over the algebraic closure of a finite field $\overline{\mathbb{F}}_q $ and let $F:G \to G$ be the standard Frobenius. On $G$ there is the Springer (perverse) ...
- 1,061
2
votes
0
answers
101
views
Number of points of parabolic Springer fibres for general reductive groups
My question is the same as this post but for an arbitrary reductive $G$ instead of just $\mathrm{GL}_n$.
Let $G$ be a connected split reductive group over a finite field $k$.
Let $P$ be a parabolic ...
- 2,751
2
votes
0
answers
268
views
Understanding a proof of a result of Schofield
I'm reading a paper of Aidan Schofield- "General Representations of Quivers" and I'm trying to understand the proof of Theorem 3.3. I'm having trouble understanding the argument that's ...
- 839
2
votes
0
answers
111
views
Tannakian recovery of a group from other tensor abelian categories
Classical Tannakian reconstruction recovers an affine group scheme $G$ over $k$ from the category of its linear representations over a field $k$ (as the automorphism group of the forgetful functor to ...
- 83
2
votes
0
answers
128
views
Kac-Peterson modular forms and shifted theta functions
Let $\Lambda$ be the root lattice corresponding to an ADE root system $R$ of rank $n$. With the ADE assumption, the weight lattice is simply the dual lattice $\Lambda^{\vee}$. Given any weight vector $...
- 1,701
2
votes
0
answers
169
views
Counting points of parabolic Springer fibers
Let $G$ be a reductive group over an (algebraically closed ) field. To each parabolic subgroup $P \subseteq G$ and $x \in G$ we can consider two types of partial Springer fibers associated to it :
$$1)...
- 1,061
2
votes
0
answers
478
views
Is the ideal of the Veronese variety $V_{d,n}$ generated by quadrics?
Maybe it sounds like a silly question to the experts but I'm not able to find a proper reference in the web. Anyone knows if the ideal $I_{d,n}$ of the Veronese variety $V_{d,n}$ is generated by ...
- 1,343
2
votes
0
answers
156
views
Semisimple covers of varieties
Let $X$ be an algebraic variety.
The finite étale covers of $X$ are measured by the étale fundamental group $\pi_1^{\rm et}(X)$.
On the other hand, the Cox ring ${\rm Cox}(X)$ of $X$ (finitely ...
- 1,585