All Questions
75 questions
33
votes
2
answers
2k
views
What do cluster algebras tell us about Grassmannians?
One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
- 1,690
26
votes
1
answer
2k
views
Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?
This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...
- 52.9k
25
votes
3
answers
6k
views
Introductory References for Geometric Representation Theory
Would anyone be able to recommend text books that give an introduction to Geometric Representation Theory and survey papers that give an outline of the work that has been done in the field? I'm ...
22
votes
2
answers
2k
views
A royal road to Coulomb branches of 3D $\mathcal{N}=4$ gauge theories
So, I've been very interested recently with the developements of the (now mathematically precise) theory of Coulomb branches - in particular because of its recent applications on representation theory ...
- 3,318
20
votes
7
answers
9k
views
Elementary reference for algebraic groups
I'm looking for a reference on algebraic groups which requires only knowledge of basic material on the theory of varieties which you could find in, for example, Basic Algebraic Geometry 1 by ...
- 15.4k
19
votes
2
answers
2k
views
Dual versions of "folding" symmetric ADE Dynkin diagrams?
Start with the Dynkin diagram of an irreducible root system, typically associated with a simple
Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE
diagrams ...
- 52.9k
18
votes
1
answer
566
views
Subgroup $\mathrm{E}_6$ generated by $\mathrm{Spin_7}$ and $\mathrm{SL}_3$
Let $\mathbb{O}$ be the octonion algebra (say over $\mathbb{R}$) and let $J_{3}(\mathbb{O})$ be the set of $3 \times 3$ hermitian matrices with octonion coefficients, that is:
$$ J_3(\mathbb{O}) = \...
- 7,300
14
votes
2
answers
3k
views
How many ways are there to prove flag variety is a projective variety?
I am looking for references talking about different ways to prove flag variety $G/B$ is projective variety. Now I have some in mind:
There is a proof in Humphreys Linear algebraic groups, he first ...
- 5,515
13
votes
2
answers
3k
views
Literature on the Springer resolution
Could you suggest me a basic reading list on the Springer resolution? Is there a textbook I can refer to? Or do I need to start with the original paper?
Unfortunately googling for "Springer" and "...
13
votes
3
answers
1k
views
Non-vanishing cohomology of line bundles on projective varieties in prime characteristic?
This is a somewhat naive question about the expected non-vanishing behavior of sheaf cohomology groups $H^i(X, \mathcal{L})$, where $X$ is a smooth projective variety of dimension $d$ over an ...
- 52.9k
12
votes
1
answer
1k
views
Reference Request for Drinfeld and Laumon Compactifications
Background
Let $X$ denote a smooth projective curve over $\mathbb{C}$ and let $G$ denote a semi-simple simply connected algebraic group over $\mathbb{C},$ which has associated flag variety $G/B.$
...
- 2,706
12
votes
1
answer
567
views
Reference for character sheaves over $\mathrm{GL}_n(q)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$I know a little bit about complex representation theory of finite reductive groups as $\GL_n(q),\SO_n(q)$ etc via Deligne-Lusztig induction and ...
- 1,061
11
votes
2
answers
977
views
Reference for combinatorics with view towards representation theory/algebraic geometry
I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
- 1,061
11
votes
2
answers
684
views
Invariants of $\mathrm{GL}_n$ representations
$\DeclareMathOperator\GL{GL}$Let $V=\mathbb C^n$ be the natural representation of $\GL_n(\mathbb C)$ and let $W=\operatorname{Sym}^2(V)$ be the symmetric square representation. Let $W^k$ denote the ...
- 673
11
votes
1
answer
875
views
An arithmetic highest weight theory?
I apologize if these questions seem naive or loaded.
Is there an analogous theory of highest weights for irreducible finite-dimensional representations of Lie algebras of algebraic group (or perhaps ...
- 436
10
votes
2
answers
955
views
Semisimplicity of étale cohomology representations
Let $K$ be a number field and $G=Gal(\overline{K}/K)$ the absolute Galois group of $K$. Let $\ell$ be a prime number.
Let $A/K$ be an abelian variety. Then the representation of $G$ on $V_\ell(A)$ is ...
- 1,673
10
votes
1
answer
299
views
Map from Bruhat stratification to Catalan stratification for the space of totally nonnegative upper-triangular matrices
$\DeclareMathOperator\SL{SL}$This question came up in a class "Total Positivity and Cluster Algebras" being taught by Chris Fraser.
