**2**

votes

**0**answers

133 views

### Octonions product: inversion in the right and identity in the left

Once octonions product is studied, together with the relations with $Spin(8)$ and $SO(8)$ geometry (see for instance Robert Bryant's notes), one realises that the key fact bringing all the phenomena ...

**1**

vote

**1**answer

206 views

### Regular embeddings of reductive groups

A regular embedding of a connected reductive linear algebraic group $G$ defined over $\mathbb{F}_q$ is a morphism $\varphi : G \rightarrow G'$ of algebraic groups which is a closed immersion where ...

**8**

votes

**3**answers

230 views

### How do small central extensions drop the dimension of a faithful representation?

Apologies in advance that this is a very soft question. I might be talking complete nonsense. But I hope I am talking about something that has even been studied...
I am interested in the phenomenon ...

**5**

votes

**0**answers

104 views

### Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B

This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...

**5**

votes

**0**answers

73 views

### How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...

**0**

votes

**0**answers

98 views

### Anticommuting operators with positive properties

Which classes of $M\in \mathsf M_k(\Bbb R)^{n\times n}$ admit solutions $N\in \mathsf M_k(\Bbb R)^{n\times n}$ such that
$$(M\otimes N+N\otimes M)(u\otimes u)=0$$ forall $u\in \mathsf D_k(\Bbb ...

**2**

votes

**0**answers

79 views

### Adding a row to a Young Tableau via Novelli-Pak-Stoyanovskii

Let $T_{\lambda}$ be the set of standard young tableaux (SYT) of shape $\lambda_1\geq \lambda_2\cdots\geq \lambda_n$. Now consider pushing a row $\mu$ with $\mu\geq \lambda_1$ onto $Y$ to give shape ...

**5**

votes

**0**answers

96 views

### Wavefront sets of irreducible representations with non-integral infinitesimal characters

Let $G$ be a complex reductive algebraic group (connected, simply connected etc), viewed as a real group. We study the representations of $G$, and we follow the notations in the paper of Barbasch and ...

**2**

votes

**1**answer

219 views

### Decomposing a reducible representation of the unitary group

Consider the representation $L_U$ of the unitary group $U(n)$ on $L(\mathbb{C}^n)$ where $L_U$: $L(\mathbb{C}^n) \rightarrow L(\mathbb{C}^n)$ is a linear operator that $L_U M=U M U^{\dagger} $, ...

**0**

votes

**0**answers

63 views

### Maximum symmetry of generic 6j symbols?

I can't get those three papers collinear...
The "classic" general reference is Butler (including conjugates and multiplicities). An example from a spherical category is Hong, an example from a ...

**2**

votes

**0**answers

92 views

### What's the relation of the Hecke algebra of a pair and the flag variety?

Let $G$ be a real semisimple Lie group and $K$ a maximal compact subgroup. Let $\mathfrak{g}$ and $\mathfrak{k}$ be the complexified Lie algebra of $G$ and $K$, respectively.
Then the Hecke algebra ...

**4**

votes

**2**answers

298 views

### Concise mathematical definition of the fusion product on the Verlinde ring?

The Verlinde ring of a (let us say) simply connected simple compact Lie group has as underlying additive group the Grothendieck group of representations of the central extension $\widehat{LG}$ of the ...

**1**

vote

**1**answer

74 views

### Explicit deformations of pseudo representations

Let $G$ be a group (which I will be glad to consider to be the absolute Galois group of a $p$-adic field, and so satisfies Mazur's finiteness condition which appears in his paper Deforming Galois ...

**2**

votes

**0**answers

123 views

### Free-field representations: how to study highest-weight submodules of the Fock module?

Suppose we have a representation of some affine Lie algebra $\mathfrak{g}=\mathfrak{n}_- \oplus \mathfrak{h} \oplus \mathfrak{n}_+$ on a Fock space $V$. The module $V$ will contain a lot of ...

**2**

votes

**1**answer

294 views

### Canonical representation of $\operatorname{SL}(2,\mathbb{R})$ on $L^2(\mathbb{R}^2)$

As a unimodular subgroup of the group of automorphisms of $\mathbb{R}^2$, $\operatorname{SL}(2,\mathbb{R})$ can be represented as a subgroup of $\mathcal{U}(L^2(\mathbb{R}^2))$ (the group of unitary ...

