All Questions
Tagged with rt.representation-theory ag.algebraic-geometry
782 questions
10
votes
0
answers
653
views
On cyclic homology of Ginzburg's DG algebra
Let $(Q,W)$ be a quiver with potential, and $D$ be Ginzburg's DG algebra associated to it (as explained in http://arxiv.org/abs/math/0612139 and other places), so that it is a 3-Calabi-Yau algebra. I ...
- 6,123
10
votes
0
answers
881
views
Invariance of Euler characteristic under base change for sheaf cohomology of flag varieties
BACKGROUND:
Over an algebraically closed field of arbitrary characteristic, most of the basic structure theory of affine (= linear) algebraic groups can be developed concretely without quoting ...
- 52.9k
9
votes
2
answers
896
views
Tannakian Formalism for the Quaternions and Dihedral Group
It is a basic fact in representation theory of finite groups over complex numbers that the character tables of $Q_8$ and $D_8$ are identical. I believe, this implies that the corresponding categories ...
- 2,751
9
votes
2
answers
770
views
Classical invariants involving exterior powers of standard representation
While investigating certain conformal blocks line bundles on $\overline{M}_{0,n}$, I was led to what seems to be an identification between two spaces of invariants, and I am curious if there is a ...
- 1,871
9
votes
3
answers
2k
views
Borel's presentation for the cohomology of a Flag Variety
If $G$ is a simple complex Lie group, $T\subset B\subset G$ is a choice of Borel and maximal torus, and $W$ is the Weyl group, then
1) $H^{*}(G/B,K)=K[T^{\vee}]/(K[T^\vee]^W_{+})$
and
2) $K[T^\vee]^...
- 1,537
9
votes
3
answers
675
views
Tannaka duality for semisimple groups
Tannakian formalism tells us that for any rigid, symmetric monoidal, semisimple category $\mathcal{C}$ equipped with a fiber functor $F: \mathcal{C} \to Vect_k$ for a field $k$ (of characteristic $0$) ...
- 3,019
9
votes
2
answers
1k
views
Relative Lie Algebra cohomology and sheaf cohomology
(I apologize in advance for the vagueness of my question). Let $G$ be a reductive algebraic group over $\mathbb C$ with Lie algebra $\frak g$ and Borel $B$. I have seen casual references to the fact ...
- 3,637
9
votes
1
answer
1k
views
Action of k* on a variety induces grading?
Let $V$ be a $\Bbbk$-variety such that $\Bbbk^\times$ (as an algebraic group) acts algebraically on $V$. Given any $f\in\Bbbk[V]$, let us call $f$ homogeneous of degree $d$ if for all $v\in V$ and all ...
- 4,902
9
votes
2
answers
1k
views
Action on the highest weight vector of a representation of a semisimple linear algebraic group
Let $G$ be a semisimple linear algebraic group, $V$ a $G$-representation and $v \in V$ a vector of highest weight $\lambda$. Is it true, that for any positive root $\alpha \in R^+$ the one dimensional ...
- 783
9
votes
2
answers
837
views
Fixed points of an involution
Let $V=\mathbb C^{2n}$ with the standard basis $\{e_1,e_2, \cdots , e_{2n}\}$ and let $\sigma$ be the involution $e_i \mapsto -e_{2n+1-i}$. This induces an involution of the Grassmannian $G(n,2n)$ of $...
- 95
9
votes
1
answer
415
views
Is the dimension of $V//G$ always the same as the dimension of $V^*//G$?
I would like to know whether there is an example of a reductive algebraic group $G$ (say, over the complex numbers $\mathbb{C}$) and a finite dimensional representation $V$ of $G$ such that dim$(V//G)$...
- 543
9
votes
3
answers
590
views
Subgroup of $\mathrm{GL}_n$ stabilizing linear subspace skew-symmetric matrices
I am currently reading "Schiffer variations and the generic Torelli theorem for hypersurfaces" by Voisin, where it is claimed that the subgroup of $\mathrm{SL}_{2m}$ ($m \geq 3$) which preserves a ...
- 7,300
9
votes
1
answer
582
views
Degree of secant varieties of Veronese varieties
Consider the degree two Veronese embedding $\mathbb{P}^n\rightarrow\mathbb{P}^N$ and let $V^n_{2}\subset\mathbb{P}^N$ be the corresponding Veronese variety.
Let $Sec_k(V^n_{2})\subseteq\mathbb{P}^N$ ...
user124234
9
votes
1
answer
346
views
Standard Monomial basis for other types
For the algebraic group $SL_n$ (type $A_{n-1}$) and for a dominant weight $\lambda$ the standard monomials are indexed by the semi-standard young tableaux of shape $\lambda$ and they form a basis for ...
- 213
9
votes
2
answers
1k
views
modularity of algebraic varieties
Hello,
Are there any examples of varieties which are not Shimura varieties or abelian varieties
and whose L-functions have been shown to be a product of automorphic L-functions?
Thanks.
