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

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Artin-Wedderburn theorem and square dimension

Let $A$ be a finite-dimensional simple algebra over $\mathbb{C}$ of dimension $n$. By Wedderburn's theorem, we have that $A$ is isomorphic to a matrix ring $M_r(\mathbb{C})$, which is of dimension ...
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Representations of a product of Lie groups

Let $G=G_1\times G_2$ be a product of two compact Lie groups. Is every finite dimensional irreducible representation of $G$ a tensor product of irreducible representations of $G_1$ and $G_2$? This ...
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34 views

Thm 5.19 (Brauer) in Isaacs Martin - Character Theory of Finite Groups

I would please like some help to understand the proof of thm 5.19 (Brauer) in Isaacs Martin - Character Theory of Finite Groups. It states: Let $\chi$ be a character of G with $[\chi,1_G]=0$. Let ...
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Decomposition of regular representation

Let $G$ be a group and consider its regular representation. We may uniquely decompose this representation into sums of irreducible components. What does it mean to find a basis for each component?
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Etingof Problem 5.1, “Field embeddings”

Recall that $k(y_1, \dots, y_m)$ denotes the field of rational functions of $y_1, \dots, y_m$ over a field $k$. Let $f : k[x_1, \dots, x_n] \to k(y_1, \dots, y_m)$ be an injective $k$-algebra ...
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$GL(V)$ representations and Schur modules.

Let $W$ be a fine dimensional complex vector space of dimension $n$ and $L_{\lambda}W$ the Schur module associate to the partition $\lambda=(\lambda_1, \cdots,\lambda_{n-1})$, where $\sum_i ...
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Size of conjugacy classes in SL(2,3)

I've been given the representations of the conjugacy classes for a group presentation $G = <x,y,z | x^2 = y^3 = z^3 = xyz>$ which is isomorphic to $SL(2,\mathbb{F}_3)$ which are: ...
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Identifying the cotangent bundle of the flag variety

Suppose $G$ is a Lie group (or I guess a linear algebraic group), $P \subset G$ a Lie subgroup with Lie algebras $\mathfrak{g}$ and $\mathfrak{p}$ respectively. In Chriss and Ginzburg's book ...
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Why is 1/2+1/2 in the weight space for SO(5)

Let's consider $\mathfrak{so}(5)$ as the Lie algebra of $\mathrm{SO}(5)$, where the symmetric bilinear form is $x_1y_5+\cdots +y_1x_5$. Then the maximal torus is given by $$\left(\begin{array}{cccccc} ...
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What are the differences between the three editions of the book “The Structure of Compact Groups”?

meta pre-clarification: I looked into another question like this but the guy didn't mark any specific tags for this type of question. Here's a link to the amazon book: ...
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Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc… & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...
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If $A$ is an abelian C*-algebra, and $\tau$ is pure then it is a character on $A$

If $A$ is an abelian C*-algebra,and positive linear functional $\tau$ is pure then it is a character on $A$. Murphy in his book(C*-algebras and operator theory) has below proof: While I think we can ...
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Invariants of $V^{\otimes N}$. [on hold]

Let $V$ be a finite dimensional complex vector space, and $G = SL(V)$ be the group of linear transformations of $V$ with determinant $1$. (a) Show that $V^{\otimes N}$ contains a nonzero ...
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Another construction of Specht module.

Let $\lambda$ be a partition of $n$, and $\lambda^*$ the dual partition (i.e. having the transposed Young diagram). Let $z_i$ be vectors in $\mathbb{C}^{\lambda_i^*}$, and$$F_\lambda = \prod_i ...
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Artin Algebra Representation Chapter Resource Request

I am working through chapter 10 of Artin's Algebra 2ed which introduces Group Representations. However, I've found that the approach to introducing groups is unlike the usual method used by more ...
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22 views

Is the coaction $\delta: A \to H \otimes A$ injective?

Let $A$ be an algebra and let $H$ be a bialgebra. Suppose that $A$ is an $H$-comodule. Then we have a coaction $\delta: A \to H \otimes A$. Is the coaction $\delta: A \to H \otimes A$ always ...
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A Representation Theory Problem in Putnam Competition

The following was the B6 problem of 1985 Putnam Competition: Suppose $G$ is a finite group (under matrix multiplication) of real $n\times n$ matrices $\{M_i\}, 1\leq i\leq r$. Suppose that ...
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35 views

Embedding so(n) in su(n)

Is there any way of embedding $\mathfrak{so}(n)$ into $\mathfrak{su}(n)$ for any $n$ other than picking the antisymmetric matrices of $\mathfrak{su}(n)$? I know that for small $n$ one can use ...
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1answer
27 views

One Dimensional Representations of $GL_3(\mathbb{F}_p)$

I am currently trying to find the irreducible one dimensional representations for the subgroup $H$ of $GL_3(\mathbb{F}_p)$ consisting of the upper triangular matrices. I know the number of ...
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2answers
87 views

Ideal of nilpotent elements in non-commutative ring.

