# Tagged Questions

**2**

votes

**1**answer

127 views

### Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)

I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form
$
\left(
...

**2**

votes

**2**answers

109 views

### Invariant planes of a nilpotent matrix with two Jordan blocks of size two

Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map
$$
N = \begin{pmatrix}
0 & 1 & & \\
0 & 0 & & \\
& & 0 ...

**3**

votes

**0**answers

101 views

### Non-linearly isomorphic non-equivalent $G-$representations?

Let $G$ be an algebraic group (or a group scheme) over a field $\Bbbk$, and let $V$ be an algebraic $G-$representation (I mean, corresponding to a homomorphism of $\Bbbk-$group schemes $G\rightarrow ...

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votes

**2**answers

347 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 ...

**0**

votes

**0**answers

61 views

### Determinant of an action and characters

In the paper of Ramanathan "Stable Principal Bundles on a Compact Riemann Surface", I read: ...where $\mu$ is the determinant of the (adjoint) action of $P$ on ...

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votes

**2**answers

190 views

### Example of linearization for GIT

Take a vector space $V$ (finite dimensional, over the complex numbers), let $G=SL(V)$. The group $G$ acts on $\mathbb{P}V$ and we can linearize its action to an action on the line bundle ...

**2**

votes

**1**answer

118 views

### Fibers of the Bott-Samelson Resolution of Schubert Varieties

Is there an explicit (perhaps visual) description of the fibers of the Bott-Samelson Resolutions of Schubert Varieties? Let's fix $G$ to be $GL_n(\mathbb{C})$.
Also, how would the answer to the ...

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votes

**2**answers

291 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 ...

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votes

**0**answers

127 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 ...

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votes

**2**answers

236 views

### Algebraic Groups, Modules, and Comodules

Background:
Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For
$$
\widehat{H} := \text{Alg}_k\{H; k\},
$$
we recall (see Abe Chapter 4 ...

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vote

**0**answers

207 views

### Relationship between algebraic groups and Lie groups? [closed]

In the literature, e.g. in representation theory, there seems to be a passage from Lie groups to (linear) algebraic groups. It is clear, particularly over $\mathbb R$ and $\mathbb C$ that they are ...

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vote

**0**answers

79 views

### G-Invariant Complete Intersection generated by G-representation

I have a smooth manifold $M$ with $G$-action and complete intersection of codimension $n$ given by ideal $I$ such that $gI=I$. I'm interested when I can choose a $n$-dimensional vector space of ...

**5**

votes

**1**answer

137 views

### “Plucker” embedding of G/N, for reductive group G, affinization of quasiaffine varieties

I'll use "affinization" to describe the natural map of schemes $X \rightarrow \text{Spec}(\Gamma(X, \mathcal{O}_X))$. For quasi-affine varieties $X$ this is an open embedding.
Let $G$ be a reductive ...

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votes

**0**answers

140 views

### Hodge Decompositions and Gamma Factors of Hasse--Weil L-Functions

Let $X$ be a projective variety over a number field $K$. If $\mathfrak{p}\unlhd\mathcal{O}_K$ is a prime ideal we can regard the Euler factor of its $m$th Hasse-Weil $L$-Function at $\mathfrak{p}$ as ...

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votes

**1**answer

148 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 ...

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votes

**3**answers

269 views

### Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...

**6**

votes

**2**answers

294 views

### When are two subvarieties of matrices conjugate?

Let $X$ and $Y$ be two subvarieties of $n\times n$ matrices. My question is that is there any condition to guarantee that there exits some matrix $g$ such that $Y=g^{-1} X g$? If such $g$ exists, then ...

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votes

**3**answers

793 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 ...

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votes

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502 views

### Hall-Littlewood functions and functions on the nilpotent cone

The following observation between the spaces of global sections of line bundles on the nilpotent cone and the Hall-Littlewood polynomials is made in a recent physics preprint 1403.0585. Is this a ...

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votes

**0**answers

134 views

### Is the formula for plethysm $S^n(S^3)$ known explicitely?

Is the formula for plethysm (in this the decomposition into irreducible GL representations of the composition of symmetric powers) $S^n(S^3)$ known explicitely? I know $S^n(S^2)$ e.g. in Macdonald's ...

**3**

votes

**1**answer

151 views

### Why is the semisimple quotient of a reductive group with semisimple rank 1 equal to PGL2?

Lets work over $\mathbb{C}$. Let $G$ be a reductive group with semisimple rank 1 (this means that a maximal torus in $G / R(G)$ has dimension 1). In section 25 of Humphreys book "linear algebraic ...

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votes

**0**answers

234 views

### Characteristic Classes in Geometric Representation Theory

Geometric respectively topological methods are widely applied in representation theory. As far as I know mainly cohomological methods are used.
I wonder if there are concrete applications of the ...

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votes

**1**answer

175 views

### Projective representation of diffeomorphism group of $S^2$ [closed]

We know that the projective representation of a group $G$ is classified by $H_{grp}^2(G,R/Z) = H^3(BG,Z)$, where $H^*_{grp}$ is the group-cohomology class.
Then do we have a classification of the ...

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votes

**2**answers

266 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 ...

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vote

**2**answers

212 views

### When representation of two different coadjoint orbits are equivalent?

Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^*$ (where ...