N
- 2,842
9
votes
2
answers
865
views
Multiplication in Peter-Weyl theorem
$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
- 2,964
9
votes
2
answers
3k
views
Resolving ADE singularities by blowing up
Let's say we have a finite subgroup $\Gamma \subseteq SL(2,\mathbb{C})$ and consider the quotient variety $\mathbb{C}^2/\Gamma$, which will have one of the well-known ADE or du Val surface ...
- 138
9
votes
2
answers
1k
views
Interesting examples of pro-algebraic completions of groups
Fix a field $k$. Given a discrete group $G$, the pro-algebraic completion (or Hochschild-Mostow completion) is the pro-algebraic group $A_{k}(G)$ with is universal with respect to finite dimensional ...
- 1,635
9
votes
3
answers
941
views
Buildings, projective geometry - what led Tits to think of "the field with one element"?
The mysterious object "field with one element" seems to appear first in J. Tits papers on buildings. It is mentioned in almost any text on $\mathbb{F}_1$.
However, I have never seen any exposition of ...
- 24.9k
9
votes
1
answer
1k
views
Geometric construcion of Proj as a quotient by a $\mathbb{G}_m$ action
I'm trying to translate the Proj construction as a kind of quotient by a $\mathbb{G}_m$ action. Here's what I have so far:
Let $X=Spec\,A$ be an affine scheme (after this case is setteled I imagine it ...
- 7,789
9
votes
1
answer
493
views
A compactification of the space of points on the affine line
I recently encountered an interesting space. It is a compactification of the space of $ n$ points in $ \mathbb A^1 $ modulo translation, $ (\mathbb A^1)^n / \mathbb G_a $.
Let $ n \in \mathbb N $ and ...
- 4,612
9
votes
1
answer
987
views
Closures of torus orbits in flag varieties
Consider the Lie group $G=SL_n(\mathbb C)$ with Borel subgroup $B$ and maximal torus $T\subset B$. I'm interested in the (Zariski) closures of $T$-orbits in the flag variety $F=G/B$.
Now, as far as I ...
- 3,513
9
votes
1
answer
400
views
Generalisations of Weyl's construction of irreducible representations
For the moment we work over the complex numbers.
Suppose that $G = \mathrm{SL}(V)$, or $G = \mathrm{Sp}(V)$, or $G = \mathrm{SO}(V)$.
Weyl gave explicit constructions of irreducible representations of ...
- 133
9
votes
1
answer
506
views
Current state of the art in geometric complexity theory
I came across this interesting question from almost 7 years ago:
What are the current breakthroughs of Geometric Complexity Theory?
My question is quite simple: Have there been any breakthroughs in ...
9
votes
1
answer
617
views
Characters of simply connected semsimple algebraic groups over local fields
Let $G$ be a semisimple algebraic group over $\mathbb{Q}_p$. Then by definition $G$ admits no non-trivial algebraic characters, i.e. homomorphisms $G \to \mathbb{G}_m$.
However, it is quite possible ...
- 21.3k
9
votes
2
answers
799
views
Automorphisms of generic complete intersections
This question concerns a seemingly folk lore result, which states that automorphism groups of generic complete intersections are trivial, under certain assumptions.
To state the question, let $r \geq ...
- 21.3k
9
votes
1
answer
754
views
"Approximating" $BGL(1)$ by projective spaces
Given a representation $V$ of a group $G$, we can think of $V$ as a vector bundle over the classifying stack $BG$, and we can define its index $\chi(BG; V)$ to be the dimension of the $G$-invariant ...
- 21k
9
votes
3
answers
1k
views
Is there a good account of D-affinity and localization theorem for partial flag varieties?
Recall that a topological space is called $A$-affine for a sheaf of algebras $A$ if taking global sections of coherent sheaves of $A$-modules is an equivalence of categories to finitely generated $\...
- 44.7k
9
votes
1
answer
335
views
Question about linear algebra in Benson's book: intersections of images or sum of kernels
I am not sure if this question is suitable in here. I asked this question in Mathematics some days ago.
The following proposition is in Benson's book “Representation theory of elementary abelian ...
- 496
9
votes
0
answers
256
views
Intersection of Springer fibre and Schubert cell
Let us consider intersections of Springer fibres and Schubert cells in type A.
Let $ Y : \mathbb C^n \rightarrow \mathbb C^n $ be a nilpotent operator. Let
$$
F_Y = \{ V_0 = 0 \subset V_1 \subset \...
- 4,612
9
votes
0
answers
247
views
Degree of a cone over the set of rank $r$ $n\times n$ matrices
Let $X_r\subset Mat_{n\times n}(C)$ denote the variety of rank at most $r$ matrices, set $k=n-r$ and assume $n\geq k^2-1$. Consider the cone over $X_r$ with vertex spanned by the first $k^2-1$ entries ...
- 835
9
votes
0
answers
389
views
Twisted Springer fibers
In the study of certain moduli spaces of $p$-divisible groups I came across the following twisted version of a Springer fiber, and I was wondering whether some expert on algebraic groups/algebraic ...
