All Questions
Tagged with rt.representation-theory ag.algebraic-geometry
782 questions
8
votes
1
answer
348
views
How fine an invariant of a representation is its quotient singularity?
This is a refinement of a question asked on MSE.
Let $G$ be a finite group and let $V$ be a finite-dimensional faithful complex representation of $G$. Consider $V$ as an affine complex variety. In ...
- 2,851
8
votes
2
answers
2k
views
Global Affine Flag Variety and Affine Flag Variety
There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the ...
- 1,719
8
votes
2
answers
798
views
When is/isn't the monoidal unit compact projective?
I am interested in developing intuition for when the monoidal unit in a closed monoidal abelian category is or isn't compact projective. As such, my question is not looking for a yes/no answer, but ...
- 54.6k
8
votes
1
answer
351
views
Irreducible $S_n$-modules and $S_n$-actions on projective spaces
Let $V$ be an $(N+1)$-dimensional vector space with an action of the symmetric group $S_n$, such that $V$ is an irreducible $S_n$ -module.
Let $\{p_1,...,p_h\}\in \mathbb{P}(V)$ be $h\geq N+2$ ...
user91659
8
votes
1
answer
530
views
Restriction to Levi Subgroups and the Affine Grassmannian
Let $G$ be a complex reductive group, $L\subset G$ a Levi subgroup and $Rep(G)$ the category of rational representations of $G$.
My Question:
What is the geometric analogue of the restriction ...
- 2,554
8
votes
1
answer
264
views
Class group of hypersurfaces of finite representation type
Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x_2,\dots,x_d]]/(f)$, where $f\in(x,y,x_2,\dots,x_d)^2$, $f\neq0$. By results of Buchweitz-...
- 411
8
votes
1
answer
599
views
Affine vs Yokonuma
Let $G=GL_n$. Let us start with the Hecke algebra $H_n$. It acts on K(constructible sheaves on $G/B$) by Hecke correpondences and on K(coherent sheaves on $G/B$) by Lusztig's construction [1]. Now we ...
- 3,772
8
votes
2
answers
386
views
regular semisimple elements on spherical varieties
Let $(G,H_1)$ and $(G,H_2)$ be spherical pairs (i.e. $G$ is a reductive group, $H_i$ are its closed subgroups and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H_i$).
What can ...
- 2,649
8
votes
1
answer
1k
views
Geometric Intuition for Big Monodromy
In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM elliptic curves. A ...
- 671
8
votes
1
answer
461
views
Existence of at least one compact orbit on the sphere
Good morning,
I've came across this question during my researches. It seems apparently very simple, however I Googled a bit and I couldn't find the answer I was looking for.
The question is as ...
- 277
8
votes
1
answer
493
views
About the strength of representation-theoretic obstructions for orbit closure problems
Let $G$ be a reductive, affine, algebraic group over $\newcommand{\C}{\mathbb C}\C$. Let $X$ be a $G$-variety. For $x\in X$, we write
$$G_x:=\{ g\in G\mid g.x=x\}$$
for its stabilizer and for any ...
- 4,902
8
votes
0
answers
388
views
Character stack and character variety
Let $\Sigma$ be a Riemann surface of genus $g$. We can consider two different type of objects associated to it parametrising representations of its fundamental groups. On one side we have the ...
- 1,061
8
votes
0
answers
150
views
Equivariant coherent sheaf category for unipotent group actions
Suppose $U$ is a complex algebraic unipotent group. Let $X$ be a projective variety with a $U$-action. For simplicity, we may assume that there are only finite many $U$ orbits on $X$. The primary ...
- 163
8
votes
0
answers
1k
views
Ramified Geometric Langlands
Is the following a reasonable formulation of (a part of) the geometric Langlands conjecture for $\mathrm{GL}_n$ over a curve $X$?
(*) Let $\mathcal{L}$ be an irreducible rank $n$ $\ell$-adic local ...
- 2,751
8
votes
0
answers
173
views
On constructible Hall algebra and instantons
I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-...
- 661
8
votes
0
answers
372
views
$p$-adic representations of the fundamental group of a smooth proper curve over a finite field
This question is very general. Let $C$ be a smooth and proper curve over a finite field ${\bf F}_p$. Are there any general results or conjectures on continuous non abelian representations
$$
\pi_1(C)\...
- 3,076
8
votes
0
answers
636
views
Chern Classes of Exterior Products of a vector bundle.
This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down $c(\wedge^k V)$ i.e. the individual summands in terms of the individual summands of $...
- 327
8
votes
1
answer
382
views
Action of the endomorphism monoid on an irreducible GL-module
Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...