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vote

**0**answers

168 views

### Tilting object in derived category

I was wondering about something concerning tilting objects... Suppose we are given a split algebraic torus $G=\mathbb{G}^n_m$, that is a linearly reductive group with no semisimple part, and let ...

**2**

votes

**1**answer

213 views

### Grothendieck group of representations

For a linearly reductive group $G$ over $k$ we consider the bounded derived category of finite dimensional representations $D^b(\mathrm{Repr}(G))$. Is the Grothendick group $K_0(D^b(\mathrm{Repr}(G))$ ...

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votes

**1**answer

161 views

### number of simple representations

For a linearly reductive Group $G$ over a field $k$ one has that the category of finite dimensional representations of $G$ is semisimple. What can one say about the number of simple representations? ...

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vote

**0**answers

100 views

### global dimension II

Suppose we are given finite dimensional semisimple $k$-algebras $A_1,..., A_r$. now we consider the matrix algebra$A=\begin{pmatrix}
A_1 & M_{1,2} & \dots & M_{1,r} \\
0 & A_2 ...

**12**

votes

**3**answers

1k 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 ...

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votes

**2**answers

243 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 ...

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votes

**1**answer

301 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 ...

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votes

**5**answers

666 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 ...

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votes

**2**answers

283 views

### Relations between affine Grassmannian and Grassmannian

Let $\mathcal K = k((t))$ be the field of formal Laurent series over $k$, and by $\mathcal O = k[[t]]$ the ring of formal power series over $k$.
Let $G$ be an algebraic group over $k$. The affine ...

**4**

votes

**1**answer

179 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 ...

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vote

**0**answers

147 views

### Derived category of representations

Suppose we are given an algebraic group $G$ (linearly reductive). Let $D^b(Repr(G))$ be the bounded derived category of finite dimensional algebraic representations of $G$. I am intressted in tilting ...

**2**

votes

**1**answer

212 views

### Is Mazur's deformation ring R integral?

Consider the absolutely irreducible Galois representation
$\overline{\rho} \colon G_{\Bbb Q} \to {\mathrm{GL}}_2({\Bbb F}_p)$. We apply the Mazur's deformation theory on the lift ${\rho} \colon ...

**0**

votes

**1**answer

231 views

### Iwasawa theory for Mazur's deformation ring R

The ideal class group $\mathrm{Cl}({\cal O}_K)$ and Mazur's deformation ring $R(\overline{\rho})$ for a number field $K$ are said to be similar to each other.
Let ${\Bbb Q}_{\infty}$ be the unique ...

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votes

**1**answer

237 views

### Is a group scheme determined by its category of representations?

More precisely, let $G$ be an affine group scheme over a field $k$, $Rep_k(G)$ be the category of finite dimensional representations of G, and $\omega_0$ be the forgetful functor from $Rep_k(G)$ to ...

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votes

**0**answers

118 views

### Find all Linear spaces on an orbit (by a connected linear algebraic group)

Suppose $V$ is a vector space over $\mathbb{C}$ and $G\subset \textrm{GL}(V)$ is a connected linear algebraic group.
Consider the orbit closure $Y = \overline{G.\mathbb{P} L}$, for a subspace ...

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votes

**1**answer

173 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 ...

**0**

votes

**1**answer

155 views

### $I/N$ is finitely presented module

Let $R$ be a commutative ring and $N = Nil(R)$ the set of its nilpotent elements. Suppose that $N$ is a divided prime ideal, i.e. for any ideal $I$ of $R$ either $I \subseteq N$ or $N \subseteq I$.
...

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votes

**2**answers

311 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 $\mathcal{G}$ ...

**3**

votes

**1**answer

263 views

### A question on algebraic loop groops

Setup:
Let $\mathcal{K}=\mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $G$ be a reductive algebraic group (over $\mathbb{C}$). Let further $\mathcal{K}_n$ denote the $\mathcal{O}$-ideal in ...

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vote

**1**answer

177 views

### Do Auslander-Reiten quivers coincide with the McKay quivers for arbitrary subgroups of GL(2,C)?

It is a theorem of Auslander that if $G< GL(2,\mathbb C)$ is a finite subgroup without pseudo-reflections, then the Auslander-Reiten quiver of $K[x,y]^G$ coincides with the McKay quiver of $G$ with ...

**0**

votes

**1**answer

102 views

### spectrum of an induced algebra

Let $G$ be a reductive group defined over an algebraically closed field $k$ and $B$ be a fixed Borel subgroup of $G$. Suppose $X=Spec(R)$ is an affine scheme with $B$ rationally acting on it; hence ...

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votes

**0**answers

130 views

### Scaling-Invariant Orbits of Semisimple Group Representations

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Note that if $V$ is the adjoint representation of $G$, then ...

**5**

votes

**1**answer

216 views

### When are orbits of semisimple group representations closed?

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Is there a nice description of those $v\in V$ for which the ...

**6**

votes

**1**answer

152 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 ...

**3**

votes

**0**answers

41 views

### points with small U stabilizer on a spherical variety

Let $(G,H)$ be a spherical pair (i.e. $G$ is a reductive group, $H$ is a closed subgroup and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H$). Let $U$ be the unipotent radical of ...