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

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Character degrees of 2B2(q^2)

Let $S \cong {}^2B_2(q^2), q^2 \ne 2$, and $\chi$ is the Steinberg character for $S$. What is the $\chi(1)$?
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1answer
34 views

Representation-theoretical reasons for positivity of product of two Schubert polynomials?

In the Wikipedia article on Schubert polynomials there is a claim that there are representation-theoretical reasons for the product of two Schubert polynomials to have nonnegative coefficients when ...
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2answers
80 views

1-dim representations of the affine Hecke algebra for $G = SL_2$

I want to count the number of (isomoprhism clases) of one-dimensional representations of the affine Hecke algebra for $G = SL_2$. I'm doing it in two ways: (1) by explicitly looking at generators and ...
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1answer
117 views

Uniformly continuous unitary representations.

Let $U$ be a unitary rep. of $\mathbb{R}^d$ on a separable Hilbert space $H$, and $H\cong\oplus L^2_{\mu_v}(\mathbb{R}^d)$ be the spectral decomposition (according to the spectral theorem for these ...
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2answers
54 views

Realizing the oscillator algebra as a matrix Lie algebra

In Hilgert's & Neeb's Structure and Geometry of Lie Groups, they introduce a Lie algebra, which they call the "oscillator algebra," as an extension of the Heisenberg algebra. They give a basis ...
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15 views

Mappings between tensor products of group representations

All of my representations are on finite dimensional complex vector spaces. Let $G$ and $H$ be finite groups. If $V$ is a representation of $G$ and $W$ is a representation of $H$, then $V\otimes W$ is ...
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34 views

Unramified Hecke character

I'm looking for a reality check here: Let $\chi$ be a character of $(F^\times\backslash \mathbb A_F^\times)^1$ where $F$ is a number field. Call $\chi$ unramified at a place $v$ if ...
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1answer
39 views

Proof check - infinite-dimensional $\mathfrak{sl}(2, \mathbb{F})$-module

Let $L=\mathfrak{sl}(2,\mathbb{F})$ with the usual basis $(x, \ y, \ h)$ and $\text{char}\,\mathbb{F}=0$. Let $Z(\lambda)$, $\lambda\in\mathbb{F}$ the infinite-dimensional $L$-module spanned by ...
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16 views

Branching rules without previous knowledge of the projection matrix?

Given a representation $R$ of some group $G$ one can find in many books and papers (e.g. page 96ff here) the decomposition under certain subgroups: $$ R= R_1 + R_2 + \ldots$$ This is often called a ...
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14 views

Inner Automorphism of Lie algebras in Terms of Roots and Weights?

An automorphism is a homomorphism of a group $G$ onto itself. For Lie groups this induces a Lie algebra $g$ automorphism, i.e. a map of the Lie alegbra onto itself that preserves the Lie bracket. An ...
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24 views

s3 representation in 2d in matrix form

Trying to arrive at how matrix form of standard representation of Symmetric Group S3 has the form $((-1,1),(0,1))$ for permutation $(1,2)$. Please let me know details. - Thanks
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47 views

Do we have $\mathbb{C}[SL_n] = \oplus_{\lambda, \text{ht}(\lambda)\leq n} V_{\lambda} $?

The coordinate algebra $\mathbb{C}[SL_n]=\mathbb{C}[x_{ij}: i, j \in \{1, \ldots, n\}]/(\det(x_{ij}) - 1)$ is a representation of $SL_n$: $(g'.f)(g)=f(g'^T g)$. Let $V_{\lambda} = \langle e_T : T ...
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2answers
48 views

How to understand that minors are matrix elements in fundamental representations of $SL_n$?

In the video, Lecture 3 of June 14, 49:00-53:00, it is said that "minors are matrix elements in fundamental representations of $SL_n$". What are fundamental representations of $SL_n$? How to ...
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1answer
78 views

Does anyone know the Burnside Matrices?

For $G$ a fine group with conjugacy classes $C_1,\dots C_k$ we introduced the Burnside Matrices $A_r$ where $1<r<k$ with entries: $$A_r := \Big(\sqrt{\frac{|C_t|}{|C_s|}}a_{rst}\Big)_{1\leq ...
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69 views

Orbit of a Weight Vector?

Given some element $\phi$ of a representation $R$ of a group $G$, the orbit $G(\phi)$ of $\phi$ is defined as the set $g \phi \ \forall \ g \in G$. We can write every element of a given ...
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24 views

Equivalence of continuity conditions of a group representation on an infinite-dimensional space

Let $V$ be an (infinite-dimensional) Banach space and $G$ a locally compact topological group (with a countable basis of neighbourhoods of $1$, and which is a countable union of compact subsets). I ...
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1answer
43 views

Representations of the form $\varphi: G \rightarrow GL(V)$ vs $\phi: G \rightarrow Aut(A)$

Standard representation theory studies homomorphisms of the form $\varphi: G \rightarrow GL(V)$ where $V$ is a vector space. How much does the focus of representation theory change if one considers ...
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38 views

What is the Wedderburn decomposition of $\mathbb{R}[D_{2n}]$?

