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

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Functorial construction of the convolution algebra of measures on a group

Let $G$ be a lcoally compact group and $C_c(G) = \lim_{K \subset G} C(K)$ its space of continuous functions with compact support endowed with the topology of the limit of banach spaces $C(K)$ with $K$ ...
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26 views

Condition for appearance of singlet in product of two irreps.

By inspecting tables for tensor products of two finite-dimensional irreps of common Lie groups, I've noticed that a trivial subrepresentation only appears when the two irreps are conjugate of ...
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31 views

Linear represenation of a group(can be infinite also)

Let G be a group and let $\sigma :G \rightarrow GL(V) $ be a representation of G. Assume $\sigma$ is reducible. That is $\sigma=\sigma_1 \oplus \sigma_2\oplus .... \oplus \sigma_k $ or interms of G ...
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14 views

Primitive Decomposition in a finite-dimensional algebra

Can you give me please, an explicit example of a primitve decomposition in a finite-dimensional algebra different than the usuals (for instance the decomposition in Mn(K))? Also, if e is an ...
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1answer
32 views

How can we compute restrictions from a character table?

I would like to how to, when given a character table, calculate the restriction. $Res_H^G : Rep(G) \rightarrow Rep(H)$. For example: Let $G=S_4$ whose character table is given below (see ...
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14 views

Bounds for the sum of the representations of the group $S_k \times S_j$

Let $S(n)$ be the sum of the degrees of the irreducible rational representations of the symmetric group on $n$ letters. I know that this number is the same as the number of involutions in $S_n$. ...
4
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31 views

Structure of k[G]/J(k[G]) when char k divides |G|

I'm self-learning representation theory, so I'd like if possible someone kind to help me with the following. Given a field $k$ whose characteristic does not divide the order of a finite group $G$, we ...
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20 views

how to show that the representation of $SL(2, \mathbb{C})$ is holomorphic

Fix an integer $n\geq 0$, and let $V_n$ be the complex vector space of polynomials in two variables $z_1$ and $z_2$ homogeneous of degree $n$. Define a representation $$\phi_n:SL(2,\mathbb{C})\to ...
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1answer
34 views

Showing that a very well-known representation is really a representation

Fix an integer $n\geq 0$, and let $V_n$ be the complex vector space of polynomials in two variables $z_1$ and $z_2$ homogeneous of degree $n$. Define a representation $$\phi_n:SL(2,\mathbb{C})\to ...
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1answer
36 views

Equivalent conditions, reductive groups

I read the book Invariant Theory by T.A. Springer. There is the following definition of a reductive group: Definition. A linear algebraic group $G$ is called reductive if for any rational ...
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19 views

Problems with Matrix of irreducible representation of SO(3)

I'm working out the irreducible representation of $SO(3)$. Let's call $R_{\theta}$ as a general rotation and $Y_{m}^{l}\left(\eta,\varphi\right)$ the spherical harmonics. I now like to have the matrix ...
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1answer
37 views

On the proof that one dimensional linear algebraic groups are either isomorphic to $\mathbb{G}_m$ or $\mathbb{G}_a$.

Let $G$ be a linear algebraic group of dimension one. The proof that I am looking at, in t.a springer's book (thm 3.4.9) proceeds by showing that $G$ must be either equal to its semisimple part ...
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19 views

Showing F-matrix representation is irreducible over $\mathbb{R}$

I have $G$ the cyclic group of order 4 and its $F$-(matrix) representation $T$ is $$\hat{T}(g) = \bigg[ \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \bigg]. $$ I am trying to show that ...
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27 views

All the Idempotent elements of a finite-dimensional algebra

Does there exist any way to determine whether or not, we have found all the idempotent elements of a finite-dimensional algebra A? In other words, if A is a finite-dimensional algebra with ...
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1answer
30 views

Product of Characters

Let $\rho:G\rightarrow\text{GL}_n(\mathbb{C})$ be a representation of a finite group and let $\chi_\rho$ be the corresponding character. If $\chi(e)>1$, then I want to show that ...
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22 views

Is $V \otimes V$ a $g \otimes g$-module?

