# Tagged Questions

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### the representation of a free group

A group $G$ is generated by $\begin{pmatrix}1&n\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\n&1\end{pmatrix}$, then we know $G\cong \mathbb{F}_2$ which is a free group generated by two ...
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### Continuous subgroup of SO(3)?

I read from a paperarXiv: cond-mat/0602109 by a theoretical physicist, Prof. Frank Bais, close subgroups of $SO(3)$ is given by ${C_n,D_n,T,O,I,SO(2)\rtimes Z_2}$, where $C_n$ is the cyclic group of ...
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### Representing natural numbers as matrices by use of $\otimes$

What I am wanting to do is to find a unique matrix representations for Natural numbers. Say I have the number $n$, how can I represent this number as a matrix in which I can do matrix multiplication ...
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### Representing numbers in unique p-adic to matrix representation, is there a way?

What I am wanting to do, if it is possible is find unique matrix representations for the p-adic representation of numbers. So for example, say I have the number 1365 = 3*5*7*13. Now I could take ...
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### $\Psi_g(A)=\Phi(g)A^t\Phi(g)$; express $\chi_\Psi$ through $\chi_\Phi$

Let $\Phi$ be a matrix n-dimensional representation of the group G. We construct a representation $\Psi$ of $G$ on the space of square matrices of order n, such that ...
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### Commuting matrices and simultaneous diagonalizability

It is a known fact from linear algebra that if a set of matrices is pairwise commutable then they are simultaneously diagonalizable. A problem in the book I am currently studying asks to prove this ...
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### Prove that the matrix A is positive definite.

A matrix $A$ is defined as: $$A := \sum_{g\space \epsilon\space G}{{D}^{\dagger}(g)D(g)}$$ Where the $D(g)$ are representations matrices of the finite group $G$ on a ...
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### elementary but confounding question about integer matrices (related to hecke operators)

Let $\Gamma(N)$ denote the kernel of the reduction map $\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ Let $p$ be a prime that is $1$ mod $N$, and let $M$ be the set of ...
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### Real representations of Lie algebra $\mathfrak{so}(3)$

How does one construct an $n$-dimensional, irreducible, real-valued and non-zero representation of the three generators of the Lie algebra $\mathfrak{so}(3)$ for a given value of $n$?
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### A quesion in Fulton & Harris book “representation theory a first course”

In Section 11.2 A little plethysm, it discusses the tensor product of two different representations of $sl_2\mathbb{C}$. It says "If $V=\bigoplus V_{\alpha}$ and $W=\bigoplus W_{\beta}$ then ...
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### Embedding of $PGL_n\mathbb{C}$ and friends

I would like the find an embedding/faithful representation from the projective linear group $PGL_n\mathbb{C}\to GL_m\mathbb{C}$ for some $m$, and likewise for the other projective groups ...
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### Invariant Inner Product on Lie Algebra

Let $G$ be a Lie group, $\frak{g}$ its Lie algebra. Suppose $\mathcal{D}$ a representation of $G$ on $V$, $d$ the associated Lie algebra representation. Suppose $V$ is endowed with an inner product. ...
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### Spinor Mapping is Surjective

I'm (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps. (1) Prove that the vector ...
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### Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
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### intuitive explanation of sparsity / references

I know it is a vague question, but I am confused by why/when we actually want sparsity of a matrix. For example, interior-point methods work better when constraint matrix is sparse. Similarly, it is ...
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### Group of Hermitian and Unitary matrices

This question is continuation of an earlier question asked in Matrices which are both unitary and Hermitian Consider the unitary group $U(n^2)$ and consider the subset $R$ of Hermitian Unitary ...
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### Irreducible Representations of Matrix Algebras

I am currently reading the book Spin Geometry by Lawson/Michelsohn to understand Dirac Operators and related topics. At some point it uses representation theory to classify Clifford Algebras. In ...
### image of symmetric matrices under representation of $GL_2(\mathbb{R})$
Let $W$ be a real vector space of dimension $2$ and let $\rho_k:GL_2(\mathbb{R}) \to GL(\mathbf{S}^kW)$ be the standard representation of $GL_2(\mathbb{R})$. Since $\rho_k$ is polynomial, it naturally ...