# Tagged Questions

**7**

votes

**0**answers

99 views

### Algebraic construction of the modular representation of $\mathrm{SL}_2(\hat{\mathbf Z})$

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if ...

**1**

vote

**5**answers

376 views

### Character table of $S_7$

Is there any reference where I can find the character table of the symmetric group $S_7$? A simple search in google gave me a GAP program that computes the character table, but I don't understand the ...

**5**

votes

**1**answer

163 views

### Equivariant Formality

Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$.
Assume further there exists a ...

**4**

votes

**1**answer

84 views

### Weyl group action on complexified Iwasawa decomposition

Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. ...

**6**

votes

**0**answers

108 views

### Zariski closure of orbits of real groups on complex flag manifolds

Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...

**1**

vote

**0**answers

39 views

### Reference Help: Matsuki duality Orbits

I'm studying the Matsuki duality of $G_0$-orbits and $K$-orbits over a flag manifold $G/P$ where $G$ is semisimple complex Lie group and $P$ is a parabolic subgroup. I would like to study some ...

**3**

votes

**1**answer

310 views

### What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?

Recall the definition of the representation ring $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can ...

**4**

votes

**1**answer

150 views

### To whom is the internal characterization of $Q$-groups due?

A group is said to be a $Q$-group if the character of any complex representation is rational valued. A well-known internal characterization of $Q$-groups is the following:
$G$ is a $Q$-group if ...

**2**

votes

**2**answers

250 views

### Density of characters

Did Harish-Chandra prove that characters of irreducible representations of a $p$-adic reductive group $G$ span a dense subspace of the space of conjugation-invariant distributions on $G$? What is the ...

**10**

votes

**3**answers

388 views

### Resource for learning quantum mechanics from the viewpoint of representation theory

Quantum mechanics is deeply connected with representation theory. Therefore, I'm looking for a textbook or article which presents quantum mechanics in a representation theoretic manner. Could anyone ...

**2**

votes

**1**answer

217 views

### The Bialynicki-Birula Stratification of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group with affine Grassmannian $\mathcal{G}r$. Fix a maximal torus and Borel $T\subseteq B\subseteq G$. I am reading "Loop Grassmannian ...

**4**

votes

**1**answer

166 views

### Reference for the Natural Ample Line Bundle on the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r:=G((t))/G[[t]]$$ be its affine Grassmannian. I have read that $\mathcal{G}r$ possesses a natural very ample line ...

**5**

votes

**1**answer

171 views

### Decomposition into irreducibles of the representation $L^2(SL_2(\mathbb{C})/\Gamma)$ for $\Gamma$ geometrically finite

I am trying to understand the decomposition
$$L^2(SL_2(\mathbb{C})/\Gamma)=\oplus_i C_i \oplus V_{temp}$$
where $C_i$ are complementary series representations corresponding to eigenfunctions of the ...

**3**

votes

**0**answers

125 views

### Generalization of Frobenius formula involving Macdonald polynomials

Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as
\begin{equation}
p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~,
...

**5**

votes

**1**answer

168 views

### Stratifications and Filtrations of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r=G(\mathcal{\mathbb{C}((t))})/G(\mathcal{\mathbb{C}[[t]]})$$ be the affine Grassmannian of $G$. We know that ...

**5**

votes

**2**answers

262 views

### Strata of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and denote by $\mathcal{G}$ its affine Grassmannian. Fix a maximal torus $T\subseteq G$. We know that $\mathcal{G}$ ...

**4**

votes

**3**answers

365 views

### Clifford's Theorem with all its aspects in modern language, looking for a textbook

I am looking for a (more or less) introductory textbook on representation theory that contains the full contents of Clifford's paper "Representations Induced In An Invariant Subgroup" in modern ...

**5**

votes

**3**answers

382 views

### What is a good introduction to branching rules in representation theory?

I'm looking for a book or introductory article, that explains branching rules in representation theory of real Lie groups.
When a Lie group has a set of irreducible representations, I'd like to know ...

**15**

votes

**0**answers

248 views

### Quasi-classical limit of representation theory

I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated ...