Let $g$ be a Lie algebra and $V$ a $g$ module. Then $V \otimes V$ is a $g$ module under the action $X.(x \otimes y) = X.x \otimes y+x \otimes X.y$, $x, y \in V$, $X \in \mathfrak{g}$. Is $V \otimes V$ ...
3
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1answer
52 views

Relations between center (fundamental group) and (co)root and weight lattices for Lie groups

I would like to find some explanation or reference for the following facts, provided they are correct, and clarify some of the assumptions. Denote by $G$ a (perhaps semisimple compact connected) Lie ...
6
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1answer
81 views

Rings that cannot be representations rings

Given a monoidal category $\mathcal{C}$ one can define the Green ring (or representation ring) $r(\mathcal{C})$ as the abelian group generated by the isomorphism classes $[V]$ of $\mathcal{C}$ modulo ...
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11 views

Finding the character table for Z_8

I am a bit confused about how to come up with the number of irreducible representations, as well to come up with the number of different conjugate classes. Starting me out would be highly appreciated ...
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15 views

Incidence algebra

Let $(I;\preceq)$ a finite poset, where $I=\{a_1,\ldots,a_n\}$ and $\preceq$ is a partial order on $I$. We define de incidence algebra $KI$ of the poset $(I;\preceq)$ with coefficients in $K$, where ...
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16 views

Positive definite functions coming from finite dimensional representations

Let $G$ be a topological group, let $\mathcal{H}$ be a complex Hilbert space, let $v\in\mathcal{H}$ be a nonzero vector, and let $\rho:G\rightarrow \mathcal{U}(\mathcal{H})$ be a unitary ...
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36 views

Classification of Specht Modules

How do you "classify" Specht Modules? As you surely know, Specht Modules are irreducible representations of the symmetric group of n letters. I have been working on finding Specht Modules of Symmetric ...
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18 views

Commutative diagram for hidden subgroup representation of graph automorphism

The hidden subgroup representation of the graph automorphism problem is defined in the section 10.2 of QUANTUM ALGORITHMS FOR PROBLEMS IN NUMBER THEORY, ALGEBRAIC GEOMETRY, AND GROUP THEORY. It is as ...
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6 views

Relationship between Casimir and index of a representation of a Lie algebra.

In several QFT textbooks (namely, those of Peskin and Shroeder and of Schwartz) there is presented an identity for representations of Lie algebras, $$ d(R) C_2(R) = T(R) d(G),$$ where $d(R)$ is the ...
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29 views

Does Fourier Algebra of locally compact group separate compact sets of the group?

Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of ...
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1answer
25 views

Pure states on $C(X)$ are exactly evaluations

Let $X$ be a compact Hausdorff space. I want to show that pure states are of the form $ \phi (f) =f(x)$. By Reisz Represenation Theorem states on $C(X)$ are of the form $\phi (f)= \int fd\mu$ where ...
3
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1answer
42 views

Using Irreducible Group Characters to Count nth Roots of Group Elements

Given $n\in\mathbb{N}$, define $\tau_n(g)=|\lbrace h\in G: h^n=g\rbrace|$. Let $\chi_i,1\leq i\leq r$ be the distinct complex irreducible characters of a finite group $G$, and let ...
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1answer
22 views

How does having a cycle in a quiver change the simple objects in the category of representations?

In theorem 1.12 on page 5 of http://www.math.utah.edu/dc/tilting.pdf, which states: Given a bounded acyclic quiver $(Q,R)$, the K-theory of it's representations is given by $\mathbb{Z}^{Q_0}$ why is ...
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12 views

What's uniform block signed permutations?

Let $[n]=\{1,2,\ldots,n\}$ and $P(n)$ the set of all partitions of [n]. A partition of $[n]$ is non-empty disjoint subsets of [n], called blocks, whose union is $[n]$. A block permutation of [n] is ...
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1answer
44 views

$R$ is isomorphic to a direct product of matrix rings over division rings

Suppose as rings, $R$ is isomorphic to a direct product of matrix rings over division rings, that is $R=R_1 \times ... \times R_n$ where $R_i$ is a two-sided ideal of $R$ and $R_i$ is isomorphic to ...
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1answer
18 views

$R$ is a direct sum of simple modules implies every $R$-modules is completely reducible.

Suppose the ring $R$ considered as a left $R$-module is a direct sum $R=L_1\oplus L_2\oplus ...\oplus L_n$ such that $L_i$ are simple modules (no nonzero proper submodules) and such that $L_i=Re_i$ ...
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1answer
42 views

Every $R$-module is injective implies $R$ has the descending chain condition.

