Questions tagged [representation-theory]

Representation theory is a broad field that studies the symmetries of mathematical objects. A representation of an object is a way to "linearize" that object as a group of matrices. It's the non-commutative analog of classical Fourier transforms.

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End, $\mathcal{B}$, $\mathcal{L}$. Notation for endomorphisms

I am writing some notes and would like to ask if dealing with algebra $*$-representations, Hilbert spaces, etc. all finite dimensional, I should use $\mathrm{End}$ or $\mathcal{B}$ or $\mathcal{L}$ ...
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Hall-Littlewood polynomials and elementary symmetric functions— Chapter III (2.8) in Macdonald's “Symmetric Functions and Hall Polynomials”

I'm confused about the proof of Chapter III (2.8), page 209 in Macdonald's book, see proof of (2.8). Here is the background. Let $\Lambda_r$ be the ring of symmetric polynomials in $r$ variables, i.e. ...
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Central cocharacter

Let $G$ be a connected reductive group over a field $K$ and let $T$ be a maximal torus and let $B$ be a Borel subgroup containing $T$. Assume that $G$ is split over $K$ with respect to $T$. Let $\...
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Representations of noncompact groups

Let $k$ be a characteristic $0$ field. An object in $k[G]$-mod is determined by its character for any compact hausdorff group $G$. What are the main tools to approach locally compact groups- the ...
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Do there exist any algebras in which we cannot take tensor product?

Let's suppose that we have an algebra $\mathcal{A}$ (I don't really care whether it's a unital one or associative). From studying Lie algebras and some of their generalizations, I am used to be able ...
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Irreducible unitary representation of compact group is contained in the left-regular representation

Given a compact metric topological group $G$ and an irreducible unitary representation $\pi$ of $G$, I would like to show that $\pi$ is contained in the left-regular representation $\lambda$ of $G$. ...
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Let A and B be Boolean algebras and f : A β†’ B is an isomorphism,and A is atomic.

Let A and B be Boolean algebras and f: A β†’ B is an isomorphism, and A is atomic. (a) B is atomic, (b) B is complete (c) A is atomless (d) B is atomless I think the answer is (A) by the representation ...
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Is the exterior square of a projective faithful representation faithful?

Let $G$ be a matrix group. (I am interested in $G=O_n$ in particular). Then we can define the projective version $PG$ of $G$ to be the group consisting of the tensor products $$PG=\{A\otimes A\mid A\...
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Each Weyl group orbit in the character lattice of $V$ contains exactly one dominant weight

Let $V = \mathbb{C}^3 \otimes \mathbb{C}^3$ be a representation of $G = SL_3(\mathbb{C})$. The weights of this representation is the set of $\varepsilon_i + \varepsilon_j$ for $i, j = 1, 2, 3$, where $...
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Unipotent upper triangular matrices with integer entries is Zariski dense

Let $N$ be the group of matrices $\begin{bmatrix} 1 & z \\ 0 & 1 \end{bmatrix}$ for $z \in \mathbb{C}$, let $\Gamma$ be the subgroup of $N$ with $z \in \mathbb{Z}$. I wish to show that $\Gamma$...
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$Ind_H^G(\pi\oplus \mu) \simeq Ind_H^G(\pi)\oplus Ind_H^G(\mu)$

Let $\pi$ and $\mu$ be representations of a subgorup $H\leq G$. ($G$ a finite group.) My question is whether or not it is true that $$ Ind_H^G(\pi\oplus \mu) \simeq Ind_H^G(\pi)\oplus Ind_H^G(\mu) $$ ...
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Definition of an algebra over a field

My professor defined an algebra over a field as a vector space over that field that is also a ring. He did not state however with regard to what operations is the algebra a ring so can I assume the ...
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The lowest weight of a representation is $w_0 \lambda$

There is the well-known fact that if $\lambda$ is the highest weight of $V$ a finite-dimensional irreducible $\mathfrak{g}$-module, then $w_0 \lambda$ is the lowest weight (here $\mathfrak{g}$ is any ...
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$\Phi$-extreme weights and the Weyl group orbit of the highest weight

$\newcommand{\g}{\mathfrak{g}}$ Let $P(\g)$ be the weight lattice of $\g$ a semisimple Lie algebra over $\mathbb{C}$, and $P_{++}(\g)$ the set of dominant integral weights. A subset $\Psi \subset P(\g)...
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Finding the set of weights for a representation of a Lie group