Let $R$ be a non-commutative ring such that every element is either invertible or nilpotent. I am trying to show that the set of nilpotent elements, denoted $I$, is a two sided ideal, but I am having ...
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33 views

Why mention the “self-conjugate” property in Tannaka duality?

Based on this Wikipedia section and this MathOverflow answer of Qiaochu, I believe I've understood Tannaka duality for finite groups. We wish to characterize a finite group $G$ as a subgroup of ${\rm ...
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Reducible Lie Algebra

I'm furthering my physics knowledge through a book called Lie Algebras in Particle Physics and am having trouble with one aspect of a problem. I believe because it's a question purely about ...
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26 views

Matrix representation of an operator

Murphy says : The pure states of $A=K(H)$ are precisely the states $\omega_x : A\to \Bbb C ~~;~~\omega_x(u) = \langle ux,x\rangle $ where $x$ is a unit vector of Hilbert space $H$ . Then he gives ...
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Questions about subrepresentations of a representation of a quiver.

Let $Q$ be the quiver $\cdot \to \cdot \to \cdot$. Then $$ \mathbb{C} \to^{f} \mathbb{C} \to^g \mathbb{C} \quad (1) \\ 0 \to^{0} \mathbb{C} \to^0 0 \quad (2) $$ are two representations of $Q$, ...
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Representations of the Lie algebra $\mathfrak{sl}(2,\mathbb{C})$

In Corollary 7.2 of http://math.uchicago.edu/~may/REU2012/REUPapers/Bosshardt.pdf, why is the set of weights an unbroken string? I understand we get a finite number of weights by looking at the ...
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1answer
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Character theory - Exercise 5.14

I am trying to solve the exercise 5.14 from the Isaac Martins Character Theory of Finite Groups. Let $G$ be a nonabelian group and let $ f=min\{\chi(1) | \chi \in Irr(G), \chi(1)>1 \}. $ Show ...
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29 views

the span of a representation's action on a vector

Consider the image of the action of a group representation $\rho: G \to V$ on some vector $v \in V$: $$ \{ \rho(g) v : g \in G \} $$ It seems that the span of this set: $$ W_v \equiv ...
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matrix Lie group embedding as a manifold

Given a Lie group of matrices, and suppose for simplicity that it is globally generated through exponential map from its Lie algebra on a element. Is there a canonical way to embed it into ...
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26 views

symmetric group acting on torus

Let $S_k$ be symmetric group of order $k$. Let $T^k=S^1\times\cdots \times S^1$. Then $T^k$ is a Lie group. For each $\sigma\in S_k$, let $\sigma$ act on $T^k$ from right in the way $$ ...
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Prove that there exists an $m$th root of unity $λ ∈ C$ such that for all $g ∈ G$, $\chi(zg) =λ \chi(g)$

Suppose $\chi$ is an irreducible character of $G$. Suppose $z ∈ Z(G)$ and that $z$ has order $m$. Prove that there exists an $m$th root of unity $λ ∈ C$ such that for all $g ∈ G$, $\chi(zg) =λ ...
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Finding the eigenvalues of the matrix corresponding to an element of finite order in a group representation

Suppose that $g$ is an element of finite order (say $n$) in a group $G$ and $\rho:G\rightarrow GL(V)$ is a degree $n$ representation of $G$. If now I know $\chi_V$, how can I find the eigenvalues of ...
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A question involving condition for an element to be in the kernel of a representation ans the kernel of the coset representation

I came across the following question. Given a representation $\rho:G\rightarrow GL(V)$ with kernel $N$, let $\rho$ have character χ : $G$ → $\mathbb C$. Then for $g ∈ G$, it first asks to prove that ...
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Finding conditions for different representations to be faithful

I'm trying to find conditions under which each of the following representations are faithful: trivial, regular, coset, sign (for $G = S_n$), defining (for $G = S_n$) and degree 1 for $C_n$, the cyclic ...
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1answer
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Young tableaux of $8\otimes 8$ in $SU(3)$