Suppose that every R-module is injective. There are a few definitions of injective that we can go by: E is an injective R-module if and only if Hom$_R(−, E)$ is an exact functor. An R-module E is ...
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18 views

Properness of isometric actions of discrete groups on affine Hilbert spaces

I've been reading Valette's introduction to the Baum-Connes conjecture and as I read the example of a construction of a (model of the) classifying space for proper actions of $\Gamma$ (discrete) given ...
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15 views

Decompose induced representation of $S_2$ and $S_3$

Let $ H=S_2 \subset G=S_3 $. Then use Frobenius reciprocity to decompose $ \operatorname{Ind}_H^G(\operatorname{sgn}_H) $ into irreducibles. $ G=S_3 $ has $ 3 $ irreps $ 1_G, ...
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Mackey's criterion and double cosets of $A_3$ and $S_3$

State Mackey's criterion $Ind_{H}^{G}$ is irreducible $\iff p$ is irreducible $p^s$ and $p$ are disjoint representations of $H \cap sHs^{-1}$ for any $s \in T $\ $ \{1\}$ Find the double ...
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20 views

Show $Res_H^G(sgn_G)=sgn_H$ where $G=S_4$ and $H=S_3$

Let $ G=S_3 $ and $ H=S_2 $. Show that $ Res^G_H(sgn_G)=sgn_H $ The symmetric group $G=S_3$ has three irreducible representations $ 1_G, sgn_G $ and $ V$ where $ 1_G $ denotes the trivial ...
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27 views

Comultiplication in a tensor algebra.

Let $V$ be a vector space. Then we have the tensor algebra $TV = \oplus_{i=0}^{\infty} T^i V$. In the webpage, it is said that the comultiplication $\Delta: TV \to TV \otimes TV$ is given by the ...
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1answer
20 views

Frobenius reciprocity and induced representations

In representation theory, we consider the restriction functor for any group $G$ and subgroup $H$. This is: $Res_H^G : Rep(G) \rightarrow Rep(H)$ This gives a representation of $H$ The Induced case ...
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15 views

Generators in adjoint representation are structure constants

Given that $g T_a g^{-1} = D^b_a T_b$ one can show that the generators in the adjoint representation of a group $G$ are the structure constants of the lie algebra satisfied by the $T_a$. Write $g$ ...
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1answer
24 views

Compute the associated induced Lie algebra action $\text{d}\pi$

Let $G=\mathrm{SL}_2(\mathbb{C})$ and consider the action of $G$ on the space of smooth functions on column vectors $\mathbb{C^2}$ given by $\big(\pi(g)\phi\big)(v)=\phi\left({g^\top}\,v\right)$ for ...
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106 views

Matrix Proof of Schur Orthogonality

It seems to me like the coordinate statement of Schur's Orthogonality relations $$ \sum_{R \in G}^{|G|} \Gamma^{(\lambda)}(R)_{nm}^* \Gamma^{(\mu)}(R)_{n'm'} = \delta_{\lambda \mu} \delta_{n n'} ...
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25 views

Question about number of irreducible representation?

From the table I attached (which is from http://chemwiki.ucdavis.edu/Core/Theoretical_Chemistry/Symmetry/Group_Theory%3A_Theory), $\Gamma$ is the reducible representation. It is then deduced that ...
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36 views

Is character of a group representation the same as trace?

If so, why cannot the Klein group's character be zero? The group element of Klein group matrices can be traceless, right?
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1answer
26 views

Is character same as one dimensional irreducible representation?

Look at character table of the Klein Group: http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_Klein_four-group Is the plus or minus ones on each row the same as one dimensional ...
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11 views

How to show that antipode is anti-Poisson and counit is Poisson?

I am reading the book Algebras of Functions on Quantum Groups: Part I by Leonid I. Korogodski and Yan S. Soibelman. I have a question about the proof of that antipode is anti-Poisson and counit is ...
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1answer
58 views

A reference for the equality $|G:H| = \sum_i \dim(V_i)\dim(V_i^H)$

Let $G$ be a finite group and $H$ a subgroup. Let $V_1, \dots , V_r$ be (equivalence class representatives for) the irreducible complex representations of $G$. Let the stabilizer subspace $V_i^H = \{ ...
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Matrix Schur Orthogonality

Where does the operator $A = \sum_S D^2(S) X D^1(S^{-1})$ in the proof of the orthogonalitie relations comes frm?
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29 views

Tilting object and mutation in Coh $(X)$

I'm studying the article "The cluster category of a canonical algebra" of Barot, Kussin, and Lenzing. I would like to understand an argument about mutation. I wrote the definition below: Let $T = ...
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1answer
57 views

composition series of a group algebra over finite field

Assume F is a finite field of characteristic 2 and G is the Klein's four-group. How many different composition series does the FG F-algebra have, as a module over itself? Is this number related ...
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31 views

Representation of a finite group and its Sylow $p$-subgroup

Let $G$ be a finite group with order $|G|=p^n \cdot m$ for some positive integers $n,m$ and $H$ be a Sylow $p$-subgroup of $G$. What relations can we say about the representations of $G$ and $H$?