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

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Properties of Group representations, duality and the derived subgroup

I am trying to understand why 1) all finite-dimensional complex representations $V$ of $G$ are self dual, and 2) How the derived subgroup $[G,G]$ is a union of particular conjugacy classes. My ...
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Question on GL(n,F) representation

Let A be the group of all invertible n x n matrices over F, A+/- the subgroups of all upper/lower matrices. F^n as an A-module is irreducible? Is this because F^n has only one orbit under A? Why is ...
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Computing quotient representations and Hom set fort wo representations

Consider the representation $M$ defined by We want to find all subrepresentations quotient representations of $M$, and $\mathrm{Hom}(M,N)$, where $N$ is a representation with $N \cong M$. I put B ...
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The indecomposable projective A-modules

Let Q be the quiver bound by $αβ = 0$, $γδ = 0$. The indecomposable projective A-modules are given by where $A=KQ/I$. This an example in Assem-Simson-Skowronski book (Elements of the ...
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quasi-split algebraic group

While reading papers, there usually an assumption "quasi-split" for reductive algebraic groups. To use their results I need to know which groups are quasi-split. For the case I am interested in ...
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Finding a basis for $sp(4,\mathbb{C})$ and related basis.

Let $$L = so_4(\mathbb{C})= \{x \in End(\mathbb{C}^4)|^txS + Sx = 0 \} \text{ where }S = \left(\begin{array}{cc} 0 & I_2 \\ -I_2 & 0 \end{array}\right)$$ Letting $x = \left(\begin{array}{cc} ...
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Example of two very ample line bundles which give different semistable locus

Let $G$ be a linear algebraic group. I'm looking for an example of a $G$-space $X$ and two very ample $G$-equivariant line bundles, $\mathcal{L}$ and $\mathcal{L}'$ such that the corresponding ...
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Exercise 5.8 from Lie Group, Daniel Bump

In the exercise 5.8 Bump has asked to prove that the group $Sp(4)$ over complex numbers, which is usual complex embedding $U(4)\cap Sp(4,\mathbb{C})$, can be described by, $$\left\{\begin{pmatrix} ...
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Questions about the bracket

In the map $\phi : L \mapsto \mathfrak {U}(L) $, where $ L $ is a lie algebra and $\mathfrak {U} $ is a universal enveloping algebra of $ L $. (1) Is the following relation true? If $[xy]=z$ in $ L ...
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Direct sum decomposition of weight spaces and relation to Tensor products.

There are 3 parts to the question that I am trying to understand, and while it is not homework it seems instrumental in decomposition modules into weight spaces and their relation to tensor products. ...
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P-adic Lie groups - Representation theory

I am quite familiar with the Representation Theory for locally compact groups and nilpotent Lie groups. I want to start with the study of $p$-adic Lie groups representation theory, in particular ...
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What is the least $n$ such that it is possible to embed $\operatorname{GL}_2(\mathbb{F}_5)$ into $S_n$?

Let $\operatorname{GL}_2(\mathbb{F}_5)$ be the group of invertible $2\times 2$ matrices over $\mathbb{F}_5$, and $S_n$ be the group of permutations of $n$ objects. What is the least ...
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Convolution and Characters

I am confused about the purpose of the Formal Character, character functions, and the convolution in representation theory of Lie algebras. Is the Character function different than just the Character? ...
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Show that if $V$ is an irreducible finite dim. representation of $A$, then $z \in Z(A)$ acts in $V$ by multiplication by some scalar $\chi_V(v)$.

Let $A$ be an algebra over a field $k$. The center $Z(A)$ of $A$ is the set of all elements $z \in A$ which commute with all elements of $A$. For example, if $A$ is commutative, then $Z(A)=A$. ...
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36 views

Endomorphism ring of finite-dimensional representation

If $G$ is any group and $V$ is a finite-dimensional representation of $G$, then we can form the endomorphism ring $E = End_G(V)$. Suppose that $V$ is indecomposable, i.e. not a direct sum of ...
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Is there anywhere some explicit Bruhat decompositions are written down?