**2**

votes

**1**answer

117 views

### cohomology of orthogonal (or general linear) group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let
$$
O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\}
$$
What is $$
...

**6**

votes

**1**answer

201 views

### In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?

Question: In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective?
Recall that an object ...

**4**

votes

**0**answers

134 views

### Counting points on Hessenberg varieties over a finite field

Let $G$ be an connected reductive group over finite field $k$. I will assume that $\text{char}(k)$ is very good for $G$ (or even larger, if preferred). Let $B\subset G$ be a Borel subgroup defined ...

**2**

votes

**0**answers

54 views

### explicit matrices for Weil ($p^2$ dimensional) representation of $Sp(4,\mathbb{F}_p)$, $p>3$

I am looking for more-or-less explicit matrices for the $p^2$ dimensional Weil representation of $Sp(4,\mathbb{F}_p)$, suitable for computer implementation. Ideally, I would like the images of the ...

**17**

votes

**3**answers

615 views

### Center of a simply-connected simple compact Lie group and McKay correspondence

Let $G$ be a simply-connected simple compact Lie group. Its center $Z(G)$ is a finite abelian group, say $Z(G) = \mathbb Z/k\mathbb Z$ for $G=SU(k)$.
I find the following interpretation of $Z(G)$ in ...

**3**

votes

**0**answers

65 views

### Linear independence of points under projection of Veronese re-embedding

Let $V$ be a complex vector space.
Let $x_1,...,x_k\in PV$. Let $v_d: PV\rightarrow PS^dV$
be the Veronese. Then $v_d(x_1),...,v_d(x_k)$ are in general linear position
as long as $k\leq d-1$.
Now let ...

**1**

vote

**1**answer

172 views

### cohomology of orthogonal group of integers

Let
$$
O(\mathbb{Z}^{\oplus k})=GL(\mathbb{Z}^{\oplus k})\cap O(k).
$$
What is $$
H^*(BO(\mathbb{Z}^{\oplus k});\mathbb{Z})?
$$
If it cannot be computed out, can we get
$$
H^*(O(\mathbb{Z}^{\oplus ...

**0**

votes

**0**answers

67 views

### On continuous part of the L^2 spectrum

Suppose $G$ is a real reductive Lie group and $\Gamma$ is a lattice in $G$ (of finite co-volume). I am reading Langlands's paper " On the functional equation satisfied by the Eisenstein series". I ...

**3**

votes

**1**answer

259 views

### Number of semi-standard tableau

What is the number of semi-standard tableau (weakly increasing on rows and strictly increasing on columns) for the partition $2n=n+n$ with entries $\{1,2, \cdots ,n\}$ such that each $i$ appears ...

**2**

votes

**1**answer

300 views

### Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism:
$$g_Y \colon Y \times_X Y \cong Y ...

**0**

votes

**0**answers

81 views

### Central extensions of SL2(R) by U(1) ?

Can somebody please tell me what are the central extensions of SL2(R) by U(1), that is, what is $H^2(SL2(R), U(1)) $ ? Thank you

**2**

votes

**1**answer

167 views

### Relation between Different Definitions of Induced Representation

I've seen two different ways to define induced representation.
One is as in the book Introduction to representation theory: If $G$ is a group, $H$ is a subgroup of it, and $V$ is a representation of ...

**2**

votes

**0**answers

70 views

### Dimension of affine Springer fiber and its functor of points as an ind-scheme

Let $k$ be a finite field and let $F = k( (t))$ with ring of integers $\mathfrak{o} = k[ [t]]$. Let $G$ be a connected linear algebraic $k$-group with Lie algebra $\mathfrak{g}$. Suppose that ...

**6**

votes

**1**answer

235 views

### Jordan decomposition of the tensor product of two matrices

I asked this question on Math.SE here, but did not get a lot of attention.
I am interested in the problem of determining the Jordan decomposition of the tensor product of two unipotent matrices over ...

**8**

votes

**1**answer

321 views

### Automorphisms of generic complete intersections

This question concerns a seemingly folk lore result, which states that automorphism groups of generic complete intersections are trivial, under certain assumptions.
To state the question, let $r \geq ...

