# Tagged Questions

309 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 ...
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### 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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### 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 ...
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### $I/N$ 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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### 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 ...
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### 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 ...
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### 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. ...
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### 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, ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...