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

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action of $SU(2)$

Let $L=\mathbb C[u,v]$ (the $\mathbb C$-algebra of polynomials over two commuting variables $u,v$. For each non negative integer $n$ let $L_n$ be the linear subspace of homogeneous polynomials of ...
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69 views

Normal basis theorem

Let $K$ be a finite Galois extension of, say, $\mathbb{Q}$. Then is known(and called normal basis theorem) that if i view $K$ as a representation of $Gal(K/\mathbb{Q})$ over $\mathbb{Q}$ it is ...
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68 views

Some questions on Langlands Classification of Irreducible Admissible Representation

I am trying to construct some representations using Langlands classification theorem. But I get confused and have some problems when constructing these representations..... i) In the classification ...
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324 views

Clifford algebra - Gamma matrices

Let's say we have $\gamma^{a}$ matrices $(a=1,2,...,D)$. They satisfy the following condition $$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\delta^{ab}I^{N\times N}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ ...
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26 views

Finding expansions for elements of a group algebra

Suppose $G$ is a finite group of order $\left|G\right|$, with an associated group algebra $K[G] = \left\{\sum_{g \in G} a_g g\right\}$ over, say, the complex numbers. Suppose we represent $G$, and by ...
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Visualisation of representations and their decomposition into irreps

A question in a Representation Theory midterm got me thinking, and made me realise I didn't really understand irreps. The question was on the subject of reps of $S_4$, and went: An obvious ...
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Why do Ad(K) orbits in the $-1$ eigenspace of a Cartan decomposition intersect the Weyl chamber once?

Let $G$ be a semisimple Lie group and let $\frak p\oplus t$ be a Cartan decomposition of $\frak g$ and $K$ the connected subgroup with Lie algebra $\frak t$. Choose a maximal abelian subalgebra ...
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68 views

Let $V=V_{1}⊕ ⋯ ⊕ V_{n}$ be semisimple. $U$ irreducible. Show that $\dim_{k} (Hom_A(U,V)) $ is equal to the number of $V_i$ equivalent to $U$.

$\DeclareMathOperator{\End}{End} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\Irr}{Irr}$ Definition. An $A$-module $V$ is ...
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89 views

Remark 3.1.3 from Introduction to Representation Theory from Pavel Etingof

$\DeclareMathOperator{Hom}{Hom}$I'm trying to prove the following proposition (remark 3.1.3 from Introduction to Representation Theory from Pavel Etingof). Proposition. Any semisimple representation ...
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75 views

Matrix representation and permutation matrices

In order to find the matrix representation associated to a permutation representation I identify each permutation with the corrisponding matrix representation. How can I prove that these matrices ...
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37 views

How does Fulton and Harris establish that the differential of a group hom respects ad?

Fulton and Harris, Representation Theory, Section 8.1 (pages 104 - 107 in my copy) is concerned with showing that group homomorphisms $\rho : G \to H$, where $G$ is connected, are completely ...
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Is $B(w_1 w_2)B = (Bw_1B)(Bw_2B)$?

Let $B$ be a Borel subgroup of $GL_n$ and $W$ the Weyl group of $GL_n$. Let $w_1, w_2 \in W$. Is $B(w_1 w_2)B = (Bw_1B)(Bw_2B)$? If this is true. How to prove it? Thank you very much. Edit: If $l(w_1 ...
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44 views

What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$?

What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$? Here $k$ is a non-archimedean local field. Thank you very much.
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What does it mean for a representation to contain a character?

I'm trying to understand the statement "The representation $\pi$ contains the trivial character of $N$ if and only if it contains an irreducible representation $\sigma$ of $B$ containing the trivial ...
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Why do we have $\{s\} = \pi\{t\}$ for some $\pi \in C_t$?

I am currently working on Bruce Sagan's The Symmetric Group. In the proof of Corollary 2.4.2, the book says Let $t = t^\lambda$ be a $\lambda$-tableau and $s = s^{\mu}$ be a $\mu$-tableau, where ...
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Angles between adjacent roots in a reduced root system.

Let $R$ be a reduced root system. ($R$ is a finite set spanning $V$, $\alpha \in R \rightarrow -k\alpha \in R$ iff $k=1$, $s_{\alpha}(R)=R$, $s_{\alpha}(\beta)-\beta=k\alpha$ whit $k$ integer). ...
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66 views

Dimension of the space of tensors obtained by making partial symmetrizations and skew-symmetrizations.

Let $A=(a_{i_1\dots i_k})_{i_1,\dots,i_k=1}^n$ be a higher order cubic tensor or hypermatrix. The following two facts are well-known and are easy to prove: ${(\bf 1) }$ The dimension of the ...
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36 views

$\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules

Let $p$ be prime. Let $\mathbb{Z}_p$ be the $p$-adic integers. I'm intersted in $\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules of low rank over $\mathbb{Z}_p$ (rank $2$ is already very ...
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A question on universal Coxeter group

In the set up of Lusztig's Hecke algebra with unequal parameters, let $W$ be a universal Coxeter group with finite many simple reflections, that is, $W=\langle s_i,i=1,2,\cdots,n | s_i^2 =1\rangle$. ...
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30 views

About the induced character of the principal character

In page 186 of Carter's Finite groups of Lie Type: Conjugacy Classes and Complex Characters, the induced representation is defined for a representation $\sigma$ of a group $H$ to a representation ...
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95 views

Is every unitary representation a direct sum of irreducible subprepresentations?

I've read that any unitary representation of a compact group decomposes as a Hilbert space direct sum of irreducible representations. In the book I'm reading this is stated as a prong of the ...
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51 views

*-homomorphism and *-representation

I understand the concept of the unitary representation of G . A unitary representation of G is group homomorphism π:G→U(H) where H is a complex Hilbert space and U(H) is the group of unitary operators ...
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50 views

Divided powers in the context of elements of the Schur algebra

I am currently reading through the paper Presenting Schur algebras as quotients of the universal enveloping algebra of $\mathfrak{gl}_2$. Here it defines the following matrices $e := ...
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58 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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62 views

Relationship between O(n)- and SO(n)-representations?

Write $O(n)$ and $SO(n)$ for the orthogonal and special orthogonal group of degree $n$ over the real numbers. Suppose that $V$ and $W$ are real, finite-dimensional and orthogonal ...
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30 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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77 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 $S_n$ ...
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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 ...
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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 ...
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100 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 $Z_2^4$, ...
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55 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 ...
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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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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 $\pi(h)v=h v$ ...
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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 ...
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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 = ...
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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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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].
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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 ...
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60 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 ...
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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 ...
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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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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 ...
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60 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 ...
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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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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 ...
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262 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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164 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 ...
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54 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 $xy(x^{10}+14x^5y^5-y^{10})=0$. ...
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192 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 ...
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151 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 $U_j$ ...