# Tagged Questions

**3**

votes

**1**answer

328 views

### What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?

Recall the definition of the representation ring $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can ...

**0**

votes

**1**answer

227 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 ...

**0**

votes

**2**answers

184 views

### 0-dimensional Gorenstein local ring.

Assume the following condition for the ring T = F_p[[X,S]]/I:
Condition 1. T is NOT a zero ring.
Condition 2. I is generated by 3 elements of F_p[[X,S]], but NOT by 2 elements.
Then, is T a ...

**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 $N \subseteq I$.
...

**4**

votes

**2**answers

383 views

### Linearisation of a group

If $G$ is a connected Lie group acting on a vector $\mathbb{C}$-space $V$ then it is well known that the algebra of invariants $\mathbb{C}[V]^G$ coincides with the algebra of invariants ...

**3**

votes

**2**answers

557 views

### Quotients in Sums of Rings

Suppose we are given a commutative ring R with unit-element. Now we have a composition of R as the direct product of two rings $R\cong R_1\times R_2$. It is now straight forward, that any ideal ...

**3**

votes

**0**answers

123 views

### ideal generated by highest weight vectors

Let $S$ be a polynomial ring which carries the action of a semi-simple linear algebraic group $G$ (I'm interested in a product of $GL$'s). Take $S$ and $G$ to be over an algebraically closed field.
...

**1**

vote

**1**answer

259 views

### Representations over $\mathbb{Z}_p$

Hi,
I would like to work with indecomposable representations over a commutative ring and start with $R=\mathbb{Z}_p$
Notations: $G$ a finite group, $S$ a subgroup, $R=\mathbb{Z}_p$ the p-adic ring, ...

**0**

votes

**1**answer

170 views

### About regular local rings and Socles

Let R be a regular local ring with $ \text{dim} R = d $. If $ 0\rightarrow R\rightarrow I_0\rightarrow ...\rightarrow I_d\rightarrow 0 $. Then why for $ 0\leq i\leq d-1 $, the socle of $ I_i $ is ...

**0**

votes

**1**answer

231 views

### Length of a module

Let R be a commutative ring, M an R-module of finite length and let N be an Injective R-module with zero socle. Then why $ \text{Hom}_R(M, N) $ is zero?

**1**

vote

**1**answer

102 views

### Can one pick generators for the ring of invariants of binary n-ic forms which have rational coefficients?

The problem of determining a set of generators of the ring of invariants of the group $\textrm{SL}_2$ acting on the complex $n+1$-dimensional vector space of binary $n$-ic forms is known to be very ...

**2**

votes

**1**answer

301 views

### Representations of $GL(n)$ containing $S^kV$

Let $V$ be a vector space of dimension $n$.
Let $S^k V$ be a representation of $GL(n)$.
I would like to know if there exists some characterization of finite dimensional $GL(n)$ modules $V_1,V_2$ such ...

**2**

votes

**1**answer

154 views

### Proving indecomposability of special modules

Hi,
I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf
On page 6 there are the definitions (in a diagrammatical way) of some $A_n$ modules, whereupon ...

**0**

votes

**1**answer

177 views

### Representation dimension of a special algebra

Hi,
I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf
I've come across a piece of information, which I don't understand, and wanted to ask, if I ...

**0**

votes

**2**answers

127 views

### Structure of Homomorphisms of commutative C^* algebra

Being new to $C$* algebra, I'm trying to understand basic properties of -homomorphisms of such algebras. Let $P$ be a set of commuting projections on Hilbert space ${\cal H}$.
Let ${\cal P}$ be the ...

**2**

votes

**1**answer

133 views

### Observable Unipotent Algebraic Subgroups of the Unipotent Upper Triangular Groups

It is well known that any unipotent algebraic group (over a field) can be embedded as a closed subgroup of $U_n$ for some $n$, where $U_n$ denotes the set of all $n \times n$ upper triangular matrices ...

**3**

votes

**3**answers

608 views

### Classify matrices up to similarity over arbitrary (commutative) ring.

One can define the K-theory space of a monoidal category $S$ in which every morphism is an isomorphism as the classifying space $B(S^{-1}S)$. Then we show that this definition coincides with the ...

