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

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Adjoint Representation of Lorentz Group

I'm thinking about the image under the adjoint representation $\mathrm{Ad}$ of the proper (identity connected component) Lorentz group $SO^+(1,3)$. Since this group has a trivial centre (it contains, ...
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56 views

Understanding the irreducible representations of $D_3$

By the dimensionality theorem, $$\sum_i d_i^2 = |G|,$$ where $d_i$ is the dimension of the $i$th irreducible representation, we can infer that the dihedral group $D_3$ has two one dimensional irreps ...
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30 views

What is a Complete Set of Weights of a Representation of a Lie Subalgebra?

In relation to Lie Group and Lie Algebra theory, I am studying about the weights of representations. I have come across the terminology "a complete string of weights" in my lecture course, but it is ...
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32 views

$\mathbb{1}\uparrow_H^{G}$ is the permutation representation on $G/H$

Is the following correct? If we have $G$ is a group with $H\leq G$ and we take $\mathbb{1}$ to be the trivial character on $H$ then I am trying to show that $\mathbb{1}\uparrow_H^{G}$ is then the ...
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39 views

Showing that $g$ and $g^{-1}$ are conjugate iff $\chi(g)$ is real

I am trying to show that for a finite group $G$ and $g\in G$, $g$ and $g^{-1}$ are conjugate iff $\chi(g)$ is real for all $\chi$ irreducible characters of $G$. I have the following: I first want ...
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16 views

restriction of spin representation to block diagonal subgroup

What is the restriction of the (complex) spin representation of $so(n+m)$ to the block diagonal subalgebra $so(n)\times so(m)$? A naive guess is that it is the (complex) tensor product of the two ...
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41 views

Lie group representatiom - quasi-equivalent representations

Let $T$ and $U$ be unitary representations of a conected simply conected nilpotent Lie group, such that all irreducible subrepresentations of $T$ and $U$ are the same. If $T$ and $U$ are finite, then ...
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304 views

Fourier Transform on compact groups

I'm trying to get my head around the concept of Fourier Transform on a compact group. The standard definition is $$\widehat{f}(\pi)=\int_Gdg\,f(g)\pi(g)$$ where $\pi\in\widehat G$, the Pontryagin ...
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25 views

Universality of restricted representations

Suppose that $H$ is a subgroup of a finite group $G$. Given an irreducible representation $\rho$ of $G$, this creates a (possibly reducible) representation $\rho'$ of $H$ obtained by restricting ...
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15 views

About representations and transformations under an $SU(n)$ Lie Group

I think my problem is that I misunderstand what "transforms under" really means. Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the ...
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26 views

Lie Algebra of Reduced Heisenberg Group Identities

I am having problems trying to understand a statement by Howe in his paper "On the role of the Heisenberg group in harmonic analysis". Here is the setting: Howe defined the (reduced) Heisenber group ...
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34 views

Schur-Weyl duality from Double Commutant Theory

Let $V$ be a finite dim complex vector space. Then $V^{\otimes n}$ carries an action by $S_n$ by permuting factors $\sigma(\pi)(v_1\otimes...\otimes v_n)=v_{\pi^{-1}(1)}\otimes...\otimes ...
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110 views

The natural representation of $SO(n)$ is irreducible for $n\ge 3$

The natural representation $(\pi,\mathbb C^n)$ of $SO(n)$ is the one for which $$\pi (g)z = g^{-1}z$$ for $g\in SO(n)$ and $z \in \mathbb C^n$ (the product $g^{-1}z$ is just the usual matrix ...
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108 views

What are the consequences of presentation of an algebra by generators and relations?

Let $A$ be a finite dimensional associative $K$-algebra, where $K$ is a field. I wonder how the presentation of $ A $ by generators and relations helps in the study of structure of the algebra ...
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45 views

Questions about the proof of generalized Poisson summation formula.

The generalized Poisson summation formula is $$ \sum_{\gamma \in \Gamma} f(\gamma) = \sum_{ \pi \in \widehat{\Gamma \backslash G}} \hat{f}(\pi), $$ where $G$ is a locally compact Abelian group, ...
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55 views

Irreducible representations of $SO(5)$

I am looking for irreducible representations of the group $SO(5)$ that can be described by a tensor of at most rank two. My own considerations have brought me to the conclusion that there is a ...
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33 views

Integration on associated vector bundle

Let $G$ be a compact lie group and $\mathfrak{g}$ be its Lie algebra then we can construct the integral on $G\times \mathfrak{g}$ by $$\int_G\int_{\mathfrak{g}}f(x,Y)dxdY$$ Where $x\in G$ and $Y\in ...
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33 views

Does the representation ring functor preserve limits?