**4**

votes

**3**answers

280 views

### Topological properties of $K$ orbits in $G/B$

I'll be working over the complex numbers.
Let $G$ be a connected reductive group, $\theta\colon G\to G$ an involution. Let $K=G^{\theta}$ be the fixed point subgroup. I am trying to track down ...

**8**

votes

**1**answer

331 views

### Action of the endomorphism monoid on an irreducible GL-module

Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...

**5**

votes

**3**answers

223 views

### Invariants in $S^n(S^k(\mathbb{C}^w)$

Are the formulas for the multiplicity of $SL(w)$ invariants in $S^n(S^k(\mathbb{C}^w)$ known? This is a very classical topic. If no, in what ranges one can compute it (for certain paramaters fixed - ...

**10**

votes

**2**answers

279 views

### Invariant differential operators on real Grassmannians

I am looking for an explicit description of the algebra of $SO(n)$- or, better, $O(n)$-invariant differential operators on the real Grassmann manifolds of $k$-dimensional linear subspaces in the ...

**0**

votes

**1**answer

88 views

### reference help: irreducible implies admissible

Let $G$ be a reductive p-adic group, $\pi$ a complex smooth representation of $G$. Then it is known that if $\pi$ is irreducible, then it is admissible.
I need help to find a reference for this fact, ...

**2**

votes

**0**answers

201 views

### Representations of the orthogonal group O(n) vs representations of the special orthogonal group SO(n), over an arbitrary field

Let $O(n)$ and $SO(n)$ denote the split orthogonal linear algebraic group and its special subgroup, over some fixed field of characteristic not two.
I am looking for a reference that explains how to ...

**3**

votes

**2**answers

243 views

### Simple representations of products of algebraic groups

I am looking for a reference for the following assertion that I believe to be true. All representations are assumed to be finite-dimensional.
Let $G_1$ and $G_2$ be affine algebraic group schemes ...

**1**

vote

**1**answer

190 views

### Holomorphic representations of complex reductive Lie groups and the boundary of orbits (Reference request)

I have difficulties finding an appropriate reference for the following question (which I hope that it to be true).
Let $U$ be a compact Lie group, $G:=U^{\mathbb{C}}$ its complexiﬁcation and $\tau: ...

**3**

votes

**0**answers

58 views

### Do purification and equivariantization commute?

Suppose that we have an action of a group $G$ on a (quasi-)Hopf algebra $H$, so that we can construct $H\rtimes G$ as in Majid's Cross Products by Braided Groups and Bosonization. It is known that ...

**0**

votes

**0**answers

68 views

### Existence of special pants decompositions for non-elementary representations into PSL(2,R)

A Theorem by Gallo, Goldman and Porter states the following:
Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation ...

**8**

votes

**2**answers

484 views

### Isomorphism between Spin(3,2) and Sp(4, R)

I've been using the fact that Spin(3,2) is isomorphic to Sp(4, R) for a while, but I've never seen a proof. Can anyone point me in the direction of a good reference?

**6**

votes

**3**answers

314 views

### Representation rings of exceptional Lie groups

Let $G$ be a compact Lie group and let $R(G)$ denote its complex representation ring. If $G$ is simply connected, such as $G_2$, $F_4$ or $E_8$, then it is known that $R(G)$ is a polynomial ring [F. ...

**8**

votes

**2**answers

367 views

### Proving that some principal series representations of SL(2,F) are irreducible

I am sorry in advance if this question is not "research level".
Let $F$ be a p-adic field.
I saw, in Bumps book, a proof which I liked, showing which principal series representations of $GL(2,F)$ ...

**5**

votes

**1**answer

234 views

### The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

I asked this question on Math.Stack but have not had any answers.
Question
What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group, $A$?
The trivial ...

**5**

votes

**0**answers

211 views

### A question about equivariant derived categories and [BBD]

Let $G$ be an algebraic group (over $\mathbb{C}$) acting algebraically on a variety $X$. Bernstein and Lunts then define in [BL94] the equivariant derived category $D^b_G(X,\mathbb{C})$ (of ...

**10**

votes

**0**answers

321 views

### A question about multiplication in $G(\mathbb{C}((t)))$ and Affine Grassmannians

I am sorry to give a bounty to such a crappy question but an answer would help me a lot.
I am stuck with the following simple (i guess but) technical problem.
Let $G$ be a complex reductive ...

