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

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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 ...
14
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310 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 ...
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252 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 ...
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111 views

Involutions, RSK and Young Tableaux

Let $S_n$ be the symmetric group on $n$ elements. The Robinson-Schensted-Knuth (RSK) correspondence sends a permutation $\pi\in S_n$ to a pair of Standard Young Tableaux $(P,Q)$ with equal shapes ...
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291 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 ...
9
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117 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 ...
8
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90 views

Absolutely irreducible representations of the absolute Galois group of Q_p

Let $p$ be a prime number. Denote by $G$ the absolute Galois group of (a finite extension of) $\mathbb{Q}_p$. Let $\ell$ be a prime number. For $\ell= p$, I guess it is well known that the ...
8
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190 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. ...
8
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141 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 ...
8
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99 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
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192 views

Generating the partners in a multi-dimensional irreducible representation.

I am trying to block diagonalize a Hermitian matrix using the irreducible representations of its symmetry group. Using the group's character table, it is straightforward to generate a set of ...
7
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92 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
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153 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 ...
6
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91 views

A little bit of Intuition for Corepresentations from Representations

Hi folks I am trying to prove what I think should be a straightforward enough result but I am having to make a somewhat unnatural definition to do it. This unnatural definition is hinted at in a paper ...
6
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119 views

The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

Question What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group? The trivial corepresentation is given by $\Delta_{|W}$ where $W$ is just the one dimensional ...
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117 views

Dimension of the space of algebraic Riemann curvature tensors

Given $n\in \mathbb N$, consider the vector space $\mathbb R^{n^4}$ whose elements I will denote by $(R_{abcd})$ with indices $a,b,c,d \in \{1, \dots, n\}$. This vector space is $n^4$-dimensional. The ...
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110 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 ...
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76 views

Characters of the symmetric group corresponding to partitions into two parts

Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character ...
6
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225 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 physical ...
6
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172 views

Invariant element in the tensor product of rectangular Specht modules?

Denote by $\mathfrak{S}_k$ the symmetric group on $k$ elements. Let $\lambda=(n^2\times n)=(n^2,\ldots,n^2)$ be a rectangular partition and $k=n^3$. Denote by $S_\lambda$ the Specht module ...
6
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399 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 ...
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48 views

Making modular representation theory and cohomology 'compelling' and 'accesible'

I'm currently putting together an application for a dissertation completion fellowship offered through my university. A part of the application includes 500-1000 words describing my dissertation. ...
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82 views

— Cartan matrix for an exotic type of Lie algebra --

(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra? For example, consider this Lie algebra: $$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = - f_{ik}{}^j Y^k \qquad\qquad ...
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304 views

Simultaneously (generalized) diagonalizable matrices

I heard the following theorem from our textbook: Given $A,B$ are two commuting ($AB=BA$) real normal matrices. There's some real orthogonal matrix $P$ such that $P^{-1}AP$, $P^{-1}BP$ are ...
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Can one reformulate tensor methods and young tableaux to account for spinor representations on $\operatorname{SO}(n)$?

Standard tensor methods and Young tableaux methods don't give you the spinor reps of $\operatorname{SO}(n)$. Is this because spinor representation are projective representations? If so, where does ...
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81 views

Deciding whether or not a class of modules is “big enough”

For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
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61 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 ...
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143 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 $n+1$ ...
5
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86 views

Connection between ramification in number fields and Clifford theory

Consider algebraic number fields $\mathbb{Q} \subseteq K \subseteq L$ with rings of integers $\mathbb{Z}\subseteq \mathcal{O}_K \subseteq \mathcal{O}_L$. If $0 \neq \mathfrak{p} \trianglelefteq ...
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130 views

Isomorphism of representations of the symmetric group

This might be a silly question, but I don't understand why the solution to the following problem implies the result: Let $A = \mathbb{C}S_d$ and let $c_{\lambda}$ denote the Young symmetrizer (with ...
5
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111 views