Question in title: most places I see Bruhat decompositions treated they're only briefly mentioned and no examples are given. Also, I calculated the following regarding the Bruhat decomposition of ...
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Tensor product of algebras which is Frobenius.

Let $A$ and $B$ be two finite dimensional algebras over a field $k$. Let us suppose that the $k$-algebra $A\otimes_{k} B$ is Frobenius (or symmetric). Is it true that $A$ and $B$ are two Frobenius ...
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Heisenberg XXX spin model

Let $\pi$ be the standard representation of $sl_2(\mathbb{C})$ on $\mathbb{C}^2$. Let $p_1,p_2,p_3$ the three Pauli matrices. Define $S^a:=\frac{1}{2}\pi(p_a)$. What does such matrices looks like?
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Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
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Show that the homomorphism $\lambda: k[X] \to End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ has a nontrivial kernel.

$\DeclareMathOperator{\End}{End}$ I'm trying to show that: Show that the homomorphism $\lambda: k[X] \to \End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ as in (see ...
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Quaternionic representation

Let $V$ be $G$-representation over quaternions $\mathbb{H}$. How to show that $$ \mathbb{H} \otimes_\mathbb{C} V $$ is canonically isomorphic to $V \oplus V$ as representation over $\mathbb{H}$? In ...
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Show that a simple ring is always an algebra over some field

Show that a simple ring $R$ is always an algebra over some field. So I need to show that there exist a field $k$ such that there exists a ring homomorphism $\phi : k \to Z(R) $. In an earlier ...
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Computing Path Algebra of a Quiver

Let $Q$ be a quiver over defined as follows Then $KQ\cong$ $\begin{pmatrix}K&K&K\\0&K&K\\0&0&K\end{pmatrix}$, where $KQ$ is just the path algebra. What the professor did was ...
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Is $A \times B$ the same as $A \oplus B$?

When $A, B$ are $K$-modules, then $A \times B$ is the same as $A \oplus B$. Let $A, B$ be two $K$-algebras, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$? Thank you very much. ...
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Why are we interested in irreducible representation but not faithful representation?

I am reading some materials of representation theory (of a group). The motivation of representation theory is to represent (by a homomorphism $h: G \to GL(V)$, from the group $G$ to a vector space ...
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Confusion in Lie algebra notes

I'm self-studying through these notes, and I ran into a roadblock on the page 38, chapter $sl(2)$ and its irreducible representations. Right after defining $U(sl(2)) \otimes_{U(b^+)} \mathbb C$ ...
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How do I map Df(w) to it's [lie] group/algebra representation?

E.G. For $p,w\in(\mathbb{R}^3,+,\times_\vartheta)$ with $(\mathbb{R}^3,+)$ a vector space and with $p=(r,s,t)$, $w=(x,y,z)$ where we have $p\times_\vartheta ...
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Easy Introduction to Representation Theory

I have a student that is interested in reading up on representation theory in her own time. She knows a small amount of linear algebra, what you would expect in a simple sophomore linear algebra ...
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Sum of degrees of irreducible complex characters for certain groups

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant do determine the dimension of a maximal torus in the group algebra. I have ...
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Complex conjugation of positive roots

I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a ...
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“Powers” of injective representations “contain” all irreducibles

Let $G$ be a finite group and let $\rho : G \to GL(V)$ be an injective representation. I need to prove that each irreducible representation of $G$ is contained in $\otimes_{i=1}^{n} \rho$ for some $n ...
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Question for recommending a good textbook in representation of quivers

I am taking representation of quivers, and the lecture notes seems not enough. So could you recommend a good textbook for this course. There is a new book "Quiver Representations, by Ralf Schiffler" ...
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isomorphism classes of representations of a quiver