Let $V = \mathbb{C}^3 \otimes \mathbb{C}^3$ be a representation of $G = SL_3(\mathbb{C})$. I want to find the weights of this representation in terms of the functionals $\varepsilon_i$ for $i = 1, 2, ...
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Reference request: homomorphisms of $\widehat{\mathfrak{sl}_2(\mathbb C)}$-modules

For a Lie algebra $\mathfrak g$ one can construct $\hat{\mathfrak g}$-modules at level $k$ by taking induced representations from $\mathfrak g$-modules, and then quotienting by the maximal submodule. ...
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2-dimensional projective unitary representation of SO(3)

I know that we can construct a continuous irreducible projective unitary representation $\pi$ with the 2-dimensional continuous unitary irreducible representation $u$ of $\mathrm{SU}(2)$. What is the ...
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Is there a known representation for this set derived from a Lie group $\subseteq \text{SU}(n)$?

Let $G \subseteq \text{SU}(n)$ be a Lie group for $n \in \mathbb{N}$, with a proper Lie subgroup $H \subset G$ having Lie algebra $\mathfrak{h}$ such that $H = e^{\mathfrak{h}}$. Let $g_0 \in G$ be an ...
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Module Over the Group Algebra of a Ring

I'm aware that if $K$ is a field, $V$ a $K$-vector space, and $G$ is a group, then a $K$-representation of $G$ in $V$ is the same thing as a $KG$-module, where $KG$ is the group algebra of $G$ over $K$...
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homomorphism of $G/N$-modules

Let $G$ be a group with $N$ a normal subgroup (not necessarily of finite index). Let $Q$ and $A$ be $G$-modules and $P$ be a $G/N$-module. I want to make sense of the term ${\mathrm{Hom}}_{G/N}(P, {\...
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2-dimensional representation of modular group

My question probably follows from something basic in character theory, but I don't see what I'm missing. Let $\rho: \Gamma \to \mathrm{GL}(2,\mathbb C)$ be a two-dimensional representation of the ...
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Normal subgroup and non-equivalent representations

I am looking for an example to illustrate that to representations are not equivalent in a particular situation. If H is a normal subgroup of G, $\pi$ a representation of $H$, and the second ...
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Two definitions of representation of a topological group

Please compare the following two definitions: Definition 1 A representation of a topological group $G$ in a vector space ${\mathbb{V}}$ over complex numbers is a continuous map $$ A\, :\quad G\times ...
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Show that the trivial representation of G occurs exactly once in $\pi_X$

I am looking at the following problem in relation to representation theory: Let $G$ be a finite group acting on a set $X$ (we denote this action by $(g,x) \mapsto g\cdot x$). We assume that $X$ has ...
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Equivalent representations

I have a few questions in relation to the definition of equivalent representations. I am looking at the following problem: Suppose $\pi : G \to GL(V)$ is a representation . For $g \in G$ define $\pi^g ...
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Existence of a one-dimensional representation

I am working on the following exercise in relation to representation theory and I have some questions. Let $G$ be a group with center $Z$. Let $\pi$ be an irreducible complex representation of G on a ...
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Representations that are below the dimension of the adjoint representations of $SU(N)$

Given $SU(N)$ group, how many kinds of representations (say, labeled by dimensions of representations or Young tableux) do we have such that they are below the dimension of the adjoint representations ...
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Can every binary functional equation be implemented algebraically in a pure way?

Consider a generic binary operation $B$. It is possible for $B$ to obey "set-independent" functional equations, that is functional equation that do not make any reference to an underlying ...
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How to prove the rep of SU(2) on homogeneous polynomials in 2 variables is irreducible?

The group $\mathrm{SU}(2)$ has a tautologous representation on the space $\mathbb{C}^2$ and thus a representation on the $d$th symmetric power $S^d (\mathbb{C}^2)$. What's the easiest way to prove ...
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Induced representations built on sections of an associated vector bundle. Questions on notations

Consider a group $\,G\,$, a vector space $\,{\mathbb{V}}\,$, and a space $\,{\cal{L}}^G\,$ of functions $\varphi$ on this group: $$ {\cal{L}}^G\;=\;\left\{~\varphi~\Big{|}~~~\varphi:\,~G\...
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Relations between real and complex irreducible representations of a finite group with the same Frobenius-Schur indicator value

I'm trying to learn some representation theory of finite groups and am struggling with the relation between complex and real irreducible representations (irrep). By a complex (resp. real) irrep of a ...
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Gauss sums in positive characteristic [closed]

Looking for references on Gauss sums when the multiplicative and additive character take their values in a finite field instead of the complex numbers. Such sums occurred in some recent constructions ...
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Sum of squares of the dimension of irreducible characters and conjugacy classes

While working through a prior problem on classifying a group of order $625$, I stumbled upon (by virtue of a mistaken answer by a user), the following problem. Since the dimension of irreducible ...
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What is the decomposition of $𝕂[G/H]$ in terms of irreducible representations?