**6**

votes

**1**answer

250 views

### Is a certain symmetric power reprsentation of GL(m) cyclically generated

Let $V_m$ be the $m$-dimensional complex vector space with basis $\{e_1, \dots, e_m\}$ and let $i\leq m$. Consider the element ${v}_0^i \in S^i(S^m(V_m))$, where ${v}_0$ is the element $e_1\dots e_m ...

**4**

votes

**1**answer

842 views

### do you know this determinant (basic commutative algebra)?

Let $\ell_1,\dots,\ell_n$ be $d+1$-variate linear forms over complex numbers in variables $X=(X_0,\dots,X_d)$. Consider the $(n-d)$-fold products
...

**1**

vote

**0**answers

637 views

### Conjugate Matrix

Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix ...

**5**

votes

**1**answer

445 views

### Where does the definition of “Tower of Algebras” come from?

A tower of algebras is a sequence of algebras
$$A_0 \hookrightarrow A_1 \hookrightarrow \cdots \hookrightarrow A_n \hookrightarrow \cdots$$
with embeddings $A_n \otimes A_m \hookrightarrow A_{n+m}$ ...

**12**

votes

**4**answers

625 views

### Conjugacy for p-adic matrices of finite order II

Question: Say $p$ is an odd prime, and take two matrices $A,B\in GL_n({\mathbb Z}_p)$ of finite order $m$. Is it true that if their reductions mod $p$ are conjugate in $GL_n({\mathbb F}_p)$ then ...

**13**

votes

**6**answers

1k views

### Conjugacy for $p$-adic matrices of finite order

Say $p$ is an odd prime, and take two matrices $A,B\in GL_n({\mathbb Z}_p)$ of finite order $m$. Is it true that they are conjugate in $GL_n({\mathbb Z}_p)$ if and only if their reductions mod $p$ are ...

**3**

votes

**1**answer

854 views

### Behaviour of Hilbert functions

Let $G$ be a complex simple reductive group. Then the set of isomorphy classes $Irr G$ is isomorphic to the set of dominant weights $\Lambda_+$ in the weight lattice of the maximal torus of a Borel ...

**9**

votes

**1**answer

903 views

### Fixing a mistake in “An introduction to invariants and moduli”

On page 13 of the book "An introduction to invariants and moduli" of Mukai
http://catdir.loc.gov/catdir/samples/cam033/2002023422.pdf there is a mistake, in the end of the proof of Proposition 1.9. ...

**6**

votes

**3**answers

525 views

### A question about an application of Molien's formula to find the generators and relations of an invariant ring

In the very beginning of the book "Introduction to Invariants and Moduli" Shigeru Mukai
proves Molien's formula for the Hilbert series of the invariant ring of a finite group action on $\mathbb C^n$. ...

**1**

vote

**0**answers

155 views

### Sum of two free o-submodules in a vector space over a local field

Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.
Given two free ...

**0**

votes

**2**answers

214 views

### Commutation of $GL_{n}$ with projective limits

Let $A$ denote a unital commutative ring. Given a system of ideals $(I_p)_{p\in P}$ indexed by a partially ordered set $P$ such that if $p \leq q$, then $I_p$ is contained in $I_q$, when is
$$GL_n ...

**6**

votes

**1**answer

277 views

### Orbits in commutative groups.

Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$
which acts on A such that $S$ is an orbit of $H$.
Can one give a simple characterization ...

**5**

votes

**0**answers

166 views

### Moduli space of modules with fixed length

Let $R$ be a (commutative) local Artinian ring, with an algebraically closed residue field $k$. I am interested in the set $L_n(R)$ of isomorphism classes of $R$-modules of length $n$.
If $R$ is a ...

**5**

votes

**1**answer

340 views

### Abelian varieties over local fields

Let $K$ be a local field of characteristic zero, $k$ its residue field, $R$ its ring of integers and $p$ the characteristic of the residue field $k$. Let $G$ be the Galois group of $K$, $I\subset G$ ...

**3**

votes

**1**answer

627 views

### What is the abstract relationship between an indecomposable representation and a sum of irreducible representations with the same character?

$\mathbb{Z}$ is the simplest example of a group with indecomposable representations which are not irreducible. Over $\mathbb{C}$, the isomorphism classes of $n$-dimensional representations of ...