If I have a diagram of groups $\{H_J\}$ and let $G$ be the limit of that diagram, how well does the representation ring functor "preserve the limit", IE: If I have $\lim_{J} H_J = G$ is it true that ...
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48 views

Permutation representations

I need a source for Permutation representations of general linear groups over finite fields. Can anyone introduce some sources?
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24 views

representation type of PI rings

A ring $R$ is said to satisfy a polynomial identity (PI for short) if there exists a polynomial $f(x_1, \ldots, x_n) \in \mathbb{Z} \langle X_1, \ldots, X_n \rangle$ such that $f(r_1, \ldots, r_n)=0$ ...
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56 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
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19 views

Meromorphic Continuation of Intertwining Operator Identities for K-finite vectors

I have a question about a "well-known result" about intertwining operators. I will restrict myself to $\text{SL}_2$ for simplicity. Let $$w = ...
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30 views

weight of a group G

how can I found the weight of a group G ( I find the fundamental weights, but I don't know how found the linear combination of fundamental weights that give me the weights ). so we found the ...
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36 views

Questions about isotypic subspaces and co-isotypic subspaces of $V$.

Let $G$ be a finite group and $\chi: G \to \mathbb{C}^*$ a character. Let $(\pi,V)$ be a representation of $G$. The $(G, \chi)$-isotypic subspace of $V$ is defined by $$ V^{\chi}=\{v \in V: ...
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52 views

How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : ...
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37 views

Monoid Ring of a Commutative Cancellative Ordered Monoid

Suppose $M$ is a commutative cancellative monoid with $0$ as the identity and $+$ as the operation, and $M$ is equipped with an order $\preceq$ defined by $$ m \preceq m' \text{ if and only if there ...
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38 views

What does “The Hilbert space carries a representation of […] group” means?

Often, in quantum mechanics I found the sentence "The Hilbert space carries a representation of $SU(2)$ group" (in particular when dealing with anglar momenta). Effectively, I know that this means ...
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31 views

Characterisations of the RSK correspondence

I know of the following three definitions of the RSK correspondence: (i) Row insertion (or more generally, plactic insertion) (ii) Viennot's construction (iii) Fomin's growth diagrams However, all ...
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22 views

Irreducible Decompositions of the Space of Sections of an Equivariant Vector Bundle

Let $G$ be a compact semi-simple Lie group, and $X = G/H$ a homogeneous space of $G$. If $V$ is a vector bundle over $X$, then is it true that $\Gamma^\infty(V)$ always has a decomposition into finite ...
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48 views

Why can't you apply the right regular representation the “wrong” way?

Let $G = GL(2,\textbf{R})^+$ be the $2 \times 2$ invertible matrices with positive determinant and given some discontinuous subgroup $\Gamma$, let $V = L^2(\Gamma \backslash G, \chi)$, the space of ...
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102 views

Questions about affine Weyl group and extended affine Weyl group for SL2.

Let $G=SL_2$. Then the Weyl group is generated by $s_1$. On page 3 of the lecture notes, it is said that the affine Weyl group is generated by $s_0, s_1$. (1) The element $s_0s_1$ can be identified ...
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31 views

Steinberg representation 1

Let $G$ the group $GL(n,F)$ where F is a locally compact non Archimedean field, and $G^{0}$ the subgroup of $G$ consists of elements $g$ in $G$ such that $\det(g)$ in $\mathcal{O}_{F}^{\times}$, where ...
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Proof of dimension formula for sl(3,C) reps

this is my first question on the forums, so forgive me if I did something wrong. I have tried to find an answer in here as well as on the usual places on the internet and was unsuccessful. So here ...
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G-space decompositions preserved by equivariant maps?

