All Questions
Tagged with rt.representation-theory ag.algebraic-geometry
782 questions
18
votes
4
answers
621
views
What are immediate applications of the classification of connected reductive groups?
After years of putting it off, I finally sat down, read, and understood the classification of connected reductive groups via root data.
That's a non-trivial theory! I'm hoping that now that I am done ...
- 557
18
votes
1
answer
1k
views
Why should algebraic geometers and representation theorists care about geometric complexity theory?
Geometric complexity theory has demonstrated that complexity theorists should care about algebraic geometry and representation theory, but, why should algebraic geometers and representation theorists ...
- 989
18
votes
1
answer
770
views
Koszul complex for non-Koszul algebras
Let $A$ be a graded, connected, locally finite, quadratic algebra over a field $k$; that is, $A$ may be presented as $T(V)/I$, where $V = A_1$ is a finite dimensional $k$ vector space, and the ideal $...
- 4,133
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:
$$ J_3(\mathbb{O}) = \...
- 7,300
17
votes
3
answers
3k
views
History and motivation for Tannaka, Krein, Grothendieck, Deligne et al. works on Tannaka-Krein theory?
I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask:
Question: What was the motivation and historical context for ...
- 24.9k
17
votes
4
answers
2k
views
Reference request: Grassmannian and Plucker coordinates in type B, C, D
Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker ...
- 6,201
17
votes
3
answers
2k
views
Variety of commuting matrices
Let $G=\operatorname{GL}(n,\mathbb{C})$ and $\mathfrak{g}=\operatorname{Mat}(n,\mathbb{C})$ and let us consider the two varieties $X,Y$ defined as $$X=\{(x,y) \in G \times G \ | \ xy=yx\} $$ and $$Y=\{...
- 1,061
17
votes
1
answer
1k
views
D-modules over algebraic curves VS differential Galois theory
Disclaimer: I know very little about both of the fields in question.
My question is pretty simple:
What's the relation between differential Galois theory and D-modules
over algebraic curves?
...
- 7,789
17
votes
1
answer
2k
views
What is the recent development of D-module and representation theory of Kac-Moody algebra?
I just started to collect the papers of this field and know little things. So if I make stupid mistake, please correct me.
It seems that there are several approaches to localize Kac-Moody algebra(in ...
- 5,515
17
votes
0
answers
402
views
Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )
Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...
- 24.9k
17
votes
0
answers
1k
views
Symmetries of local systems on the punctured sphere
Let $X=S^2\setminus D$, for $D\subset S^2$ some finite set of points, say with $|D|=n\geq 1$. The category of locally constant sheaves of $\mathbb{C}$-vector spaces on $X$ (equivalently, complex ...
- 23k
17
votes
0
answers
704
views
When is the determinant an $8$-th power?
I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...
- 7,300
16
votes
6
answers
1k
views
Are there any known criteria for quadratic mapping from R^n to R^n being surjective?
Let quadratic mapping be the function from $\mathbb{R}^n$ to $\mathbb{R}^n$, where each coordinate is a quadratic form of $n$ variables. Are there any known criteria for it being surjective? May ...
- 161
16
votes
1
answer
695
views
Multiplicativity twisted Hochschild Kostant Rosenberg isomorphism
Let $X$ be a smooth projective variety over $\mathbb{C}$. I call (following Swan) Hochschild cohomology of $X$ the graded algebra:
$$ \mathrm{HH}^{\bullet}(X) := \mathrm{Ext}^{\bullet}_{X \times X}(\...
- 7,300
16
votes
0
answers
530
views
Comparing the Kazhdan-Lusztig and Steinberg pre-orders
Both Kazhdan-Lusztig and Steinberg have defined pairs of preorders on $S_n$. Kazhdan and Lusztig's preorders come from their basis:
We write $x\leq_L y$ if any left ideal spanned by K-L basis ...
- 44.7k
16
votes
0
answers
605
views
Division fields of abelian varieties over function fields
Let $k$ be a finitely generated field (for example a finite field or a number field) and $K/k$ a finitely generated regular extension with $trdeg(K/k)=1$. Let $A/K$ be a principally polarized abelian ...
- 1,673
15
votes
2
answers
2k
views
Why is Drinfeld's Zastava space called Zastava?