Let $N^+$ denote the space of uni-upper-triangular ...
- 24.2k
10
votes
2
answers
896
views
Applications of classifying thick subcategories
So, relatively recently, Balmer introduced this notion of a spectrum for a tensor triangulated category and used it to prove a generalization of a classification theorem done in several areas of ...
- 13.5k
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
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
589
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
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
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
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
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
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
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
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
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
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
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
0
answers
597
views
Reference for shtuka and trace formula
I really want to learn the work of Laurent Lafforgue and the joint work of Zhiwei Yun and Wei Zhang. They both involve shtuka and trace formula, which I only know the basic idea. So I would like to ...
- 409
6
votes
1
answer
401
views
Stratifications and Filtrations of the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r=G(\mathcal{\mathbb{C}((t))})/G(\mathcal{\mathbb{C}[[t]]})$$ be the affine Grassmannian of $G$. We know that $\...
- 4,920
6
votes
0
answers
268
views
duality between quiver variety and affine Grassmannian
Let $\frak{g}$ be a ADE type simple lie algebra. There are (at least) two geometric ways to get highest weight irreducible representations of $\frak{g}$. One is by considering constructible functions ...
- 849
6
votes
0
answers
455
views
Cohomology of Bott-Samelson varieties?
How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there ...
- 1,719
5
votes
2
answers
2k
views
Canonical reference for Chern characteristic classes
I'm a little uncertain about the definitions for
Chern roots
Chern classes
Chern characters
From perusing several discussions, I gather that if one correlates the nomenclature with that of ...
- 10.5k
5
votes
1
answer
436
views
Is the Veronese variety "enough" to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?
I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.
Let $V$ be a complex vector space of dimension $n$, ...
- 2,527
5
votes
1
answer
261
views
A "prequestion" about meromorphic representations of algebraic groups
In a comment exchange around an answer to Is a group scheme determined by its category of representations? there arose the issue of Tannakian reconstruction for non-affine algebraic groups (e. g. ...
- 18.4k
5
votes
1
answer
514
views
Reference for the Natural Ample Line Bundle on the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r:=G((t))/G[[t]]$$ be its affine Grassmannian. I have read that $\mathcal{G}r$ possesses a natural very ample line ...
- 4,920
5
votes
0
answers
140
views
Classification of visible actions for *reducible* representations?
Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
- 3,230
5
votes
0
answers
293
views
On the deformation theory of associative algebras
Let us start by recalling the notion of a formal deformation:
Let $K$ be a field of characteristic zero and $A$ be an associative $K$-algebra. Consider a commutative augmented $K$-algebra $R$, with ...
- 541
5
votes
0
answers
263
views
Reference/list of reductive subgroups of reductive groups?
Let $G$ be a (say, connected) reductive group over an algebraically closed field of characteristic zero (say, $\mathbb C$).
I am looking for simple examples of (ideally) complete characterizations of ...
- 3,799
5
votes
0
answers
118
views
Where to read about the toric variety coming from a principal nilpotent element of a (semi)simple algebraic group?
Given a principal (regular) nilpotent element $e$ in the Lie algebra $\mathfrak g$ of a complex semisimple algebraic group $G$, let $\mathfrak s=(e,f,h)$ be an $\mathfrak{sl}_2$-triple for $e$. Then ...
- 18.4k
5
votes
0
answers
92
views
Question concerning the representation dimension of a special algebra
I would like to know, if the following problem is still open:
Let $k$ denote an algebraically closed field of characteristic 3.
Determine the representation dimension of $k(C_3\times C_3)$, where $...
- 1,755
5
votes
0
answers
324
views
A question about equivariant derived categories and [BBD]
Let $G$ be an algebraic group (over $\mathbb{C}$) acting algebraically on a variety $X$. Bernstein and Lunts then define in [BL94] the equivariant derived category $D^b_G(X,\mathbb{C})$ (of $\mathbb{C}...
- 2,554
5
votes
0
answers
227
views
Is there a brief name for the symmetric space $SL_{2n} / Sp_{2n}$?
Let $V$ be a complex vector space of even dimension. Then the homogeneous space $SL_{2n}(V) / SP_{2n}(V)$ is known to parametrize the space of non-degenerate skew-symmetric bilinear forms on $V$.
(1)...
4
votes
1
answer
783
views
Three dimensional representations of Alternating group
The alternating group $A_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations ...
- 125
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
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