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

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2
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2answers
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(Double) Coset of $GL(n, q^2)/GL(n, q)$

I am trying to understand a particular coset/double coset of the finite group $G = GL(n, q^2) = GL_n(\mathbb{F}_{q^2})$. It has a natural subgroup $H = GL(n, q)$, which can also be viewed in the ...
0
votes
0answers
39 views

A kind of permutations and possible relation to cyclic groups.

Any permutation that moves $n$ elements in some fashion never revisiting the same until all others have been visited, in other words so that: $${\bf P}^n = {\bf I}, \text{ but no } 0<m<n \text{ ...
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0answers
11 views

Cluster algebra of finite type

It is proved in the paper that a cluster algebra is of finite type if its Cartan counter part of the principal part of its seeds is a Cartan matrix of finite type. If the initial quiver of a cluster ...
2
votes
0answers
21 views

What is number of irreducible characters in modular representation?

In ordinary representation, I know that the number of irreducible characters is the number of conjugacy classes, what about the modular representation? Can I find a good and simple book on modular ...
0
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1answer
29 views

Linear representations of projective groups

Does the projective linear group $PSL_2(\mathbb{R})$ admit faithful linear representations? In other words, does there there exist a homomorphism $SL_2(\mathbb{R}) \to GL_n(\mathbb{R}),$ for some $n$, ...
0
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1answer
18 views

Table of e8 representations

I want to understand the representation theory for the (complex-valued) $e8$ exceptional Lie algebra. An ideal answer to this question would contain a link to a text file (or any other format) ...
0
votes
1answer
19 views

Tensor product of representations of a Lie algebra (or Lie Superalgebra)

Let $V$ and $W$ be finite dimensional irreducible representations of a Lie Algebra or a Lie Superalgebra. If $V$ is one dimensional, is $V\otimes W$ necessarily irreducible? I know this to be true ...
4
votes
2answers
77 views

Trace identities for $\text{SO}(n)$

The Green-Schwarz mechanism in Type I string theory involves certain identities relating traces in the vector and adjoint representations of $\text{SO}(n)$ of dimension $n$ and $n(n - 1)/2$ ...
2
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0answers
54 views

Is the assignment of a root system to a semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
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0answers
30 views

Commutators in the context of local Lie groups.

Let $G$ be a local Lie group in the neighbourhood $V \subseteq \mathbb{C}^d$ with identity element denoted by $e \in G$. Also, let $$ t \mapsto f(t) = (f_1(t), \dots, f_d(t)) \quad \forall t \in \...
2
votes
1answer
30 views

What is the simplest example of the tame representation type?

What is the simplest example of the tame representation type? I tried to find simple example could help me to understand the tame representation type. I know the definition of tame is like: A ...
0
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0answers
14 views

If $a\in IBr(G/N)$, then $a\in IBr(G)$? [closed]

If $a\in Irr(G/N)$, then $a\in Irr(G)$. How about replacing Irr by IBr?
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0answers
19 views

How to compute the number of modular/Brauer characters in a p-blocks of a finite groups, for example $A_5$ or $S_3$?

I do not know how to compute the modular character in a p-block of finite group.I want to know some skills for computing the number of modular characters in a p-block of a finite or some material ...
0
votes
0answers
44 views

Attempt to represent gaussian integers with matrices over ${\mathbb Z_+}^{4\times4}$

Let us first consider the generating element for $C_2$ : $$M_1 = \left[\begin{array}{cc}0&1\\1&0\end{array}\right], \text{ and } P_1 = ({M_1})^2 = I_2 = \left[\begin{array}{cc}1&0\\0&1\...
2
votes
1answer
30 views

Irreducible representations of the fundamental group of a closed surface in $SU(2)$

For a compact Lie group $G$, consider the map $f : G^{2n} \to G$ given by $f(A_1, B_1, \ldots, A_n, B_n) = \displaystyle\prod_{i = 1}^{n} A_i B_i A_i^{-1} B_i^{-1}$ A theorem of Goldman (from the '...
4
votes
1answer
37 views

If $H<G$ is abelian, and $\chi(1)=[G:H]$ for irreducible $\chi$, then $H$ contains a nontrivial normal subgroup?

Suppose $\chi$ is an irreducible character of a finite group $G$, and $H$ is a nontrivial abelian subgroup such that $\chi(1)=[G:H]$. Why does $H$ contain a nontrivial normal subgroup? I understand ...
3
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52 views
+50

Decomposition of An Induced Representation of $GL(2, q)$

Denote $G = GL(2, q) = GL_2(\mathbb{F}_q)$, $B$ its Borel subgroup of upper triangular matrices, $T$ its splitting torus of diagonal matrices. The object I am interested in is $Ind_B^G\rho$, where $\...
0
votes
1answer
16 views

How could I check the closedness under multiplication of the ring of symmetric functions?

