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

learn more… | top users | synonyms (1)

2
votes
0answers
75 views

Representation theory& module

$V$ is a left $R$ module, how do you understand the ring homomorphism $$\rho_{V}:R \to End_Z(V)$$ I know that it is like a group acting on sets, but it is very easy to understand like a group $S_n$ ...
2
votes
0answers
76 views

Representations of Nilpotent Lie Algebras

Let $\mathfrak{g}$ be a rational, nilpotent Lie algebra. Then its adjoint representation will consist of elements which are nilpotent matrices over rationals. But this representation generally is not ...
2
votes
0answers
35 views

The embedding $L^2(\Gamma(N)\backslash\text{SL}_2(\mathbb{R})) \hookrightarrow L^2(\text{SL}_2(\mathbb{Q})\backslash \text{SL}_2(\mathbb{A})))$?

Let $\mathbb{A}$ be the ring of adeles. Let $N$ be a natural number. Let $\Gamma(N)=\ker(\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})))$. I saw the following embedding of ...
2
votes
0answers
95 views

Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
2
votes
0answers
51 views

A construction of $\mathfrak{e}_8$ in Fulton and Harris

In section $22.4$ of "Representation Theory: A First Course" by Fulton and Harris, the exceptional Lie algebra $\mathfrak{e}_8$ is constructed using a method of Freudenthal. For background, I will ...
2
votes
0answers
38 views

On an invertible element for equivariant K-theory

Fix a positive integer $m$. Let $G = \lbrace h\in\mathbb C | h^m = 1\rbrace$ and $(X,\pi)$ the standard representation of $G$. Namely $X = \mathbb C$ and $\pi:G \to GL(X)$ is defined by $\pi(h)v=h v$ ...
2
votes
0answers
52 views

Learning representation theory of real reductive lie groups

I am interested in any sources that can be helpful for learning the representation theory of real reductive groups. I am currently reading Wallach book, but I feel that I don't understand the subject ...
2
votes
0answers
31 views

What do diagonal matrices do in irreducible repns of SL$_2(\mathbb{Z}/N\mathbb{Z})$?

Let $N \in \mathbb{N}, \mathbb{Z}_N := \mathbb{Z}/N\mathbb{Z}$. For every $a \in \mathbb{Z}_N^\times$ put $R_a = \begin{pmatrix} a^{-1} & 0 \\ 0 & a\end{pmatrix}$ and also set $T = ...
2
votes
0answers
29 views

Rational representation of a 136 order group

Let G be the group of order 136 = 8 * 17 with presentation $$<x,y : y^{8}=x^{17}=1 \quad yxy^{-1}=x^{4} >$$ Find the simple summands of the group ring Q[G].
2
votes
0answers
57 views

What is the Lie algebra of $G=\mathbb{R}$

The question is updated as following. 1. Let $(\Phi,L^2(R))$ be left regular representation of $\mathbb R$ given by $$ \Phi(g)f(x)=f(x-g). $$ It is unitary representation of $\mathbb R$. 2. For ...
2
votes
0answers
57 views

Skew polynomial algebra and deformation

Let $R$ be an associative unital $k$-algebra. If $\alpha \in End_k(R)$ and $\delta$ is a $\alpha$-derivation, then one can define the skew polynomial algebra $R[x; \alpha,\delta]$ by letting $ax = x ...
2
votes
0answers
52 views

A question in the proof that the weight of a finite dimensional module is W-invariant

Recently I'm reading Humphrey's book "Introduction to Lie algebra and representation theory", section 21 on the finite dimensional module of a semisimple lie algebra, and I have a question here which ...
2
votes
0answers
50 views

question on $\mathbb{Q} \otimes R[G]$ his maximal ideals, the action of a Galois group on it

Reasoning on a question a friend posed me, i've found a question in the following setup: Suppose you have a finite group G, now you can pass to the algebra $\mathbb{Q} \otimes R[G]$ where the second ...
2
votes
0answers
57 views

Fulton-Harris Lemma 3.35

In the proof of Lemma 3.35 in Fulton--Harris, Representation Theory, it is claimed that the identification $H(\phi^2(x),y)=H(x, \phi^2(y))$ implies that $\lambda$ is a positive real ($\phi^2$ is known ...
2
votes
0answers
51 views

Why will we never be able write down all automorphic representations “explicitly”?