**1**

vote

**2**answers

138 views

### References request: representations of Heisenberg algebra.

Let $p_1, p_2, \ldots$, be the power sum symmetric functions. Let $p_n^* = n \frac{\partial}{\partial p_n}$. Then $$ p_n^* p_m - p_m p_n^* = \delta_{m, n} 1. $$
Where could I find this result in some ...

**5**

votes

**1**answer

330 views

### Wrong-way Frobenius reciprocity for finite groups representations

This is a typical lazy mathematician question, so do not hesitate to close it and recommend me to do my homeworks...
Let $H$ be a subgroup of a finite group $G$, and let $Res_H^G$ and $Ind_H^G$ the ...

**10**

votes

**2**answers

274 views

### Temperley-Lieb algebras for other Weyl groups?

The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen ...

**3**

votes

**3**answers

355 views

### Decomposition into irreducibles of symmetric powers of irreps.

Suppose I have an irreducible representation of a simple Lie algebra, say $\mathfrak{sl}(n)$ or $\mathfrak{so}(n)$ i.e., $A$ and $D$ type. Given such a representation, $\Gamma_\lambda$, indexed by its ...

**0**

votes

**1**answer

116 views

### Free resolution for Lie algebras (reference)

What is a reference for the subject of "free resolutions for Lie algebras"?
Does the term "standard resolutions" means "free resolutions"?
What is a "bar resolution"?
Is there only one way to talk ...

**2**

votes

**1**answer

236 views

### How to detect if a subgroup lands inside an orthogonal group?

Equivalently, my question may be phrased as, "Are there defining characteristics of representations of orthogonal (symmetric form-preserving) groups?"
Here I am working with a unitary representation ...

**2**

votes

**1**answer

118 views

### Reference request : dimensions of real representations of Lie groups

What is a good reference to learn about real representations of Lie groups ? I've parsed through the very enlightening book of Fulton and Harris, but it is extremely (if not exclusively) ...

**7**

votes

**2**answers

314 views

### Good effective versions of theorems of Artin and Brauer

The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups.
For example, Artin's theorem is the statement that for every character $\chi$ of ...

**3**

votes

**1**answer

136 views

### Plancherel formula for non-second-countable (non-unimodular) groups

The Plancherel formula for unimodular, second-countable, type 1 groups can be found in A Course in Abstrict Harmonic Analysis by Gerald Folland (theorem 7.44) or here. It states that we can get a ...

**8**

votes

**0**answers

389 views

### Reference/quote request: “All of combinatorics is the representation theory of $S_n$”

I think I remember reading somewhere a glib (or is it deep?) quote, perhaps due to Rota?, which was something like
"All of combinatorics is essentially [or can be reduced to?] the representation ...

**0**

votes

**1**answer

90 views

### Continuation of homomorphisms of representations…

Hi all.
If $G$ is a finite group and $\varrho : G \to \text{GL}(V), \eta : G \to \text{GL}(W)$
are finite dimensional representations, $V_0$ is a $G$-invariant subspace of $V$
and $f : V_0 \to W$ is ...

**4**

votes

**4**answers

508 views

### A catalog of faithful representations of finite groups?

I want a reference that catalogs the smallest-dimensional faithful representation of every noteworthy finite group. Specifically, I want representations on $\mathbb{R}^n$ and $\mathbb{C}^n$.
Where ...

**11**

votes

**0**answers

332 views

### Source of a formula for tensor product multiplicities?

This is a follow-up to a recent question by Allen Knutson here, involving a special type of tensor product multiplicity for a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (or other ...

**4**

votes

**1**answer

191 views

### Expression of basis vectors of permutation modules in different bases.

This is a cross-post from math.se, because I did not get any answer there:
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that ...

**0**

votes

**0**answers

121 views

### The use of $Ext^{1}_A(M, N)$.

In the paper Prime Representations from a Homological Perspective. The authors show that $Ext^{1}_{\hat{\mathcal{F}}}(V, V)$ is one-dimensional if and only if $V$ is prime for some modules $V$ of ...