Classify all isomorphism classes of representations of dimension vector 1 and 2 of the following quiver The professor briefly did the solution, but I could not understand what was going on. What he ...
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Irreducible Representations and Direct Sums

I am learning about representation theory, and my professor stated the following as a remark: Let $A$ be a $k$-algebra. Every finite dimensional representation of $A$ is a direct sum of ...
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Reference request for properties of harmonic polynomials

I am reading this paper, and on page $4$ it takes a non-degenerate quadratic form $q$ on a finite dimensional complex vector space $V$, defines the laplacian assosciated to $q$, $\Delta$, acting on ...
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left regular representation of a group thru group action

Let G be a group and let $g\cdot:G\to G$(i.e.$g\cdot g'=gg'$) . This induces a permutation representation of the group. I was trying to walk thru the problems in dummite and foote. One of the problem ...
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Sum of an irreducible character over all $G$

Let $\phi: G\to GL_n(\mathbb C)$ be an irreducible representation of a finite group $G$. Let $\chi: G\to \mathbb C$ be the character of $G$. Prove that: $$\displaystyle \sum_{g\in G}{\chi(g)}=0 $$ I ...
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Does the induced $K^G$-comodule correspond to the induced $KG$-module?

Let $G$ be a finite group with group multiplication $m\colon G\times G \to G$ and $K$ a field. Then $K^G$ (the set-maps from $G$ to $K$) is a commutative algebra with pointwise multiplication. Because ...
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Failure of the Krull-Schmidt Theorem?

Theorem 1.19 of Representation Theory of Finite Groups: Algebra and Arithmetic is the Krull-Schmidt theorem, which I screenshotted and uploaded it here, I don't have any problem with this theorem and ...
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Lattices in Lie Algebras

I am having a little confusion with the different types of lattices involved with Lie algebras. Root system: represented as euclidian vector arrows. However I have seen the same arrangement with ...
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Prove that if $\rho,\phi$ are irreducible representations then so is this representation

Let $G_1$ and $G_2$ be finite groups and let $G = G_1 \times G_2$. Suppose $\rho: G_1 \to GL_m(\mathbb{C})$ and $\phi: G_2 \to GL_n(\mathbb{C})$ are representations. Let $V =M_{mn}(\mathbb{C})$. ...
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regular representation of algebras

Let suppose we have universal enveloping algebra, what is the meaning of the notion of the right regular representation of that? How can we determine the right regular representation of universal ...
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Intuition behind PBW

The PBW theorem states: $\omega:\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras. Where $\mathfrak {S} $ is the symmetric tensor algebra of a Lie algebra $ L $. Where $\mathfrak ...
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Reduced Group algebras

Take a finite group and a field of characteristic zero. The group algebra is due to Maschke's theorem semisimple so that its a finite direct sum of matrix algebras over division algebras. I like to ...
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Orbits of $Sp(n,R)$ under action of $Gl(2n,R)$ by conjugation

These questions arose from a question related to K-theory, I am hoping for (big) results from the theory of linear algebraic groups to be helpful. Maybe somebody with a better background there can ...
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Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and let $H$ be ...
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The Heisenberg group over $\mathbb{Z}/2\mathbb{Z}$

This is inspired by a problem from from Dummit and Foote. It asked me to calculate the order of every element in the Heisenberg group over $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$, which is defined as ...
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123 views

Verify that two linear representations are equivalent

I've a problem in verifying that two linear representations are equivalent. First of all, I have two permutation representations of the group $G=\langle\alpha ,\beta ,\gamma\rangle$ on the set ...
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Matrix representation associated to a permutation representation

I have just begun studying group representation theory. I don't understand how I can find the matrix representation associated to a permutation representation. Should I identify each permutation ...
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Why do we study representations of groups but not fields?

Groups are great objects to work with as we all know. With surprisingly little structure, we can say fairly general things. However groups can be difficult to manage and so we look to representations ...