In a recent article (p. 7), J. Arthur writes In fact, it is pretty clear that we will never be able to write down all automorphic representations explicitly. I have often heard similar remarks ...
2
votes
0answers
161 views

irrep over $\mathbb R$

Let $G$ be an abelian group such that $|G|=8.$ How many irreducible representations $G$ over $\mathbb R$ are there? I know that dimension of those representations is $\leq2.$ AndThe number of ...
2
votes
0answers
53 views

Equation on the vertices of regular polyhedra

I found in this book, on page 6 that the equation on vertices of icosahedron inscribed in sphere considered as $\mathbb{CP}^1$ by means of stereographic projection is $xy(x^{10}+14x^5y^5-y^{10})=0$. ...
2
votes
0answers
178 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
2
votes
0answers
127 views

A representation of $SU(2)$ is self dual

Let $SU(2)$ be a set of $2 \times 2$ unitary matrices over $\mathbb{C}$ with determinant $1$. Let $H_j$ be a $2j+1$ dimensional vector space with basis $x^ay^b$ with $a+b=2j$. A representation $U_j$ ...
2
votes
0answers
72 views

Characters of subgroups of finite abelian groups

Let $G$ be a finite abelian group. Let $H$ be a subgroup of $G$. Let $\hat{G}$ be the group of characters of $G$. Is there a character $\chi \in \hat{G}$ such that $\chi(g) = 1$ iff $g \in H$?
2
votes
0answers
45 views

Volume of a ball for SO(n).

Let us equip the special orthogonal group $SO(n)$ with a normalized Haar measure $\theta_n$ and let $G_r$ be the subset of rotations $\Omega$ which differ from the identity by (sufficiently small) $r$ ...
2
votes
0answers
49 views

Square-integrable representations of noncompact groups

In Marc Rieffel's paper "Square-integrable Representations of Hilbert Algebras," he establishes (Corollary 5.12) that a nonfinite discrete group has no square-integrable, irreducible representations. ...
2
votes
0answers
335 views

Spinor representation and Clifford modules

Let $V$ be an even-dimensional real inner product space. We denote the Clifford algebra of $V$ by $C(V)$ and the spinor representation by $S$. For a finite-dimensional $\mathbb Z_2$-graded complex ...
2
votes
0answers
53 views

Indecomposable vector space

Let us define $V$ as a quotient space $\mathbb{K}[t]/(p^m)$, where $p$ is an irreducible polynomial. Condsider the linear operator $\phi\in Hom_{\mathbb{K}}(V,V)$, which sends each $q+(p^m)$ to ...
2
votes
0answers
48 views

Questions about Lusztig's $\mathbf a$-function

In chapter 13 of Lusztig's Hecke Algebra with Unequal Parameters, the function $\mathbf a$ is defined to be $$\mathbf a(z) = \max_{x,y} \deg h_{x,y,z},$$ for any $z$ in the Coxter group, where the ...
2
votes
0answers
193 views

Wedderburn decomposition of $D_{5}$

I'm wondering how to solve this question. Find the Wedderburn decomposition of $D_{5},$ the dihedral group of order 10, over the field $\mathbb{F}_{3}.$ I have shown that the irreducible ...
2
votes
0answers
29 views

The character of $\mathbb{C}[S_n]b_\lambda$

Let $b_\lambda = \sum_{g \in Q_\lambda} (-1)^{sgn(g)}g$, where $Q_\lambda$ is the subgroup of elements that permute the numbers in columns, $\lambda$ a Young diagram corresponding to partition ...
2
votes
0answers
287 views

Construction of line bundles on the flag variety

Edit: The following is phrased in terms of algebraic geometry, but can be thought of analytically as well. Hence I added some tags... I am a bit confused about the subject in the title. For ...
2
votes
0answers
88 views

If $S$ is a simple $\mathbb{C}G$-module and $U$ is a one-dimensional $\mathbb{C}G$-module, then $S \otimes U$ is simple

Show that if $S$ is a simple $\mathbb{C}G$-module and $U$ is a one-dimensional $\mathbb{C}G$-module, then $S \otimes U$ is simple. I've tried to prove this fact, but I think ther's a mistake ...
2
votes
0answers
68 views

Simple components and the irreducible characters of the group ring $K[G]$

Let $K$ be an algebraically closed field of characteristic $0$ and $G$ a finite group. I know that the group ring $K[G]$ is a semisimple and so decomposes as a direct sum of $m$ simple components ...
2
votes
0answers
168 views

Computing ext over graded rings

This question came up as I was reading Beilinson, Ginzburg, Soergel paper Koszul Duality Patterns in Representation Theory. Suppose that $A$ is a Koszul ring (for the definition of Koszul ring ...
2
votes
0answers
40 views

explicit realization of irreducible representations of simple lie algebras

I know explicit realization of irreducible representations of simple lie algebra $sl_n$ when the highest weight of that representation is a fundamental weight.Is there any explicit realization of any ...
2
votes
0answers
126 views

The matrix form of a representation of $S_3$.

