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

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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 ...
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392 views

Finding a matrix representation for two Grassmann numbers.

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
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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 ...
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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 $n$, ...
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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) ...
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+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 $\...
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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 ...
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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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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 $n(n - 1)/2$ ...
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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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Compatibility of Yetter-Drinfeld modules.

Let $H$ be a Hopf algebra. A Yetter-Drinfeld module over $H$ is a triple $(V, \cdot, \delta)$, where $\cdot : H \otimes V \to V$ , $\delta : V \to H \otimes V$ are actions and coactions respectively, $...
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111 views

Representing natural numbers as matrices by use of $\otimes$

What I am wanting to do is to find a unique matrix representations for Natural numbers. Say I have the number $n$, how can I represent this number as a matrix in which I can do matrix multiplication ...
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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 \...
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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 ...
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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\...
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If $a\in IBr(G/N)$, then $a\in IBr(G)$? [on hold]

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

$d\pi(X)$ is skew-symmetric. What does it mean?

This is from a lemma in Lang $SL_2$ If $\pi$ is a unitary representation of G, and $X \in \mathfrak g$, then $d\pi(X)$ is skew symmetric on $H_\pi^\infty$ What does skew symmetric mean here? And ...
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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 ...
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Prove $V$ is simple $\iff$ all non-zero vectors are cyclic

I am working on some Representation Theory practice questions and I think I have given a valid proof of : Prove $V \ne 0$ is a simple A-Module$\iff$ all non-zero vectors are cyclic $"\leftarrow"$ ...
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Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...
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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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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 ...
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An example of a discontinuous “$\ell$-adic Galois representation”

Let $\mathbb{F}_p$ be a finite filed with $p$ elements, and $G=\mathop{\mathrm{Gal}(\mathbb{F}_p^s/\mathbb{F}_p)}$ be its absolute Galois group. $G$ is a pro-finite group, with the Krull topology, see ...
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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 ...
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Reference needed for Determinant of convex combination of two matrices as a function [on hold]

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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323 views

Conceptual description of the isotypical component

This is probably rather simple but I have not found it in the literature. Consider the category $C$ of representations of a finite group $G,$ over a field $k$ of characteristic not dividing the order ...
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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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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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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?
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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 ...
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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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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{...
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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.
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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 ...
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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 ...
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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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Haar Measure of a Topological Ring

A topological ring is a (not necessarily unital) ring $(R,+,\cdot)$ equipped with a topology $\mathcal{T}$ such that, with respect to $\mathcal{T}$, both $(R,+)$ is a topological group and $\cdot:R\...
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166 views

Representation Theory Symmetric Group Book?

I'm looking for a nice book that discusses the representation theory of the symmetric group. My background is an introductory class in representation theory.
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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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Finite dimensional representations of the Weyl algebra in characteristic $p>0$

I'm working through representation theory course notes of P. Etingof. In problem 1.26 it is asked to find all finite dimensional irreducible representations of the algebra $A=\frac{k[x,y]}{\left\...
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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 ...
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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 ...
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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?
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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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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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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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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 ...