Let $\Lambda$ be the ring of symmetric functions, which is defined as the subspace of the power series ring over $\mathbb{C}$ generated by monomial symmetric functions. Now, the monomial symmetric ...
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0answers
29 views

Representation of $A_5$

Can someone give me a proper reference (a book probably)for how a 3 dimensional representation of the Alternating group $A_5$ is related to the reflection group $H_3$ or the Icosahedral group ? Thanks
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votes
0answers
109 views

Reference needed for Determinant of convex combination of two matrices as a function [closed]

What can one say about the function $(t,A,B) \mapsto \det(tA + (1-t)B)$, with $t \in [0,1]$, $A$, $B$ square matrices, in my case, say, permutational matrices? Where such a function shows up? Hoping ...
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0answers
64 views

Growth of the characters of finite permutation groups in the number of symbols

I have the following questions. When can the characters of the irreducible representations of the elements of a finite permutation group increase exponentially in the number of the symbols the ...
0
votes
1answer
51 views

why the algebra $A = k[x]/(x^n)$ has finite representation type? [closed]

Suppose that $k$ is algebraically closed. Then why the algebra $A = k[x]/(x^n)$ has finite representation type? please clarify the answer.
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0answers
32 views

does algebra over algebraically closed field has isomorphism classes of irreducible modules?

Let $F$ be an algebraically closed field and $M$ be an F-algebra. Which are the conditions that make $M$ has finitely many isomorphism classes of irreducible $M$-modules?
2
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0answers
26 views

Finite Dimensional Representation of Lie Algebra.

Let $V, W, U$ be finite dimensional representations of a lie algebra $\mathfrak{g}$. Show that $\hom(V \otimes W, U) \cong \hom (V, U \otimes W^*)$. I think I have to use the enveloping algebra of ...
0
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0answers
21 views

Affine and linear reflections

Let $\gamma$ - affine reflection in complex space, which is transformation with properties: (1) $\gamma$ is a motion (thus linear part of $\gamma$ : $\mathbf{Lin} \gamma \in U(V)$), (2) $\gamma$ ...
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votes
1answer
33 views

Tensoring over the group ring versus tensoring over the ring in view of group representations.

I was reading a chapters homology with local coefficients. Where one of the preliminary sections asks us to compute $$\mathbb{Z}_{+} \otimes_{\mathbb{Z}[\mathbb{Z}/2]}\mathbb{Z}_{-}$$ Here $\mathbb{...
1
vote
1answer
28 views

Schur Multipliers in Finite Simple Groups

I heard that Schur multiplier's played important role in classification of finite simple groups. By means of simple example, can one illustrate how the Schur multiplies played their role in the ...
2
votes
0answers
37 views

Singular Locus of a Schubert variety

I am trying to compute the singular locus of the schubert variety $X_w$ in $G_{2,7}$ where $w=(4,7) \in I_{2,7}$. Following the notation in the book "The Grassmannian Variety: Geometric and ...
2
votes
1answer
34 views

irreducible unitary reflection group

Let $G$ be a finite irreducible unitary reflection group (i.e. without G-invariant subspaces). Given orthonomal basis, we have that $g_1 \in GL(V)$ commutes with every element of $G$. It is said that ...
0
votes
1answer
26 views

About Structure of Free Algebra over $K$

In MIT Course No. $18.712$, Associative Algebra $A$ is defined as a vector space over a field $K$ with a bilinear associative map $A \times A \to A$, $(a,b) \to ab$. Then some examples are given, ...
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2answers
30 views

Is the tensor product of two Yetter-Drinfeld modules a Yetter-Drinfeld module?

Let $U,V$ be two Yetter-Drinfeld modules over a bialgebra $H$. Is $U \otimes V$ a Yetter-Drinfeld modules over $H$? Thank you very much.
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0answers
20 views

When $H$ is a Yetter-Drinfeld module over itself? [closed]

Let $H$ be a bialgebra. When $H$ is a Yetter-Drinfeld module over itself? Thank you very much.
0
votes
0answers
31 views

Name for quiver representation

Let $Q = (Q_0, Q_1)$ be a quiver, and pick some $i \in Q_0$. Define the quiver representation $M$ by $$M_j = \begin{cases} k & \text{ if there is a path from $i$ to $j$,} \\ 0 & \text{ ...
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0answers
22 views

Representation of the group of rotations on the space of spherical functions

On a project on how Representation theory can help improve the complexity of shape matching, I couldn't understand this result : If $V$ is the space of spherical functions, consider the ...
4
votes
0answers
145 views

