Questions tagged [representation-theory]

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

98
votes
0answers
2k views

Application of Hilbert's basis theorem in representation theory

In Smalo: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand: Two orders are defined on the set ...
66
votes
0answers
1k views

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
45
votes
0answers
1k views

Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In some ...
14
votes
0answers
156 views

If $g$ is commutator then so is $g^m$ for $(m,o(g))=1$

There are certain theorems in finite group theory whose proofs involve character theory and for which there are still no character-free proofs. Among such is Frobenius theorem on transitive ...
13
votes
0answers
651 views

Character theory of $2$-Frobenius groups.

Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this? Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\...
12
votes
0answers
592 views

Lowest dimensional faithful representation of a finite group

How does one compute the lowest dimensional faithful representation of a finite group? This question originated in the context of given a finite group $G$: trying to find the lowest dimensional shape ...
12
votes
0answers
284 views

The division algebras arising in the Wedderburn decomposition of a finite group modulo its radical in characteristic $p$

The following question is probably straightforward for those who know. However, I am used to working either over splitting fields or in characteristic zero. Question. Let $G$ be a finite ...
12
votes
0answers
240 views

Subgroup structure of $\mathrm{SL}(2, p^2)$ and its irreducible characters

I am taking a course in representation theory of finite groups, and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and $\...
11
votes
0answers
154 views

$\text{SL}(2, \mathbb{F}_q)$, for which characters is the $G$-representation irreducible?

Followup to here. Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = \text{SL}_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes ...
11
votes
0answers
214 views

Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\...
11
votes
0answers
463 views

Representation theory of the general linear group over a finite prime field

The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely classified and well-understood via Schur-Weyl duality, the algebraic Peter-Weyl theorem and the entire ...
11
votes
0answers
1k views

Induced representation, Ind(Res(U))

I am reading a book of Fulton and Harris "Representation theory, a first course". Now it's all about representation theory of finite groups, and there is one exercise, which I can't solve: If $U$ is ...
11
votes
0answers
592 views

cohomological proof of Maschke's theorem

I have been working on the following problem.. I have spent plenty of time trying to solve it myself. I am, however, unable to prove one small step in the argument. Beneath you can find my attempt. ...
10
votes
0answers
193 views

How do I know if an irreducible representation is a permutation representation?

I have a vague question, a less vague question and a lot of vaguer questions about permutation representations of a finite group $G$. Vague question. Recall that if $G$ acts on a finite set $X$, we ...
10
votes
0answers
174 views

A characterization of the “Direct Integral” construction in terms of the properties it satisfies?

Fortunately there's a wonderful thing called the "direct integral" which enables one to make sense of direct sums of uncountably infinite families of Hilbert spaces. Unfortunately I've tried to read ...
10
votes
0answers
135 views

Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
10
votes
0answers
92 views

Relationship between exterior power of representation and variance?

I was reading the question: Symmetric and exterior power of representation regarding how to determine the character of an exterior power of a representation from the original representation. One of ...
9
votes
0answers
168 views

Spherical Schubert Variety

I am studying Schubert variety and I came across a problem understand a particular detail. Let $G$ be a reductive group, and $\mu\in X_{\bullet}(T)$ a coweight i.e. $\mu\in Hom(\mathbb{G}_m,T)$, ...
9
votes
0answers
456 views

A question about the article 'You can't hear the shape of a drum'

I have read the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
9
votes
0answers
109 views

Theorem about the subgroup of a Lie group fixed by an involution

When trying to do Lie-theoretic calculations on Lie groups (finding the Bruhat decomposition, etc.) I've often come across expositions that seem to be implicitly using a result something like the ...
9
votes
0answers
107 views

Reciprocity for branching rules of $\mathrm{GL}_n(\mathbb C)$

[Separated from another question] If I have information about the restriction of representations of the general linear group, can I make any statements about the induction (by Frobenius reciprocity)? ...
9
votes
0answers
223 views

Weyl character formula for locally compact Lie groups.

I was just wondering if there exists such a formula. Specifically I need to calculate characters of irreducible representations of GSp$(4,\mathbb{C})$. I know how to do it for the compact Lie group ...
9
votes
0answers
178 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
8
votes
0answers
105 views

Calculating the Ring of Invariant Polynomials for the Action of a Compact Simple Lie Group

Let $G$ be a compact connected simple Lie group and let $V$ be a real $G$-representation. How does one go about computing the ring of $G$-invariant (polynomial?) functions $V \to \mathbb{R}$? I've ...
8
votes
0answers
84 views

Missing $\{2,p\}$-Hall subgroups in finite non-abelian simple groups

Can anybody tell me how to prove this theorem? Theorem: Suppose that $G$ is a finite non-abelian simple group. Then there exists an odd prime $p\in\pi(G)$ such that $G$ has no $\{2,p\}$-Hall ...
8
votes
0answers
153 views

Duality of $Z(G)$ and $[G,G]$ in representation?

This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group. I was thinking about its manifestation in group ...
8
votes
0answers
513 views

Young Tableaux as Matrices

These questions are motivated only by curiosity. Take a Young tableau of shape $(\lambda_1,\lambda_2,\ldots,\lambda_n)$, where $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n$. Is there any ...
8
votes
0answers
299 views

Generating function for characters of representations

One example of such a generating function that I know how to derive is for $SU(2)$, $\frac{1}{(1-tx)(1-\frac{t}{x})}$. The coefficient of $t^n$ in the above function is the character in the $n+1$ ...
7
votes
0answers
157 views

Local factors determine Weil representations - proof of the cyclic case

I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and Vladimir Dokchitser: Theorem 1 Every Frobenius-semisimple Weil representation $...
7
votes
0answers
101 views

Can this puzzle be solved using the representation theory of quivers?

