# Tagged Questions

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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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### 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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### 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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### 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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### The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, ...
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### History of the matrix representation of complex numbers

It is well-known to many that $\mathbb{C}$ can be represented by matrices of the form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$. For example, see this question or this ...
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### Positive definite functions generated by irreducible representations — what do people call them?

Let $G$ be a group and $\pi:G\to B(H)$ be its irreducible unitary representation (one can endow $G$ with topology and claim that $\pi$ is continuous in some sense, this doesn't matter). For a given ...
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### Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
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### Reference request for Modular representation theory

I am trying to learn modular representation theory. I would be thankful if any one tell me a good reference to start with?
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### Representation of topological groups

I am looking for a good book of topological representation. I have a very good insight of representation theory of finite groups, and I want to explore topological representations. I saw a book by ...
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### The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

Question What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group? The trivial corepresentation is given by $\Delta_{|W}$ where $W$ is just the one dimensional ...
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### Number of indecomosbale $\mathbb{Z}_p[G]$ modules finite

Is there a theorem like those of Jones, which tells if the number of different $\mathbb{Z}_p[G]$ modules is finite, where $G$ is a finite group and $\mathbb{Z}_l$ the $p$-adic ring?
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### Very basic question on continuous representations

Suppose $G$ is a topological group, i.e. $$G \times G \to G, (g,h) \mapsto gh ~~~~ G \to G, g \mapsto g^{-1}$$ are continuous. Suppose $K$ is a field with norm $|\cdot|$ on it and $V$ is a normed ...
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### Applications of representation theory in physics

The notes of a lecture on basic group and representation theory I attended last semester begin with a bit of motivation for the argument. They give the following examples for applications in physics: ...
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### Learning Roadmap for Borel - Weil - Bott Theorem

Next semester I may study a course where the ultimate goal is to get to the Borel - Weil - Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have ...
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### Online course in representation theory or differential geometry

Are there any courses in representation theory that are available online? I'm looking for a course including videos, notes as well as assignments. I'd also be interested in a course in differential ...
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### How to write down the maximal subgroups of $GL(9, \mathbb{C})$

I am wondering about the maximal subgroups of the group $GL(n^2, \mathbb{C})$. My motivation for wondering about these groups is a project (in its most general form) I am working on where I am trying ...
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### Book recommendation for associative algebras

Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment. ...
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### Introduction to the trace formula for people outside number theory

I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula. I am aware that there are "standard references" on the ...
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### Connection between $\mathbb{Q}_p[G]$ and $\mathbb{Z}_p[G]$

In this post there was the comment, that having $\mathbb{Q}_p[G]$ modules, it is possible to construct $\mathbb{Z}_p[G]$ modules. How is it possible to find out when there is a bijection between ...
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### Expression of basis vectors of permutation modules in different bases.

Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
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### Frobenius reciprocity

I would like to ask a question on Theorem 8.6 on page 246 in this book. There is the claim that the multiplicity of $F$ in $E^G$ is equal to the multiplicity of $E$ in $F_H$. Why is this just ...
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### Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?

Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have $$d_{\alpha}|\#G,$$ where $d_\alpha$ is the degree of the representation and $\#G$ ...
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### Why are (representations of ) quivers such a big deal?

Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...
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### Induced representation is isotypical?

Is there a theorem like this for the induced representation? Let $N$ be a normal subgroup of a finite group $G$ and $\rho$ be an irreducible linear representation over any field $k$. Then one of ...
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### Subspaces stabilized by representations of $\mathrm O(9)$

I am trying to figure out what representations of maximal subgroups of $\mathrm{GL}_{n^2}$ stabilize one dimensional subspaces in $\mathrm{GL}(\mathrm{Sym}^n(\Bbb C))$. More precisely, let the setup ...
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### Representations over $\mathbb{Q}_p$

I would like to understand representations over the $p$-adic field $\mathbb{Q}_p$ and find simple $\mathbb{Q}_p[G]$ modules for a finite group $G$. Is there some famous literature like for ...
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### Good book on representation theory after reading Rotman

I'm about to finish Rotman's "Introduction to the Theory of Groups" and I would like to continue my study of group theory with a book on representation theory. The book should give a broad overview ...
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### Topic for presentation on Group Representations, Young Tableaux, Symmetric Group

I need to do a presentation relating to group representations/Young tableaux/symmetric group; however, for all my searching, I cannot find a cool topic that I find personally interesting (and that is ...
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### Definition of induced representation by tensor product

Suppose there is a finite group $G$ with a subgroup $H$ an some field $K$. If one has a representation of the subgroup, one can construct the induced representation $\rho:=Ind_H^G$ according to ...