- 4,902
7
votes
1
answer
749
views
Number of conjugacy classes of finite reductive groups
Let $G$ be a connected reductive group over $\mathbb{Z}$. Let $c_{G(\mathbb{F}_q)}$ be the number of conjugacy classes of $G(\mathbb{F}_q)$.
Question: Is it true that $c_{G(\mathbb{F}_q)}$ is a quasi-...
- 2,751
7
votes
2
answers
1k
views
Identifying $T^* \mathrm{Bun}_G$ with Higgs bundles
Let $G$ be a semisimple algebraic group, $C$ be a smooth projective curve, and $\omega$ be the canonical line bundle.
The stack $\mathrm{Higgs}_{\omega}$ is defined as the stack associating to each $S$...
- 2,216
7
votes
3
answers
1k
views
The relation of nilpotent orbits and simple singularities, for orbits smaller than subregular ones
Pick a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. There is a partial ordering among nilpotent orbits defined by $O\geq O'$ iff $\bar O\supset O'$.
The unique maximal element under this ...
- 6,123
7
votes
2
answers
775
views
Why is the generalized flag variety a “variety”?
In several places (for example, Chriss & Ginzburg’s book “Representation Theory and Complex Geometry”), the author says that the set $X$ of Borel subalgebras of a semi-simple Lie algebra $\...
- 159
7
votes
1
answer
872
views
What are local spaces and what are they good for?
Factorization structures have been popular in the past decade. Recently a variant of this structure has been suggested by Ivan Mirkovic (and possibly collaborators). This variant, which goes under the ...
- 2,751
7
votes
3
answers
716
views
Nice algebraic approximations of classifying spaces
Let $G=GL_k(\mathbb C)$ be the complex linear group. Then the infinite Grassmannian is a model for the classifying space $BG$.
We can write the infinte Grassmannian as a colimit of the finite ...
- 13.2k
7
votes
3
answers
595
views
GIT quotients for linear representations of $SL(2,\mathbb C)$
Let $V$ be the standard two-dimensional representation of $SL(2,\mathbb C)$ and let ${\rm Sym}^2V$ be its symmetric square. Let $n$ be a positive integer and consider the following two representations ...
- 14.3k
7
votes
1
answer
613
views
Are all representations of the geometric étale fundamental group subquotients of representations of the arithmetic étale fundamental group?
Let $X$ be a variety over a field $k$. The étale fundamental group of $X$ fits into the exact sequence:
$$1 \to \pi_1^{\text{geom}}(X) \to \pi_1^{\text{arith}}(X) \to \text{Gal}(\overline{k}/k) \to 1,$...
- 2,907
7
votes
1
answer
555
views
Examples of function fields Langlands for small genus (<= 2)
See Edward Frenkel's article "Lectures on the Langlands program and conformal field theory" for an exposition of the function fields Langlands correspondence (now a theorem of Drinfel'd, L.Lafforgue &...
- 2,216
7
votes
1
answer
900
views
Bruhat decomposition for G(R), R local ring or R=Z/p^r
Is there an invariant, which encodes the failure of the Bruhat decomposition to hold for a reductive group with coefficients in a local ring like the p-adic integers or the ring $\mathbb{Z}/\mathfrak{...
- 11.2k
7
votes
1
answer
490
views
Equivariant perverse sheaves and orbit stratification
Let $X$ be a complex algebraic variety with an action of a connected algebraic group $G$.
The forgetful functor from the category of $G$-equivariant perverse sheaves on $X$ to the category of perverse ...
- 3,446
7
votes
2
answers
593
views
Whitney stratification and affine grassmanian
Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by $\lambda\...
- 3,472
7
votes
2
answers
1k
views
Strata of the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and denote by $\mathcal{G}$ its affine Grassmannian. Fix a maximal torus $T\subseteq G$. We know that $\mathcal{G}$ ...
- 4,920
7
votes
2
answers
999
views
Kostant's theorem on invariant polynomials in positive characteristic
Let $k$ be an algebraically closed field and let $G$ be a reductive linear algebraic group over $k$ with Lie algebra $\mathfrak g$. If the characteristic of $k$ is $0$ then, by a classical result of ...
- 3,637
7
votes
1
answer
616
views
l-adic Turrittin
What would be an l-adic analogue of the Turrittin-Levelt decomposition theorem?
Turrittin-Levelt is a structure theorem of meromorphic connections on a complex curve in the formal neighbourhood of a ...
7
votes
1
answer
978
views
Decomposition of induced representations in S_n
Let C be a cyclic subgroup of S_n.
How does the representation $Ind_C^{S_n}\rho$, where $\rho$ is some representation of $C$, decompose into irreducible components?
Is there are a way to know which ...