Let $G$ be a finite group and $𝕂$ be a field. Any $G$-set can be linearized to give a $𝕂$-representation of $G$. Each $G$-set is decomposed into a coproduct of indecomposable (transitive) $G$-sets $...
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Proof that $\operatorname{Ext}^n_{kG}(M,N) = \operatorname{Hom}(\Omega^nM, N)$

Let $G$ be a finite group and let $k$ be a field of modular characteristic. Write $\mathsf{St}_k(G)$ for the stable module category of $G$. It is the triangulated category obtained from $\mathsf{Mod}...
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Example for a faithful and finite dimensional representation over a sovable and finite dimensional Lie-Algebra

I am new to StackExchange. I am learning about Lie-Algebras and I was wondering whether somebody can give me an example for a finite dimensional and faithful representation of a sovable and finite ...
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Trace orthogonality of representations of Lie group generators

Typically, given a (simple) Lie group $G$ we choose a basis for the associated Lie algebra $\mathfrak{g}$ so that in the fundamental, or defining, representation, the basis is trace-orthogonal, $$ \...
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presentation of a generic algebra

In the book $\ulcorner$Reflection Groups and Coxeter Groups$\lrcorner$ written by J. E. Humphreys, in the beginning of chapter 7 $<$Hecke algebras and Kazhdan-Lusztig polynomials$>$, it defines ...
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Bilinear form respect to the representation $V/W$ of Lie algebra

I got a problem from the book An Introduction to Lie Groups and Lie Algebras written by Kirillov. The exercise 5.1 says $$\begin{array}{l} \text { (1) Let } V \text { be a representation of } \...
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Role of wavelength in the representations of the translation group

In the representation theory of the translation group we have $U^p(x)=e^{ipx}$ where $p=2\pi/\lambda$. I know in quantum mechanics this ends up being momentum somehow. And we can also see that $p$ is ...
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Real characters of odd degree

Suppose that $G$ is a finite group and $\chi$ is an irreducible real character, namely that $\chi(g) \in \mathbb{R}$ for every $g \in G$. Is it true that if $\chi(1)$ is an odd number, then $\chi$ is ...
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Character triples isomorphism and real characters

A character triple is a triple of the form $(G,N,\theta)$ where $G$ is a finite group, $N$ is normal in $G$, $\theta \in Irr(N)$ and $\theta$ is $G$-invariant. For the concept of character triple ...
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Roots of the same length differ by a permutation by the Weyl group

I am to show that for $ G \subset GL(n,\mathbb{C})$ a classical group, $(\cdot, \cdot)$ the usual inner product on $\Phi$ the root system of $G$, and $(\alpha, \alpha) = (\beta, \beta)$, there exists $...
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Function that takes Wigner D matrix and complex vector and maps to real valued rotated vector

I am looking for a function $f$, that takes a the product of a Wigner D matrix $D_{l=1}(\boldsymbol{R})$ and a complex vector $\boldsymbol{c} \in \mathbb{C}^3$ as input, and outputs a rotated real ...
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Harmonic analysis and characters

In the following video, at the linked time, https://youtu.be/HhTGyDuNI_w?t=762 there is an argument presented for characters and the discrete fourier transform. Is there some book where I can find ...
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Finding change of basis with only the knowledge of a set of homomorphisms

I came across two different sets of matrices of the same group representation. Naturally, I wanted to see how they relate, i.e. find the explicit change of basis. All the matrices have the same ...
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Why are G modules using same notations as group actions?

I may be missing something important , but i don't understand why group action and G modules are using the same notation , even in some texts i feel like some authors call "action" a G ...
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Definitions of the quadratic Casimir of $SU(N)$

Particle physics, especially QCD, deals a lot with $SU(N)$ and therefore also with $\mathfrak{su}(N)$. In the QCD literature it is normal to define the (quadratic) Casimir element in a representation ...
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A simple equality related to an irreducible representation of a finite group

Let $G$ be a finite group of cardinal $|G|$. Let $u \colon G \to M_n$ be an irreducible unitary representation of $G$. How to show that for any matrix $A \in M_n$ we have $$ \frac{1}{|G|} \sum_{g \in ...
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Number of inequivalent one-dimensional representations of a group of order $p^n$

This is from an old comprehensive exam: Prove that any group with $p^n$ elements has a nontrivial center and use this to prove that any group with $p^n$ elements has at least $p$ inequivalent one-...

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