Tensor product decomposion in Lie algebras

As a chemist, I do this all the time...for symmetry groups. Which are finite, luckily :-) For knot theory purposes, I'd like to have a complete list of Lie group irreps R with the property that the ...
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95 views

How to prove “The maps factoring through an injective object are precisely the null-homotopic maps”

Thanks for your attention, I'm an undergraduate. I'm reading the book of Dieter Happel, Triangulated categories in the representation theory of finite dimensional algebras, I cannot prove this ...
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62 views

Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...
4
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63 views

Correspondence of representation theory between $\mathrm{GL}_n(\mathbb C)$ and $\mathrm U_n(\mathbb C)$

If I know something about the representation theory of the general linear group $\mathrm{GL}_n(\mathbb C)$, what can I say about the representation theory of the unitary group $\mathrm U_n(\mathbb ...
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79 views

Making the definition of dual root unambiguous

In 5.4 of his book Lectures on Invariant Theory, Igor Dolgachev introduces the dual of a root by requiring that $\check\alpha(t) f_\alpha(x) \check\alpha^{-1}(t)= f_\alpha(x)$ ...
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55 views

Doubt about Proposition 2.39 in Dana Williams' crossed product book

You can see the proposition in a google books preview here. First and foremost, my question is: Question: Am I correct to interpret Proposition 2.39 as setting up a bijective correspondence ...
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78 views

on the simple group $M_{11}$

As we know, the simple group $M_{11}$ is a important group,it has order $7920$, how can we prove the simple group of order $7920$ is isomorphic to $M_{11}$ ?
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77 views

Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
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55 views

Verma modules and delta function

What is the relationship between Verma modules and delta functions? Here's the quote from Woit's notes on Lie theory (http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf): The ...
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140 views

How to write $SO(2n)$ characters in terms of rotation angles?

Say one is working in a representation of $SO(2n)$ such that it has the highest weights $(h_1,...,h_n)$. And let $\{H_i\}_{i=1}^{n}$ be a basis in the Cartan of $so(2n) = Lie(SO(2n))$. Now one says ...
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38 views

Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
4
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112 views

Mackey criterion for normal subgroups

I am wondering how Mackey's criterion works for arbitrary fields. If there is a representation $\vartheta$ of a subgroup of $G$, then the induced representation $\operatorname{Ind}(\vartheta)$ is ...
4
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114 views

Irreducible representation of $S_4$

Could one please point out an irreducible representation of degree 2 of the group $S_4$. Thank you.
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117 views

Representations of non-semisimple Lie algebras

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple. An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with ...
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Problem 4.3, I. Martin Isaacs' Character Theory

Let $G=H\times K$ be the direct product of finite groups. Let $\varphi\in Irr(H)$ and $\eta\in Irr(K)$ be faithful. Show that $\varphi\times\eta$ is faithful if and only if $(|Z(H)|,|Z(K)|)=1$. Here, ...
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Correspondence between summands of a module and primitive idempotents of the endomorphism ring of the module

I am reading this paper and have the following questions. Let $A$ be a finite-dimensional algebra over a fixed field $k$. Let the finitely-generated $A$-module $M$ be a generator–cogenerator for ...
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396 views

Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.

I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question. To start with we work with the $\mathbb{Q}$ version of ...
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116 views

Basic semi-invariants

Let $G$ be a (finite) group and $\chi$ be a linear character corresponding to an irreducible representation. A polynomial $f_{\chi}$ is called semi-invariant (of type $\chi$) if $\sigma\circ ...
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An algebraic algorithm for finding inverses in the group algebra

This is an extension to my earlier question. Is there a purely algebraic algorithm to find inverses in the group algebra? For example, in the group algebra $\mathbb{C}S_{4}$, how would one go about ...
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107 views

image of symmetric matrices under representation of $GL_2(\mathbb{R})$

Let $W$ be a real vector space of dimension $2$ and let $\rho_k:GL_2(\mathbb{R}) \to GL(\mathbf{S}^kW)$ be the standard representation of $GL_2(\mathbb{R})$. Since $\rho_k$ is polynomial, it naturally ...