- 1,645
9
votes
0
answers
543
views
Status of Borho and Brylinski's irreducibility conjectures?
In Differential Operators on Homogeneous Spaces, III, Section 6.7, Borho and Brylinski make a long series of equivalent conjectures which they show are all equivalent. Perhaps the easiest statement ...
- 44.7k
9
votes
0
answers
668
views
Role of nontrivial component groups in Springer Correspondence?
Set-up for classical Springer Correspondence:
$G$ = reductive group (usually assumed to be semisimple of adjoint type) over $\mathbb{C}$, with Borel subgroup and
maximal torus $B \supset T$, Weyl ...
- 52.9k
8
votes
3
answers
803
views
Why can I divide an affine variety by the action of the general linear group?
Let $G\subseteq\mathrm{Gl}_n(\mathbb{C})$ be a subgroup of the general linear group and assume that $\rho:G\to\mathrm{Gl}(V)$ is a representation. Understand the complex vector space $V$ as an affine ...
- 4,902
8
votes
3
answers
1k
views
Further reading in algebraic geometry
I recently finished reading W. Fulton's "Algebraic Curves" and also attended a lecture series on moduli spaces and am interested in learning about them as well. I looked for a few books to ...
8
votes
3
answers
1k
views
Is there an analogue Beilinson-Bernstein localization for quantized enveloping algebra
I am completely a beginner in this field. I wonder know whether there is appropriate notion for quantum flag variety of finite dimensional Lie algebra. If so, what is the correspondent notion for &...
- 81
8
votes
3
answers
2k
views
Smoothness properties of the Springer fiber
The Springer fiber, recall, is defined (briefly) with reference to a chosen unipotent matrix $U \in \mathrm{GL}_n$, and consists of all flags $0 = F_0 \subset F_1 \subset \dots \subset F_n = \mathbb{C}...
- 7,273
8
votes
2
answers
577
views
Faithful flatness and non-commutative algebras
$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following:
Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $...
- 541
8
votes
2
answers
1k
views
Is every subgroup of an algebraic group a stabilizer for some action?
Suppose G is an algebraic group (over a field, say; maybe even over ℂ) and H⊆G is a closed subgroup. Does there necessarily exist an action of G on a scheme X and a point x∈X such that ...
8
votes
3
answers
529
views
Intuitive reason that the regular representation is a uniform function
Corollary 12.14 of Digne-Michel's book Representations of finite groups of Lie type gives various decompositions of the regular representation $\operatorname{reg}_G$ in terms of the Deligne-Lusztig ...
- 721
8
votes
1
answer
987
views
Steps in Geometric Complexity Theory
GCT purports to provide a program to show that $NP \not \subset P/poly$.
At the high level what are the steps involved in the program and what stage is each step in?
What difficulties currently are ...
- 13.9k
8
votes
1
answer
549
views
Ring of invariants for the regular representation
The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ...
- 195
8
votes
1
answer
935
views
T-bundles and the Borel-Weil-Bott theorem
Hi,
Let $G$ be a reductive, connected group, $T$ a maximal torus, and $B$ a Borel subgroup containing $T$ with unipotent radical $U$. Then it turns out that the functions on the algebraic variety $G/...
- 647
8
votes
1
answer
1k
views
The Bialynicki-Birula Stratification of the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple group with affine Grassmannian $\mathcal{G}r$. Fix a maximal torus and Borel $T\subseteq B\subseteq G$. I am reading "Loop Grassmannian ...
- 4,920
8
votes
2
answers
868
views
If a representation has enough reductive stabilizers, is it a direct sum of characters?
Suppose $G\to GL(V)$ is a linear representation of an irreducible algebraic group over a field $k$.
Suppose $C\subseteq V$ is a $G$-invariant closed cone that spans $V$, and that the stabilizer of ...
8
votes
2
answers
557
views
Quasi-affineness of the base of a $\mathbb{G}_a$-torsor
Let $\mathbb{G}_a$ be the additive group over an algebraically closed field $k$ of any characteristic. Let $X \to Y$ be a $\mathbb{G}_a$-torsor of $k$-schemes (of finite type - in case that is ...
- 1,645
8
votes
1
answer
517
views
Conformal blocks in genus zero
In section 10.4 of "Vertex Algebras and Algebraic Curves", Ben-Zvi & Frenkel (second edition), the authors claim that for any vertex algebra V, the space of one-pointed conformal blocks with ...
- 5,958
8
votes
1
answer
1k
views
Conformal Field Theory and Langlands
I'm a Mathematics masters student currently
studying some aspects of TQFT. I'm interested in Langlands, mainly
because it sounds oppressive! Is anyone familiar with any links between
CFT and Langlands,...
- 379
8
votes
1
answer
548
views
Why is the $\operatorname{GL}_n$ character variety "cohomologically" the product of the $\operatorname{PGL}_n$ character variety and a torus?
$\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\GL{\operatorname{GL}}$This question is about an assertion in Mixed Hodge polynomials of character varieties, by Hausel and Rodriguez-Villegas. Fix positive ...
- 156k