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

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Multiple reps if $g$ not conjugate to $g^{-1}$

If $g \in G$ is not conjugated to $g^{-1}$, how do I prove that $G$ has irreducible non-equivalent representations of the same order? I think the multiple representations are going to be in some way ...
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21 views

Representation/Character theory of $S_3$: What is the Vector space $V$?

This is a basic question that I may have a misunderstanding on. When we study the character table of a group, say $S_3$, what vector space are we looking at? I understand that a linear ...
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16 views

Is there a direct sum decomposition of the tensor product of two representations of two group elements?

I know that I can decompose $\rho_a(g) \otimes \rho_b(g)$ into $U^\dagger \left[ \rho_c(g) \oplus \rho_d(g) \right] U$. Is there a similar way to decompose $\rho_a(g_1) \otimes \rho_b(g_2)$ into $U^\...
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24 views

Composition series of a regular module.

Suppose $A$ is an $k$-algebra with basis ${1,e,s,t}$ and multiplication is given by $$ e^2 = e, es = s, te = t, s^2=t^2=se=et=st=ts=0. $$ I am trying to find the composition series for ...
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33 views

Finite dimensional algebraic representation of $SL_2(\mathbb{C})$

I heard that for each $n\in \mathbb{N}$, there is the unique algebraic irreducible representation of $SL_2(\mathbb{C})$ with dimension $n$ over $\mathbb{C}$. Would you let me know what is such ...
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Matrix representations of particular generators of the full octahedral group

I want to find matrix representations of the generators $a, b, c$ of the full octahedral group in the presentation $$\{a,b,c \mid a^2=1,b^3=1,(ab)^4=1,ac=ca,bc=cb\}.$$ Is there a recipe to write the ...
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69 views

Prove that the sum of all simple roots is a root

Let $\Delta$ be an indecomposable root system in a real inner product space $E$, and suppose that $\Phi$ is a simple system of roots in $\Delta$, with respect to an ordering of $E$. If $\Phi = \{\...
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30 views

Functoriality of the adjoint representation

Just a simply question. I came across the following statement which is used for deriving Weyl's integral formula: ''$\text{Ad}_G(h)|_{\mathfrak{h}} = \text{Ad}_H(h)$ due to functoriality in the Lie ...
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25 views

Irreducible representation restricted to index 2 subgroup

Suppose $G$ is a (not nec. finite) group with index 2 subgroup $H$ and $k$ is a field (possibly of positive characteristic). Suppose $$\rho:G\to\mathrm{GL}_2(k)$$ is an irreducible 2-dimensional ...
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About decompositions of induced characters

Suppose $G$ is a finite group, $H\leqslant G$ is a subgroup. $\chi_1,...,\chi_s$ are all the irreducible characters of $G$ and $\psi$ is an irreducible character of $H$. Prove that if $$\psi\uparrow G=...
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Every irreducible representation of $G_2$ appears in some tensor power of the standard representation

In the Book "Representation Theory" by Fulton and Harris, this fact ist stated on page 353 after looking at the weight diagrams of the complex Lie-Algebra $G_2$. The authors deduce that with $V=\...
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20 views

Modules generated by primitive idempotent elements

Assume that A is a finite dimensional k-algebra, and $e \in A$ is a primitive idempotent element. Is it true that the submodule of $A$ namely $<e>$ is simple $A$-module? If it is, how do we ...
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22 views

references of modular representations for finite group

What is modular representation for finite groups? I tried to find a book to understanding that but I could not find a good one. Are there any useful references?
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31 views

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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76 views

Every irreducible representation is either even or odd. [closed]

Let $V$ be any $n$-dimensional complex vector space and $SL(2,\Bbb{Z})$ is special linear group. Let $\rho:SL(2,\Bbb{Z}) \rightarrow GL(V)$ be a representation. It is even if $\rho(-I)=\Bbb{id}_V$ and ...
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23 views

Dual of a faithful representation

A representation $\sigma$ of a finite group G is said to be faithful if Ker$\sigma={1}$. Then is it true that dual of a faithful representation is also faithful?
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Composition Series of the regular A-module

Assume A is a finite-dimensional algebra over field K. How can we prove that any simple A-module occurs, as a composition factor (up to isomorphism) of an arbitrary composition series of A, as module ...
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18 views

How to represent the function of variables?

I have a function as $$E=\int_\Omega -\log\big( p_i(x)\big) dx$$ where $p_i(x)$ is density distribution which estimated by Parzen window method. $p_i(x)=\frac{1}{\Omega_i} \int_{\Omega_i}K_\sigma\big(...
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63 views

Why do the characters of an abelian group form a group?

I was reading through Serre's Linear Representation Theory book and encountered a question to show that the set of all irreducible characters of an abelian group form a group. The proof of closure ...
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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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27 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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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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36 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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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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21 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 GL(...
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35 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 GL(...
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1answer
38 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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20 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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38 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 $G_s=\...
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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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37 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 $\chi_\rho\cdot\...
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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 $V \otimes V$ ...
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80 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 ...
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87 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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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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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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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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38 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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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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25 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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34 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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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 $\...
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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 $\gamma_n(\chi_i)=\...
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25 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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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
56 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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47 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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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 ...