- 2,179
7
votes
2
answers
922
views
What is the geometric meaning of reconstruction of quantum group via Ringel Hall algebra
If I remembered correctly. There are some work done by C.M.Ringel,he defined so called Ringel-Hall algebra on abelian category and then show that Ringel-hall algebra is isomorphic to positive part of ...
- 5,515
7
votes
2
answers
595
views
Representation theory of Discrete Subgroups of Lie groups
My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good ...
- 327
7
votes
1
answer
443
views
Integral lattices in Lie group representations
Let $G$ be a split semisimple algebraic group scheme over $\mathbf{Z}$ (I'm mostly interested in the case $G = Sp_4$).
Let $V$ be an irreducible representation of the generic fibre $G_{\mathbf{Q}}$, ...
- 37k
7
votes
1
answer
472
views
Spin manifolds with one parallel spinor
Are there any examples of D-dimensional Ricci-flat Riemannian (spin) manifolds of dimension D= 2,3,4,5 with the dimension of the space of parallel spinors equal to 1? And the same question for the ...
- 371
7
votes
2
answers
545
views
Generators for the algebra of GL(n)-equivariant maps from M_n + M_n to M_n
Let $M_n$ be the set of $n$-by-$n$ matrices with complex entries, viewed as a variety over $k=\mathbb{C}$. Equip $M_n$ with the conjugation action of $\mathrm{GL}(n)=\mathrm{GL}(n,\mathbb{C})$. ...
user9154
7
votes
1
answer
710
views
Bialynicki-Birula Decomposition and moment polytopes/graphs
Let $X$ be a possibly singular projective scheme which admits a torus $T$ action and has finitely many $T$ fixed points and one-dimensional $T-$orbits. There are many such schemes in the Grassmannian/...
- 1,719
7
votes
1
answer
477
views
Conjugation of the quotient of $\mathrm{SL}(n,\mathbb{C})$ by a finite subgroup
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$EDITED Let $G=\SL_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$.
Let $H\subset G$ be a finite subgroup.
Set $X=G/H$ be the ...
- 14.2k
7
votes
1
answer
180
views
Complexity of rational $\mathrm{GL}_{n(r)}$-modules
Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G=\mathrm{GL}_n(k)$ for some natural number $n$. For any integer $r\ge 1$, let $G_{(r)}$ denote the $r$th Frobenius ...
- 768
7
votes
1
answer
254
views
Group-like elements in quotients of group rings
$\DeclareMathOperator\Gr{Gr}$Let $R$ be a local ring, let $A$ be a finite abelian group, and let $I$ be a Hopf ideal of the ring $R[A]$. The quotient $R[A]\twoheadrightarrow R[A]/I$ induces a map on ...
- 308
7
votes
1
answer
489
views
Combinatorial meaning of Kazhdan-Lusztig-Stanley polynomial
This question is motivated by
Why do combinatorial abstractions of geometric objects behave so well?
The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials
Kazhdan-Lusztig-Stanley polynomials ...
- 5,230
7
votes
1
answer
384
views
Why do non-equioriented A<sub>n</sub> quivers have singularities identical to the singularities of Schubert varieties?
In the general case, quiver cycles are of the form of orbit closures of $GL\cdot V_{\vec{r}}$, where $GL= \prod_{i=0}^n GL_{r_i}$ is the possible changes of basis on all of the vector spaces on each ...
7
votes
0
answers
1k
views
What's the point of geometric representation theory?
Please forgive the provocative title, what I mean is the following:
One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...
- 157
7
votes
0
answers
259
views
Status of the Luna's conjecture
In the famous IHES paper <Variétés Sphériques de Type A> of D. Luna, he proposed a conjecture asserting that wonderful varieties of an adjoint semisimple group $G$ are bijective to spherical ...
- 291
7
votes
0
answers
250
views
$D(\mathcal{O}(n))$ via generators and relations
Let $V$ be a complex vector space. Consider the algebra $D(\mathbb{P}(V),\mathcal{O}(n)))$ of global differential operators from line bundle $\mathcal{O}(n)$ to itself, here $n \in \mathbb{Z}_{\...
- 143
7
votes
0
answers
237
views
$X$ with $H^*(X)=$affine Verma module?
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbf{C}$, and $\widehat{\mathfrak{g}}_\kappa$ the associated affine Lie algebra. It is the central extension of the loop algebra $...
- 5,701
7
votes
0
answers
261
views
Explicit form of raising and lowering operators in spherical gl(n) DAHA
I am working with polynomial representations of spherical subalgebra of double affine Hecke algebra (DAHA) for $\mathfrak{gl}_n$.
Let's call this algebra $\mathfrak{A}_n$ for short. Typically we think ...
- 765