I have been looking everywhere and can't seem to find a general formula for the Wedderburn decomposition of the real group ring of the dihedral group ring of order $2n$, $\mathbb{R}[D_{2n}]$. Does ...
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2answers
60 views

Representation of $sl(2,R)$.

I am interested in the unique (up to isomorphism) $5$-dimensional representation of the Lie algebra $sl(2,R)$. I understand that one can choose the module $V_4 = ...
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0answers
26 views

List of simple roots in the H-basis for various Lie algebras?

There are four usual bases one can use to express the roots and weights of a given algebra. The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$. The ...
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28 views

Question concerning a correspondence between basis elements of the Schur algebra and some matrices

I have the following question: Let $k$ be an infinite field and let $S_k(n,r):={A_k(n,r)}^{∗}=\text{Hom}_k(A_k(n,r),k)$ and $A:=A_k(n):=\text{polynomial functions on}\ \Gamma:=\text{GL}_n(k)$ and ...
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1answer
32 views

Explicit Representation of the SU(N) Simple Roots in with redundant coefficents?

Commonly the simple roots for $SU(n)$ groups are given as $n$ dimensional vectors, although root-space is $n-1$ dimensional. The $SU(n)$ Wikipedia article explains: Here, we use n redundant ...
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56 views

Is ${(k^n)}^{\otimes r}$ a faithful $k\Sigma_r$-module for $n\geq r$?

I have the following question: Let $k$ be an infinite field. Let $E:={(k^n)}^{\otimes r}$ and let the symmetric group $\Sigma_r$ act from the right on $E$ by place permutations. It is well-known that ...
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66 views

Is this parabolic induction?

In one of my previous questions, @PL. explained the idea of parabolic induction. In my bachelor thesis I used a technique quite like this, to find all irreducible representations of a certain group. I ...
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1answer
45 views

General form of a matrix $M$ commutes with the unitary representation $U^{\otimes m},~ \forall U\in U(n)$

My question is about the general form of a $n^m\times n^m$ positive definite matrix $M$ where $$[M,U^{\otimes m}]=0,~ \forall U\in U(n)$$ or in other words, M commutes with all members of the the ...
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74 views

Question concerning a property of polynomial functions on $\Gamma:=\text{GL}_n(K)$ and the Schur algebra

I'm reading Green's book ''Polynomial Representations of GL_n. with an Appendix on Schensted Correspondence and Littelmann Paths''. Consider theorem 2.4b) part (i) on page 14: Consider the map $e : ...
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36 views

How to prove that the Schur algebra is isomorphic to a certain endomorphism ring?

I'm reading Green's book ''Polynomial Representations of GL_n. with an Appendix on Schensted Correspondence and Littelmann Paths''. Consider theorem 2.6c) on page 18: Let $K$ be an infinite field ...
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58 views

Question concerning a self-injective algebra and a faithful module

I'd like to know how corollary 2.11 of http://www.sciencedirect.com/science/article/pii/S002186930098726X# follows from theorem 2.10 from the same reference. 1) I know that $A$ being self-injective ...
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52 views

Dominant dimension $\geq 2$ implies a certain double centralizer property

let $A$ be an Artin algebra and $M$ in $\mathfrak{mod}\ A$. Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent: $\bullet$ $Ae$-dom.dim.$(A)\geq 2$ ...
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1answer
27 views

Question concerning a faithful module over an Artinian ring

Let $A$ be an Artinian ring and $M$ in $\operatorname{\mathfrak{mod}} A$. Is it true that $M$ is faithful if and only if there is an exact sequence of the form $0\rightarrow A \rightarrow M^r$ for ...
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1answer
45 views

Representation theory of locally compact groups

My knowledge about representation theory of locally compact groups is rather scattered. As I got more interested with this subject, I would like to know some good references, where I could learn the ...
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16 views

Questions about distributions on $l$-spaces.

I am reading the paper. I have some about distributions on $l$-spaces. On page 7, Section 1.7. Let $X$ be an $l$-space. Locally constant complex-valued functions on $X$ with compact support are ...
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114 views

Representation of an abelian group

Without using the structure theorem, how do I prove b? I struggle with the proof of injectivity. Any tips? Problem: Let $G$ be a finite Abelian group. (a) Prove that the group homomorphisms $\chi ...
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Matrix representation of function concatenation using other basis than polynomials.