I'm trying to get an idea of Drinfeld's Zastava space. It seems to be an infinite-dimensional version of the flag variety, for affine Lie algebras.
But, first of all, why is it called Zastava (...
- 6,123
15
votes
2
answers
2k
views
Why are coroots needed for the classification of reductive groups?
As we know reductive groups up to isomorphism corresponds to root data up to isomorphism. My question is why in the definition of root data do we need the coroots?
Let's break it down to two questions:...
- 2,071
15
votes
1
answer
2k
views
What is the relation between L. Lafforgue and Frenkel-Gaitsgory-Vilonen results on Langlands correspondence ?
What is the relation between Lafforgue's result on Langlands
and Frenkel-Gaitsgory-Vilonen ? ( http://arxiv.org/abs/math/0012255 , http://arxiv.org/abs/math/0204081 )
Does one imply other ? If not ...
- 24.9k
15
votes
1
answer
321
views
Are triangulated equivalence detected at compact level?
Suppose that $D$ and $E$ are compactly generated triangulated categories, even algebraic (i.e. equivalent to derived categories of small dg categories) if we want, and asume that their subcategories $...
- 151
15
votes
0
answers
541
views
Applications of character sheaves
There are many important recent works (for example, by Lusztig, Bezrukavnikov-Finkelberg-Ostrik, Ben-Zvi-Nadler, Boyarchenko-Drinfeld, Lusztig-Yun, Vilonen-Xue) on character sheaves (which are certain ...
- 2,964
14
votes
8
answers
2k
views
Applications of the idea of deformation in algebraic geometry and other areas?
The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show ...
14
votes
2
answers
571
views
Number of d-Calabi-Yau partitions
This problem arises from algebraic geometry/representation theory, see https://arxiv.org/pdf/1409.0668.pdf (chapter 2).
We call a partition $p=[p_1,...,p_n]$ with $2 \leq p_1 \leq p_2 \leq ... \leq ...
- 26.5k
14
votes
2
answers
1k
views
Invariants of matrices (by simultaneous $\mathrm{GL}_n$ conjugation) over arbitrary rings
$\DeclareMathOperator\GL{GL}$Let $R$ be a commutative ring, let $R[n] := R[M_d^{\oplus n}]$ be the polynomial ring on $nd^2$ variables corresponding to the coordinates of $n$-many $d\times d$ matrices....
- 7,527
14
votes
2
answers
1k
views
Information from Moment Polytopes
Let $T$ be a compact real torus, and $X$ a Hamiltonian $T$-manifold (which you may take to be a smooth complex projective variety) with moment map $\mu:X\rightarrow\frak{t}^*$. If $\dim(T)=\frac{1}{2}\...
- 4,920
14
votes
3
answers
1k
views
The conjugacy classes of diagonalizable $2 \times 2$ matrices can be identified with their eigenvalues, what about pairs?
For sake of simplicity, let's say that we live in $G = SL(2, \mathbb{C})$. Every conjugacy class of diagonalizable matrices $$[A] := \{gAg^{-1} \mid g \in G\}$$ can be identified with its set of ...
user76098
14
votes
1
answer
3k
views
When does a perverse sheaf occur in the decomposition theorem?
Suppose I am in the setting of the decomposition theorem, i.e., we have the decomposition of the direct image $f_*\mathbb Q_\ell$, where $f:X\to Y$ is proper. Then the direct image decomposes into a ...
- 3,799
14
votes
3
answers
2k
views
Naïve definition of parahoric subgroup
Background
Let $F$ be a $p$-adic local field, and let $G$ be a connected reductive group over $F$. Recall that there is a rich theory of compact open subgroups of $G(F)$ which is, essentially, ...
- 843
14
votes
2
answers
781
views
Interpretation of the cohomology of compact lie groups and their classifying spaces in DAG?
I'll be using homological grading throughout this question.
Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods:
$$H^{\bullet}(...
- 7,789
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
14
votes
1
answer
1k
views
Factorization and vertex algebra cohomology
A chiral algebra on a smooth curve $X$, in the sense of Beilinson-Drinfeld, is a right $D_{X}$-module with a chiral bracket, which is a map $\mathcal{V}^{\boxtimes 2}(\infty\Delta)\rightarrow \Delta_{*...
user108998
14
votes
0
answers
891
views
Local proof of Grothendieck-Riemann-Roch theorem
There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem.
Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of ...
- 7,377
13
votes
3
answers
1k
views
Decomposition of k[G]
There's a well-known decomposition of $L^2(G)$, a regular representation of compact complex group Lie $G$, called Peter-Weyl theorem.
Turns out for some reason I automatically think that there is a ...
- 15.1k
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
2
answers
2k
views
Non-Abelian Hodge theory
Let $X$ be a compact Riemann surface. I would like to find a somehow complete reference for the proof of the so called non-Abelian Hodge correspondence relating Dolbeaut, Betti and Higgs bundle moduli ...
- 1,061
13
votes
3
answers
1k
views
$\ell$- vs. $p$-adic and de Rham vs. Betti in the geometric Langlands correspondence
In Emerton–Gee–Hellmann’s IHÉS notes on the categorical $p$-adic local Langlands programme, one finds the following remark:
The differences between the $\ell$-adic and $p$-adic settings are ...
- 607
13
votes
1
answer
698
views
Counting representations of $k[x,y]$ when $k$ is finite
$\newcommand{\GFq}{\mathbf F_q}$
Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial $p_n(...
- 5,654
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
13
votes
0
answers
2k
views
Calculation-free proof of the Weyl Integral formula for U(n)
The Weyl integral formula states that if $f$ is a class function on $U(n),$ $T$ is the torus of diagonal matrices in $U(n)$, and $dU(n)$ and $dT$ are the standard Haar measures on $U(n)$ and $T,$ then:...
- 28.1k
13
votes
0
answers
943
views
Beilinson-Bernstein localization in positive characteristic
This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...
- 3,637
12
votes
3
answers
1k
views
construct scheme from quivers?
I heard from some guys working in noncommutative geometry talking about the idea that one can construct the noncommutative space from quivers. I feel it is rather interesting. However, I can not image ...
- 1,305
12
votes
3
answers
920
views
Schemes of Representations of Groups
Let $G$ be a group, say finitely presented as $\langle x_1,\ldots,x_k|r_1,\ldots,r_\ell\rangle$. Fix $n\geq 1$ a natural number. Then there exists a scheme $V_G(n)$ contained in $GL(n)^k$ given by ...
- 16k
12
votes
3
answers
3k
views
Zariski tangent spaces to representation varieties
In Bill Goldman's paper "The Symplectic Nature of the Fundamental Groups of Surfaces" (Advances, 54, 200-225, '84) it is stated that the "Zariski tangent space" to a representation space Hom$(\pi, G)/...
- 8,803
12
votes
1
answer
1k
views
Recovering classical Tannaka duality from Lurie's version for geometric stacks
In Lurie's paper Tannaka Duality for Geometric Stacks, it is essentially shown that specifying a morphism of geometric objects
$$ f \colon X \to Y$$
is equivalent to giving a corresponding pullback ...
- 805
12
votes
2
answers
929
views
Are representations of a linearly reductive group discretely parameterized?
Suppose $G$ is a linearly reductive group over a field (say $\mathbb C$). Does somebody know of a proof that any flat family of finite-dimensional representations of $G$ must be locally constant?
In ...
12
votes
2
answers
674
views
Cohomology of representation varieties
Perhaps this question is too general then I am sorry about this.
My question is the following.
Let $\pi$ be the fundamental group of a compact surface of genus $g$ (with if necessary $n$ punctures) ...
- 121
12
votes
2
answers
634
views
Coordinate ring of universal centralizer (BFM space)
In the paper titled Equivariant (K-)homology of affine Grassmannian and Toda lattice, the authors, Roman Bezrukavnikov, Michael Finkelberg, and Ivan Mirković, derived the coordinate ring of each ...
- 303
12
votes
1
answer
842
views
What kind of algebraic object is $\mathcal{D}_X$? (algebra of diifferential operators). What's special about modules over it?
Let $R$ be a regular ring over a field of char 0. Let $X=Spec R$ and $D=\mathcal{D}_X$
the algebra of differential operators over it.
The overall vague question is what kind of algebraic object is $...
- 7,789
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
2
answers
658
views
What is the correct notion of representation for abelian varieties?
Zeroth question - am I right that in the "ordinary" sense an abelian variety does not possess any representations at all?
More precisely, a representation of an algebraic group $G$ (over an ...
- 18.4k