**6**

votes

**0**answers

223 views

### What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...

**2**

votes

**1**answer

103 views

### Projection and representation

Suppose that $\rho : G \longrightarrow U_n(\mathbb C)$ is an irreducible representation of group $G$. Suppose that $P$ is a projection of $\mathbb C^n$ into a subspace of small codimension (i.e. of ...

**3**

votes

**1**answer

170 views

### Which group algebras in analysis are “true group algebras”?

Let $G$ be a group, $A$ a unital associative algebra over ${\mathbb C}$, and let us call a representation of $G$ in $A$ an arbitrary map $\pi:G\to A$ such that
$$
\pi(1)=1,\qquad \pi(a\cdot ...

**1**

vote

**0**answers

115 views

### counting how many boxes from a given Young tableau contribute to hook length made out of two YTs

Think of a Young diagram as a collection of rows with numbers of elements
$\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ ...

**5**

votes

**0**answers

139 views

### Euler Subgroups and Automorphic L-functions

Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...

**2**

votes

**2**answers

106 views

### admissible characters for $PGL_{2}(F)$

What are the irreducible admissible representations of $PGL_{2}(F)$ for $F$ a local nonarchimedean field and do we have formulas for their characters?

**3**

votes

**3**answers

380 views

### classifying space and cohomology of integer general linear group

I have obtained that the classifying space
$$
BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty)
$$
is the Grassmannian.
I have also obtained that the mod 2 cohomology is the polynomial ...

**5**

votes

**0**answers

114 views

### LS paths construction

Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on ...

**3**

votes

**2**answers

313 views

### symmetric 2-cocycle / many projective representations

Let $G$ be a finite group, $k$ the field of complex numbers.
Are there (cohomologically nontrivial) group 2-cocycles $\sigma\in Z^2(G,k^\times)$ such that for all $g,h\in G$:
...

**-2**

votes

**1**answer

162 views

### Is there any Lefschetz-like principle for representations of finite groups?

Representation theory (at least the origin of this terminology) aims to exhibit a model (a represetative) in the group of matrices for an abstract group which is known by only its group law. So ...

**3**

votes

**0**answers

322 views

### A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups.
This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved:
Theorem: $\mathcal{L}(H\subset G)$ ...

**3**

votes

**0**answers

50 views

### Signs associated to self-dual simple objects in a fusion category

Every self-dual simple object $X$ in a fusion category can canonically be assigned a number $a$, from its "snake" associator element:
The square of $a$ equals Muger's "squared dimension" of $X$, an ...

**0**

votes

**0**answers

110 views

### Weyl group representation

Let $G$ be a reductive p-adic group.
Let $W$ be a weyl group. if $w$, and $w_o \in W$.
I want to know in which case we have $w w_o w^{-1}= w_o$ ?
in case if $w_o(\theta)=\theta $ where $\theta$ is a ...

**5**

votes

**0**answers

180 views

### Conjugation of the quotient of $SL(n,\mathbb{C})$ by a finite subgroup

EDITED Let $G={SL}_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$.
Let $H\subset G$ be a finite subgroup.
Set $X=G/H$ be the corresponding homogeneous space, it is a complex variety.
...

**2**

votes

**1**answer

176 views

### An expectation of the product of random unitaries

I want to find the answer of
$$\int dU \ U^m X \ U^{\dagger m}$$
Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint ...

**2**

votes

**0**answers

41 views

### “Prime” fusion rings

Surely this concept is known! (But I don't recall seeing it - maybe under another name? But "prime" is the obvious name choice.)
Example. Open the Gepner/Kapustin paper at ...

**5**

votes

**2**answers

346 views

### Expectation of trace of nth power of unitary matrices

I am trying to find the answer of
$$\int dU \ |Tr(U^m)|^2$$
where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...

**19**

votes

**1**answer

440 views

### How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity
To ...

**4**

votes

**0**answers

227 views

### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...

**1**

vote

**0**answers

167 views

### What is the spectrum of $L^1(G:H)$?

Let $H$ be a compact subgroup of a locally compact topological group $G$ and
$$ L^1(G:H)=\{f\in L^1(G): R_h f=f\;\text{ a.e. }\; \forall h \in H\}$$ and $\widehat{(G:H)}=\{\xi\in ...