I have now familiarized myself with the Carleman-matrices which represent function composition of polynomials (actually taylor series terms) and built some of my own. I noticed that for any finite ...
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19 views

$SO(n)$ algebra relations in the vector rep

The $\mathfrak{so}(n)$ algebra has some relations between generators always indicated as $$\left[T_{ij}, T_{kl}\right] = \delta_{ik}T_{jl} - \delta_{jk}T_{li} - \delta_{jl} T_{ik} + ...
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1answer
26 views

Function composition as representable by matrices?

I know from linear algebra that for different sets of functions differentiation can be expressed using matrix multiplication on a vector representation of the function. For instance polynomials and ...
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1answer
23 views

Root space $L_\alpha$ is completely contained in simple ideal?

I'm having trouble understanding a section in Humphrey's Lie algebras on page 74. Suppose $L$ is a semisimple Lie algebra which decomposes as a direct sum of simple ideals $L_1\oplus\cdots\oplus ...
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1answer
61 views

How can I show that the characters in sense of irreducible representations are the same as the character maps from the burnside matrices?

My Task is: Let G be a finite group. 1. Let $C_1 = \{e\}, C_2,..., C_k$ be the conjugacy classes, and let $v_1,..., v_k$ be the normalised eigenvectors of the Burnside matrices of G, then for all s ...
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9 views

Weights system corresponding to reflected Dynkin diagram?

Given a set of weights corresponding to the $SO(10) Dynkin diagram How can I transform these weights into weights that correspond to the Dynkin diagram ?
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1answer
29 views

Dihederal Group $D_{2n}$ Where $n$ is even/odd

I know that the group presentation of $D_{2n}$ is the following $$D_{2n} = \big<a,b: a^n=b^2=1,b^{-1}ab =a^{-1} \big>$$ Now if we consider the case where $n$ is even and we write $n =2m$ for ...
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1answer
29 views

Calculating the Lie algebra representation of the regular representation on subspace of functions on $\mathbb R$.

Let $G = \mathbb R$ and let $\pi$ be the regular representation of $G$ on $L^2(\mathbb R)$, that is, $\pi(g)(f)(x) = f(x-g)$ for $g \in G$. Let $V = \{f \in \mathcal C_c^\infty | supp f \subseteq ...
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72 views

Invariant subspaces of Lie group vs invariant subspaces of Lie algebra

I am starting to study infinite-dimensional representations of Lie groups and I am wondering about the following: Let $G$ be a connected Lie group with Lie algebra $\mathfrak g$ and with a ...
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Map for roots of a Lie group to roots of a special subalgebra?

For regular subalgebras $h$ of some group's Lie algebra $g$, $$ h \subset h $$ the root system of the subalgebra is a subset of the root system of the original's group algebra. Subalgebras whose ...
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What can we do about the indecomposable representations of wild quivers and wild algebras?

I know that using finite number of parameters we can not describe indecomposable modules of wild quivers, but is it possible for us to describe them using infinitely many parameters for at least some ...
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Locally compact spaces that are not first-countable and continuity of functions on locally compact groups and continuity of group representation

If $X$ is a topological space that is first-countable, then a function $f: X \to Y$ into another topological space $Y$ is continuous if and only if $f$ is sequential continuous. Only the implication ...
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How to transform roots/weights from the simple root basis to the H-basis?

Often the roots and weights of some Lie algebra are written in terms of the simple root basis $$ r =(a_1,a_2,a_3,\ldots)=a_1 \alpha_1 + a_2 \alpha_2 + a_3 \alpha_3 +\ldots,$$ where $α_i$ denotes the ...
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1answer
35 views

Simple Roots of E6 in Coordinates?

There are several possibilities how one can write simple roots in terms of coordinates. Firstly, they depend on the numbering of the nodes in the Dynkin diagram. Let's fix the choice for $E_6$ to be ...
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fulton and harris representation theory exercise 2.5 solution

Trying to solve Exercise 2.5: If $V$ is a permutation representation associated to the group $G$ on a finite set $X$, show that $\chi_\nu(g)$ is the number of elements fixed by $g$. Looking for hint ...
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21 views

Using Peter Weyl theorem to decompose an orbit

Let $G$ be a finite group and let $\pi$ be a unitary representation of $G$ on a Hilbert space $H$. Since $G$ is finite, we have that for every $v \in H$, the orbit of $v$, $\pi (G).v$, is of dimension ...
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Why is $Ind^G_H(M)=Ind^{G/H}_{\{e\}}$?

I was looking at some representation theory notes and found the following statement: $Ind^G_H(V)=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V=\mathbb{C}[G/H]\otimes_\mathbb{C} V$. Now, this makes intuitive ...