Restriction of irreducible unitary representation to normal subgroup of finite index [migrated]

Let $G$ be a Lie group (or more generally a locally compact group), let $N$ be a closed and normal subgroup of $G$ of finite index. Let $H$ be an infinite dimensional complex Hilbert space, and let $\...
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0answers
21 views

Irreducible representation of $1$-transposition groups

I would like to know the theory of irreducible representation of $1$-transposition groups. Could anyone provide me a pointer from where I can proceed?
1
vote
3answers
54 views

Faithful representation of the Heisenberg group

I have been trying to solve a problem concerning the Heisenberg Lie group $H$. Show that there does not exist a faithful representation $\rho:H\to\text{GL}(2,\mathbb{R})$. Any ideas about how to ...
1
vote
0answers
29 views

Definition of $k$-transposition group

In A monster tale: a review on Borcherds’ proof of monstrous moonshine conjecture, a $k$-transposition group is defined as follows. Recall that a $k$-transposition group $G$ is one generated by a ...
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0answers
43 views

Smoothness of Schubert Variety

Consider the Schubert variety $X(s_3s_2s_1s_4s_3s_2)$ in $SL_5/P_2$, where $P_2$ is the maximal parabolic corresponding to the simple root $\alpha_2$. In one line notation this permutation can be ...
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0answers
20 views

A $*$-closed algebra of compact operators is completely reducible

In page 13 of Lang's $SL_2$ there is a proof that for a $*$-closed algebra $\mathscr A$ of compact operators on a Hilbert space $H$, $H$ is completely reducible. The proof follows by taking the ...
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vote
0answers
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Is $\pi^1:C_c(G)\rightarrow \operatorname{End}(H)$ a homomorphism of the convolution algebra when $G$ is not unimodular?

Let $G$ be a Hausdorff locally compact group and $H$ a Banach space. Let $\pi:G\rightarrow \operatorname{GL}(H)$ be a representation and define $$\pi^1(\phi)v = \int_G\phi(x)\pi(x)vdx$$ for $v\in H$ ...
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0answers
23 views

Special case of Pieri-Rule

is there an "elementary" (read: short combinatorial) proof for the rule $$ s_\lambda \cdot s_{(1)} = \sum_{\mu} s_{\mu} $$ where $\mu$ ranges over all partitions obtained from $\lambda$ by adding a ...
0
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0answers
22 views

representations of SL_n(R) or SL_n(C)

Could someone point me to a reference for the finite dimensional representation theory of $G=SL_n(F)$ where $F = \mathbb{R}$ or $\mathbb{C}$? In particular, I want to know what this "highest weight" ...
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0answers
13 views

Irreducible representations of locally compact semigroups

What class of semigroups have finite dimensional irreducible representations. For example, If we have a compact groups $G$ then every continuous irreducible representation of $G$ is finite ...
2
votes
1answer
36 views

Is $\operatorname{Stab}(\lambda)$ generated by the simple reflections it contains, for $\lambda\in A_0$?

For a finite Weyl group, the stabilizer of an element in the fundamental domain is generated by the simple reflections of the Weyl group that is contains. Does the same still hold for the closure of ...
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0answers
40 views

Reference Request: Lie Theory For Quantum Field Theory

I have encountered the section on non-Abelian gauge theories in Peskin and Schroeder's QFT textbook, and although I am comfortable with the derivation of the Yang-Mills Lagrangian they present, the ...
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0answers
23 views

(B,N) pair and Steinberg idempotent

Let $q=p^f$ where $p$ is prime and $G$ be a finite group with a $(B,N)−$pair ($T=B\cap N$ and $W=N/T$), and assume that $B=UT$ with $U\triangleleft B$ and $U\cap T=1$. Define $$e=\dfrac{1}{[G:U]}\...
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0answers
13 views

Tannaka Krein duality for finite groups, explicit

Tannaka-Krein duality theory says that the natural mapping $G\rightarrow Aut^{\otimes}(F)$ (see http://mathoverflow.net/questions/155743/can-one-explain-tannaka-krein-duality-for-a-finite-group-to-a-...
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votes
1answer
27 views

(B,N) pair and normal subgroup

I am trying to prove the following: Let $G$ be a finite group with a $(B,N)-$pair and assume that $B=UT$ with $U\triangleleft B$ and $U\cap T=1$. Let $\widetilde{G}\triangleleft G$ such that $U\le \...
2
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0answers
20 views

Simple groups and irreducible characters of degree 3

The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}_5$ and $PSL(2,7)$. That seems to be a result coming from Blichfelt's work on $GL(3,\mathbb{C})$, which I ...