Let $X,Y$ be topological $G$-spaces, with (left) $G$-invariant probability measures $\mu_X,\mu_Y$ respectively, and let $f:X \to Y$ be a surjective $G$-equivariant map preserving the measures, i.e. ...
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28 views

Baily Borel Compactification: choice of boundary

In Borel/Ji " compactifications of symmetric and locally symmetric spaces " the Baily Borel compactification of a locally symmetric space is defined as $$\Pi\backslash(X\coprod_{\bf{P}}X_{P,h})$$ ...
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106 views

Decomposition of representation of symmetric group

Let $V$, $\dim V=n-1$ be the standard representation of the symmetric group $S_n$ and let $V'= \langle x_1,x_2,\ldots,x_n \rangle$ be its natural representation. Then ( see. Fulton, Harris, 4.19) ...
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59 views

Each irreducible representation is a subrepresentation of induced one

I'm learning what irreducible representation is and need some examples. One of them is as follows: Let $G$ be any group and $H$ - it abelian subgroup. How to prove that each irreducible representation ...
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41 views

Irreducible representations and abeliean subgroups

There is a theorem in representation theory which is surprising to me: the dimension of irreducible (complex) representation of finite group is not greater that $(G:H)$ - an index of abelian subgroup ...
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25 views

Questions about real-valued measure on a vector space.

I am reading the lecture notes on representation theory. I have some difficulty in solving Exercise 1.8 on page 4. Let $K$ be a non-archimedian local field and $v : K \to Z \cup \{\infty \}$ a ...
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55 views

What are the “irreducible factors” in an algebra?

What are the irreducible factors in an algebra? In the book "Spin Geometry" by Lawson and Michelsohn, page 35, line 1, there occurs "take irreducible factors of...", but I don't know what irreducible ...
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37 views

Dual representation matrix “recycling”

Imagine we have $V$, a finite dimensional vector space endowed with an inner product and its dual space $V^*$. We have also a matrix Lie algebra and a representation of it, $\pi$, that acts on $V$. ...
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43 views

Intertwiner for $U(n-1) \subset U(n)$

I'm using the notation of Vilenkin and Klimyk, ''Part3: Representations of Lie Groups and Special Functions''', chapter 18. Given an irreducible representation $T_m$ of the complex Lie algebra $U(n)$ ...
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18 views

The isomorphism of representive functions

$G$ is a compact Lie group with closed subgroup $H$ and $\mathscr{T}(G),\mathscr{T}(H)$ are the sets of their representative functions respectively (with real or complex representation). If the ...
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45 views

Isomorphism types of stabilizers of vectors in linear representations of the special linear group

Suppose we have a linear representation of the group $SL_d$ over $\mathbb{C}$. i.e. a finite dimensional vector space $V$ with a linear action of $SL_d$ on it. Let $v\in V$ be some vector and let ...
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Weight space for a finite-dimensional $\mathfrak{g}-$module $M$.

Let $\mathfrak{g}$ a semisimple Lie algebra, $M$ finite-dimensional $\mathfrak{g}-$module, $\mu\in\mathfrak{h}^*_{\mathbb{Z}}$ and $s_i$ simple reflection such that ...
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24 views

Finding an orthonormal basis for a gl(3) module

I'm trying to find an orthonormal basis for gl(3)-module V(ε1-ε3), where ε1-ε3 is the weight (1,0,-1) of the highest-weight vector. Using Gelfand-Tsetlin (/Zetlin/Zeitlin) patterns, I'm at the point ...
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90 views

Irreducible modules - semisimple algebras and endomorphism rings

Let $A$ be a finite dimensional, semi-simple $k$-algebra and $V$ and irreducible $A$-module. I am trying to prove the following claim: If $B = \text{End}_A(V^{\oplus r})$ then $$W = ...
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30 views

Exactness of Hom functor for torus representations?

Given a reductive algebraic group $G$ and a maximal torus $T$. Is it true that the functors $$ Hom_T(-,\lambda) $$ are exact, where $\lambda$ denotes one of the the simple one-dimensional ...
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24 views

A question about Cayley-Chow forms

I'm reading some papers about $k$-stable theory and I have a question about Cayley-Chow forms. Maybe this question looks silly. Let X be a variety of $\mathbb{P}^N$ with dimension n and degree d. ...
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45 views

Finite simple nonabelian groups with the same character table

Conjecture: If $G_1$ and $G_2$ are finite simple nonabelian groups, and if $G_1$ and $G_2$ have the same character table, then $G_1\cong G_2$. I am looking for a proof, or at least some intuition. Or ...