I am going through some notes on group theory, and one problem states: Consider the three-dimensional representation of $S_3$ constructed as follows: Choose a basis $v_1,v_2,v_3$ of ...
2
votes
0answers
54 views

Irreducible component of representation of symmetric group modulo $p$

Let $p$ be a prime, $n$ an integer divisible by $p$. Consider the natural representation of the symmetric group $S_n$ with coefficients in $\bar{\mathbb{F}}_p$ (namely to each permutation we associate ...
2
votes
0answers
63 views

If $\mathbb{C}[G]$ is Noetherian and $G$ has a representation on $V$, when must $V$ be finite-dimensional?

I know this is a bit vague, but please bare with me here. Let's assume that $G$ is a finitely-generated torsion group. I want to show that $G$ is a finite group if I add some conditions. I suspect ...
2
votes
0answers
71 views

Group's algebra of finite group (representation theory)

Let $G$ be a finite group, and let $r$ - the number of it's conjugate classes, so our group has $r$ classes of irreducible representations. $\varkappa_1,..,\varkappa_r$ - irreduciible characters, and ...
2
votes
0answers
128 views

Comparing two character tables

Suppose that you are given two finite groups, for example, via their Cayley tables. One can efficiently compute their character tables (efficiently = polynomial time in the order of the group), this ...
2
votes
0answers
81 views

A question about almost split sequences

On page 124 of the book Elements of representation theory of associative algebras, volume 1, Proposition 3.11, I have a question about the proof. On Line 8 of the proof, it is said that if $u: R \to ...
2
votes
0answers
70 views

A technical problem on constructing smooth vectors in a representation

Let $G$ be a linear Lie group, say $\mathrm{SL}(2,\mathbb{R})$, and $\pi:G\to\mathrm{GL}(H)$ a continuous representation of $G$ in a Banach space $H$. Let $\pi^1:\mathcal{C}_c(G)\to\mathrm{End}(H)$ be ...
2
votes
0answers
105 views

Character of half-spin representation

Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum $$\sum x_1^{\pm ...
2
votes
0answers
55 views

Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?

Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have \begin{equation} d_{\alpha}|\#G, \end{equation} where $d_\alpha$ is the degree of the representation and $\#G$ ...
2
votes
0answers
101 views

Representations of $\text{GL}_2(\mathbb{Q})$

Let's say that as a representation theorist I am naively interested in representations of $G(\mathbb{Q})$, where $G$ is an algebraic group defined over $\mathbb{Q}$. For the purposes of this question, ...
2
votes
0answers
57 views

Subspaces stabilized by representations of $\mathrm O(9)$

I am trying to figure out what representations of maximal subgroups of $\mathrm{GL}_{n^2}$ stabilize one dimensional subspaces in $\mathrm{GL}(\mathrm{Sym}^n(\Bbb C))$. More precisely, let the setup ...
2
votes
0answers
62 views

Why $\text{top}h$ is an isomorphism?

I am reading the book Elements of representation theory of associative algebras I have a question about from Line -9 to Line -6 of page 29, the proof of Theorem 5.8. How to show that ...
2
votes
0answers
100 views

If $\mathfrak{g}$ admits a decomposition then it is semisimple

I want to show: If $\mathfrak{g}$ is a Lie algebra that has an abelian subalgebra $\mathfrak{h}$ such that $\mathfrak{g}$ has a Cartan decomposition $\mathfrak{g}=\mathfrak{h}\oplus(\bigoplus_{\alpha ...
2
votes
0answers
59 views

Molecular vibrations and a generalisation of Wigner's rule for (non-finite) compact groups

years student of mathematics and write my script for my bachelor. The topic is "Representations of groups and applications in physics". I understand the representations very good but now i want to ...
2
votes
0answers
158 views

Young tableaux: evaluate action of permutation

Consider the irreducible representation $V$ in the symmetric group $S_5$ corresponding to the Young diagram (these are meant to be boxes): $$[\;\;][\;\;] \\ [\;\;][\;\;] \\ [\;\;]\;\;\;\;$$ (a) List ...
2
votes
0answers
148 views

Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...
2
votes
0answers
140 views

DFT shift theorem generalizations?

The DFT shift theorem implies that any circular shift in the input space is equivalent to a phase change in the frequency domain, while the absolute values are preserved. $$ ...
2
votes
0answers
89 views

4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...