# Questions tagged [irreducible-representation]

An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces.

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### Matrices commuting with a completely reducible representation

By Schur's lemma, a matrix commuting with an irreducible representation (of a group over complex numbers, say) is a multiple of identity. What about a direct sum of irreducible representations (a.k.a. ...
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### Example of an irreducible homogeneous Markov chain, that possesses an invariant measure but is not recurrent

I’m currently studying the topic of Markov chains and how invariant measures are connected to recurrence. I now know that an irreducible homogeneous Markov chain that possesses invariant measures must ...
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### Unique irreducible complex representation of Clifford algebra implies isomorphism with matrix algebra

Consider the Clifford algebra $\mathrm{Cl}(n)$ over Euclidean space $\mathbb{R}^n$ (with the standard inner product). Now, in the case that $n$ is even, it is known (cf. [1]), then $\mathrm{Cl}(n)$ ...
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### Tensor product of two representations of $D_4$

Let $(\tau,\mathbb C^2)$ be the irreducible representation of $D_4$ by matrix multiplication, namely for every $v\in\mathbb{C}^2$: \begin{bmatrix}\tau(s)\end{bmatrix}v=\begin{bmatrix}-1 & 0\\ 0 &...
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