In Georgi's Lie Algebras in Particle Physics, one finds the following Young tableaux for $8\otimes 8$ in $SU(3)$: I am unsure of all the cancellations. Let us number the canceled tableaus increasing ...
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Finding normal subgroup from a character table

I have the following character table. Note I assume that $\chi_i$'s are all irreducible. $$ \begin{array}{|c|c|c|c|c|} \hline & C_1 & C_2 & C_3 & C_4 & C_5 \\ \hline \chi_0 & ...
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1answer
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Finding irreducible subrepresenations of modular representation in GAP

Recently, I have been fiddling with modular representations in GAP. First from what I can tell, GAP does not have a good way built in to find things like Brauer characters of a given non-solvable ...
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1answer
59 views

Vector spaces over quaternions

Let $V$ be an $n$-dimensional vector space over the quaternions $\mathbb{H}$, and let $G$ be the multiplicative quaternion group. How would one show that $V$ would then be a $4n$-dimensional ...
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THE positive half-spin space of quaternion vector space

I have the following information: $T$ is the one-dimensional quaternion vector space with the canonical action of $\Gamma$, a finite subgroup of SU$(2)$. This makes sense as SU$(2)$ is the unit ...
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Question about using two associative algebras irreducible modules to prove the algebras are isomorphic

If two associative algebras $A_1$, $A_2$, over some field admit a bijection $f$ between irreducible modules such that dim $M_1$= dim $f(M_1)$ where $M_1$, ($f(M_1)$) are irreducible $A_1$ ...
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37 views

Complete the character table of group of order $21$

You are given the incomplete character table of a group $G$ with order $21$ which has $5$ conjugacy classes, $C_1,\dots,C_5$, which have sizes $1,7,7,3,3$. $$ \begin{array}{|c|c|c|c|c|} \hline & ...
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prove diagonalizablity using argument from representation theory

Let $\{X_1,X_2,...,X_n\} \subseteq GL_d$ be a subgroup of commutating matrices then show this matrices are simultaneously diagonalizable (using some argument from representation theory)
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Finding the character table of $Q_8$

Assume $G$ is a finite group. I am trying to construct the character table of $Q_8$, which is defined by $$Q_8=\{\pm 1,\pm i, \pm j,\pm k \}, \ i^2=j^2=k^2=-1, \ ij=k,jk=i,ki=j$$ By considering the ...
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What is the difference between the representation of a group and an algebra?

Sometimes, I come across this idea in physics -> the representation of Lorentz group: SO(3,1) and the representation of Lorentz algebra: so(3,1). At times, I mix them up. Is there a good intuitive way ...
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Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation show that $|\operatorname{tr} X| \leq \dim \rho$

Let $G$ be a finite group. Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation, pick $g \in G$ and write $X=\rho(g)$. Prove that all eigenvalues of $X$ are roots of unity, and deduce that ...
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How to rigorously show tensor identities using symmetry arguments?

I am wondering how to rigorously show tensor identities such as the following. Let $n$ denote the radial unit vector in $D$ dimensions. Then $\langle n_i n_j \rangle = \frac 1 D \delta_{ij}$ and ...
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What is the notion of “character” in the context of Cayley graphs?

I am looking at these notes, http://www.eecs.berkeley.edu/~luca/books/expanders.pdf On page 37, Lemma 5.16, the notion of "character" defined seems to be any map from the finite Abelian group to ...
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Example of $\det \rho(g)=\det \sigma(g)$ for all $g\in G$, but $\rho \not\simeq \sigma$

Give an example of a group $G$ and two representations $\rho$ and $\sigma$ of $G$ such that $\det \rho(g)=\det \sigma(g)$ for all $g\in G$, but $\rho \not\simeq \sigma$. At the moment but ...
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88 views

Ten dimensional representation of $S_6$

Let $S=\{1,2,3,4,5,6\}$. For every three-element subset $A\subset S$ and $B=S\setminus A$ consider the symbol $e_{(A|B)}$ for which we assume that $e_{(A|B)}=e_{(B|A)}$. Then the vector space $V$ ...
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Reducibility of Cyclic groups

Let $G$ be the cyclic group $C_{4}$ and consider the 2-dimensional representations of G. Why does extending scalars to the complex numbers let this representation become reducible? I understand how it ...
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Decomposing some representations as a direct sum of irreducibles

I'm taking $V$ to be the standard representation of $S^3$. I'm looking for the decomposition of the following representations as a direct sum of irreducibles. (a) $V\bigotimes V$ (b) $Sym^2$ $V$ (c) ...