# Tagged Questions

Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.

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### Composition series of a regular module.

Suppose $A$ is an $k$-algebra with basis ${1,e,s,t}$ and multiplication is given by $$e^2 = e, es = s, te = t, s^2=t^2=se=et=st=ts=0.$$ I am trying to find the composition series for ...
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### Finite dimensional algebraic representation of $SL_2(\mathbb{C})$

I heard that for each $n\in \mathbb{N}$, there is the unique algebraic irreducible representation of $SL_2(\mathbb{C})$ with dimension $n$ over $\mathbb{C}$. Would you let me know what is such ...
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### Matrix representations of particular generators of the full octahedral group

I want to find matrix representations of the generators $a, b, c$ of the full octahedral group in the presentation $$\{a,b,c \mid a^2=1,b^3=1,(ab)^4=1,ac=ca,bc=cb\}.$$ Is there a recipe to write the ...
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### How does having a cycle in a quiver change the simple objects in the category of representations?

In theorem 1.12 on page 5 of http://www.math.utah.edu/dc/tilting.pdf, which states: Given a bounded acyclic quiver $(Q,R)$, the K-theory of it's representations is given by $\mathbb{Z}^{Q_0}$ why is ...
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### What's uniform block signed permutations?

Let $[n]=\{1,2,\ldots,n\}$ and $P(n)$ the set of all partitions of [n]. A partition of $[n]$ is non-empty disjoint subsets of [n], called blocks, whose union is $[n]$. A block permutation of [n] is ...
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### $R$ is isomorphic to a direct product of matrix rings over division rings

Suppose as rings, $R$ is isomorphic to a direct product of matrix rings over division rings, that is $R=R_1 \times ... \times R_n$ where $R_i$ is a two-sided ideal of $R$ and $R_i$ is isomorphic to ...
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### $R$ is a direct sum of simple modules implies every $R$-modules is completely reducible.

Suppose the ring $R$ considered as a left $R$-module is a direct sum $R=L_1\oplus L_2\oplus ...\oplus L_n$ such that $L_i$ are simple modules (no nonzero proper submodules) and such that $L_i=Re_i$ ...
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### Every $R$-module is injective implies $R$ has the descending chain condition.

Suppose that every R-module is injective. There are a few definitions of injective that we can go by: E is an injective R-module if and only if Hom$_R(−, E)$ is an exact functor. An R-module E is ...
I've been reading Valette's introduction to the Baum-Connes conjecture and as I read the example of a construction of a (model of the) classifying space for proper actions of $\Gamma$ (discrete) given ...