This riddle originates in the youtube video here. It's mathematical content was summarised here as follows: There's a $5\times 5$ grid of nodes, all nodes are (bidirectionally) connected to their ...
7
votes
0answers
79 views

dimension of $\{x \in F^n \mid Ax=x^p \}$ for general $A \in M_n(F)$

Let $F$ be an algebraically closed field with char${}=p>0$, so $x \mapsto x^p$ is an automorphism of $F$ hence induces a $\Bbb F_p$-linear morphism $ x \mapsto x^p: F^n \rightarrow F^n$. For $A \in ...
7
votes
0answers
155 views

Is there a linear injection $ \Lambda^k V^* \otimes \Lambda^k V^* \to \Lambda^k (V^* \otimes V^*)$ which preserves decomposability?

Let $V$ be an $n$-dimensional real vector space, and let $2 \le k \le n-2$. Definitions We say an element $\omega \in \Lambda^k V$ is decomposable if $\omega=\alpha_1 \wedge \dots \wedge \alpha_k$, ...
7
votes
0answers
142 views

Reference request: indefinite orthogonal groups $O(p,q)$, spin groups $\mathrm{Spin}(p,q)$, and projective orthogonal groups $PO(p,q)$

The indefinite orthogonal group $O(p,q)$ is the orthogonal group preserving the standard scalar product of signature $(p,q)$ on $\mathbb{R}^{p+q}$. Are there any good references that discuss the ...
7
votes
0answers
90 views

Cuntz algebra and Schur-Weyl duality

Let $\mathcal{O}_n$ be the Cuntz algebra with generators $a_1,...,a_n$. We can define an action of $U(n,\mathbb{C})$ (the group of $n\times n$ unitary operators) on $\mathcal{O}_n$ in a very natural ...
7
votes
0answers
93 views

How can I find the monodromy of a cyclic galois cover of the affine line minus a few points?

Consider the following cyclic covering of the affine line minus a few points: $$ \text{Spec}(\mathbb{C}[t,x]/(x^n - t(t-1)(t-2))) \to \mathbb{A}^1_t - \{ 0, 1, 2 \} $$ How can I find the local ...
7
votes
0answers
137 views

Decomposition of An Induced Representation of $GL(2, q)$

Denote $G = GL(2, q) = GL_2(\mathbb{F}_q)$, $B$ its Borel subgroup of upper triangular matrices, $T$ its splitting torus of diagonal matrices. The object I am interested in is $Ind_B^G\rho$, where $\...
7
votes
0answers
771 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) \...
7
votes
0answers
223 views

Expression of basis vectors of permutation modules in different bases.

Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
7
votes
0answers
148 views

Clifford theory and induction

in the answer to this post there was the statement that a representation $\vartheta$ of a subgroup $\langle z\rangle$ can extend to a representation of the whole group $D_{2n}$. If I start the other ...
7
votes
0answers
1k views

Positive definite function zoo

A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$. For a definition and discussion of ...
6
votes
0answers
70 views

Is the matrix $\sum_{g \in G} a_g \rho(g)$ normal and what further properties does it have?

Let $\rho$ be the regular representation of $G$. $S \subset G$ a generating set, $|g| := |g|_S=$ word length with respect to $S$. Then I construct such a matrix, where we have some ordering $g_i$ of ...
6
votes
0answers
84 views

Supercuspidal representation from compact induction

Note: this is a homework problem. Any hints would be great. Consider the group $G=GL_n(\mathbb{Q}_p)$ and an open subgroup $K$ that is compact modulo center. Suppose we have a smooth representation $\...
6
votes
0answers
804 views

Irreducible 2-Brauer characters of $S_5$

Beginning with the ordinary character table of the symmetric group $S_5$, one immediately gets the following Brauer characters in characteristic two: $\begin{array}{c|c|c|c} S_5 & () & (...
6
votes
0answers
1k views

Some concrete examples of $M_q(2)$ points

Given $q \in C$ invertible, Kassel says that a $M_q(2)$ point of an $R$ algebra is a $m=\left(\begin{array}{cc} A & B\\ C & D \end{array}\right)\in R^{4}$ such that $A,\,B,\,C,\,D \in R$ ...
6
votes
0answers
156 views

Reference for flag varieties G/P

Is there a good reference for learning about flag varieties $G/P$? I'm already comfortable with the algebraic geometry and the example of Grassmannians, but I am not so comfortable with algebraic ...
6
votes
0answers
154 views

Upper Bound Lemma implies the Ergodic Theorem for Random Walks on Groups?

Cross posted on Mathoverflow Ergodic Theorem A random walk on a finite group $G$ driven by a probability $\nu\in M_p(G)$ is ergodic if $\operatorname{supp}(\nu)$ is not concentrated on a proper ...
6
votes
0answers
210 views

Application of SU(2) in physics

How can we interpret the representations of SU(2) and $\mathfrak{su}(2)$ in physics? I have studied a lot of mathematics including representation theory and differential geometry, so I understand SU(...
6
votes
0answers
102 views

Different notions of upper / lower indices

In differential geometry and tensor analysis, lower and upper indices appear naturally through covariant and contravariant transformations. One uses the metric tensor and its inverse to lower and ...
6
votes
0answers
151 views

Understand representations of c*-algebras from a categorical point of view

In my lecture on von Neumann algebras we have defined a representation of a c*-algebra $\mathcal{A}$ as a *-homomorphism $\pi$ into $\mathcal{B}(\mathcal{H})$ for some Hilbert space $\mathcal{H}$. ...
6
votes
0answers
311 views

Frobenius determinant theorem

Can anyone please recommend a paper or a book that gives a detailed proof of the Frobenius determinant theorem? I have read some few papers I saw online but their information are not sufficient for my ...