3
votes
1answer
56 views

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 ...
3
votes
0answers
44 views

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 ...
0
votes
0answers
46 views

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 ...
0
votes
0answers
21 views

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 ...
0
votes
1answer
74 views

$\Psi_g(A)=\Phi(g)A^t\Phi(g)$; 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 ...
5
votes
1answer
146 views

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 ...
0
votes
0answers
36 views

Prove that the matrix A is positive definite.

A matrix $A$ is defined as: \begin{equation} A := \sum_{g\space \epsilon\space G}{{D}^{\dagger}(g)D(g)} \end{equation} Where the $D(g)$ are representations matrices of the finite group $G$ on a ...
2
votes
0answers
44 views

Direct Sums of Matrix Algebra

This is the first half of the question introduced in Representations of direct sums of matrix algebras Let $A_1, A_2....A_n$ be n algebras with units $u_1, u_2,...u_n$ respectively. Let $A = A_1 ...
1
vote
0answers
26 views

Embedding of $GL_n(F)$ inside another matrix groups

We can embed $GL_n$ inside $SL_{n+1}$ easily. is there any other such embedding of $GL_n$ or its subgroups inside any other group of invertible matrices? Thanks in Advance.
1
vote
0answers
46 views

Matrix exponent and representations of $\mathbb{R}$

It is well known that $\exp(C^{-1}AC)=C^{-1}\exp(A)C$, where $C$ is an invertible matrix. Really, $$\exp(A)=\sum{}\frac{A^k}{k!} \quad \text{and} \quad (C^{-1}AC)^k=C^{-1}A^kC,$$ so the first ...
1
vote
4answers
358 views

Questions about the subgroups of $SU(2)$ and relevant problems?

This question is based on the invariant gauge groups in condensed matter physics( ...
5
votes
0answers
332 views

Simultaneously (generalized) diagonalizable matrices

I heard the following theorem from our textbook: Given $A,B$ are two commuting ($AB=BA$) real normal matrices. There's some real orthogonal matrix $P$ such that $P^{-1}AP$, $P^{-1}BP$ are ...
4
votes
3answers
345 views

Matrix which commutes with permutation matrix

I'm trying to show that if $A$ commutes with all $3\times 3$ permutation matrices, then $A$ has to be of the following form: $ A = \begin{pmatrix} a & b & b \\ b & a & b \\ b & b ...
2
votes
1answer
90 views

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 ...
0
votes
1answer
190 views

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 $n$?
1
vote
1answer
162 views

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 ...
1
vote
1answer
69 views

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 ...
5
votes
1answer
267 views

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. ...
2
votes
1answer
70 views

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 ...
6
votes
2answers
262 views

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 ...
0
votes
1answer
30 views

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 ...
3
votes
0answers
228 views

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 ...
3
votes
1answer
481 views

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 ...
2
votes
3answers
325 views

Matrix Representation of Octonions

Since quaternions $\mathbb{H}$ have a matrix representation as elements of $\text{SU}(2,\mathbb{C})$ as the following $$ 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\quad \mathrm ...
2
votes
1answer
58 views

Eigenvalues of the matrix $(-1)^{i_1+i_2+\cdots+i_k+j_1+j_2+\cdots+j_k}$

$M_{[i],[j]}=(-1)^{i_1+i_2+\cdots+i_k+j_1+j_2+\cdots+j_k}$, where $1\le i_1<i_2<\cdots<i_k\le n$ and $1\le j_1<j_2<\cdots<j_k\le n$, can be taken to be an $\left(n\atop ...
1
vote
1answer
43 views

An Embedding of $PGL_n \Bbb C$

I have a question about the projective general linear group. How does one realize it as a matrix group? Specifically, what is an embedding of $PGL_n \Bbb C \to GL_k \Bbb C$ for some $k$? In this case, ...
2
votes
1answer
137 views

Projective Tetrahedral Representation

I can embed $A_4$ as a subgroup into $PSL_2(\mathbb{F}_{13})$ (in two different ways in fact). I also have a reduction mod 13 map $$PGL_2(\mathbb{Z}_{13}) \to PGL_2(\mathbb{F}_{13}).$$ My question is: ...
6
votes
2answers
141 views

Embedding $GL_2(\mathbb{F}_l)$ into $GL_2(\mathbb{C})$

I recently learned that $GL_2(\mathbb{F}_3)$ can be embedded into $GL_2(\mathbb{C})$; specifically, $$ \left(\begin{array}{cc} -1 & -1 \\ -1 & 0 \end{array} \right) \mapsto ...
4
votes
0answers
109 views

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 ...