**5**

votes

**0**answers

135 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

104 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

368 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

301 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

158 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 ...

**2**

votes

**0**answers

246 views

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

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

**3**

votes

**0**answers

48 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

107 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

176 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

170 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

324 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

433 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

220 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

164 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 ...

**1**

vote

**1**answer

123 views

### Isomorphisms of Positive and Negative Spinor Bundles

Here is an extract of the doctoral thesis of C. Lewis under the supervision of D. Joyce (https://people.maths.ox.ac.uk/joyce/theses/LewisDPhil.pdf, 1998):
2.6 Spin Bundles and the Dirac Operator
...

**4**

votes

**1**answer

184 views

### Continuous-piecewise-linear versus piecewise-linear

Some authors use the term "continuous piecewise-linear" where other authors use the shorter term "piecewise-linear" (with continuity tacit).
I'd be interested in people's thoughts about this ...

**2**

votes

**0**answers

114 views

### Invariant generalized sections of dual vector bundles

Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My ...

**2**

votes

**1**answer

79 views

### Is a matrix element of a norm continuous representation always a trigonometric polynomial?

I asked a similar question for the case of compact groups not long ago in math.stackexchange. Now I understand that the answer was "yes", and I want to modify that question. This is also related to my ...

**6**

votes

**1**answer

131 views

### Trigonometric polynomials on non-compact and non-abelian groups

I asked this initially in math.stackexchange, but it disappeared almost immediately, so I hope it will be proper to aks this here.
Hewitt and Ross define trigonometric polynomial on a locally compact ...

**6**

votes

**3**answers

470 views

### Identity for Power Series and Binomial Coefficients

This question concerns a combinatorial identity obeyed by power series coefficients. Throughout we let $[x^{M}]\{\phi(x)\}$ denote the coefficient of $x^{M}$ in a power series $\phi(x)$.
Let $k$ be ...

**2**

votes

**1**answer

161 views

### Rankin-Selberg convolution and product of degrees

As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as ...

**3**

votes

**1**answer

134 views

### the number of indecomposable modules of finite groups over finite fields of a fixed dimension

I am interested in determining the the number of indecomposable modules of finite groups over finite fields of a fixed dimension. Specifically, I have the following conjecture:
Conjecture. Suppose we ...

**3**

votes

**1**answer

131 views

### Do the following two filtrations of the affine Grassmannian agree?

Let $H = L^{2}(S^{1},\mathbb{C}^{n})$, $H_{0}\subseteq H$ the subset of maps that extend holomorphically to the unit disc, and $H_{m} = z^{m}H_{0}$. Consider the affine Grassmannian for $GL_{n}$ in ...

**1**

vote

**0**answers

65 views

### research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\pi_j\circ \pi_i^{-1}$

Is there any research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\pi_j\circ \pi_i^{-1}$, where $\{\pi_1,\pi_2,...,\pi_{k!}\}=S_k$?
I know if we apply the ...

**6**

votes

**2**answers

243 views

### Global Affine Flag Variety and Affine Flag Variety

There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the ...

**4**

votes

**1**answer

333 views

### Tannakian fundamental group of two explicit tensor categories

Let $K/k$ is a field extension and $G$ an affine group scheme over $K$. What are the Tannakian fundamental groups of these two $k$-tensor categories (with trivial fiber functors over $k$):
1. The ...

**0**

votes

**1**answer

91 views

### Graph lifts and representation theory

Is there any connection known between the two?
One can naturally define lifts of graphs by groups like $\mathbb{Z}_k$ and hence I wonder if representation theoretic properties can be used to say ...

**2**

votes

**0**answers

55 views

### Mellin transform of Plancherel measure

Let $G$ be a reductive p-adic group with a chosen Haar measure $dg$. The Plancherel measure is the measure $\mu$ on the set of (tempered) irreducible representations of $G$ such that for any locally ...

**3**

votes

**0**answers

101 views

### Invariant Laurent polynomials under cyclic group action

Start with the cyclic group $G:=\mathbb{Z}/p$ of prime order $p$ and and an integer lattice $P:=\mathbb{Z}^p$. Let $G$ act on $P$ by cyclic permutation of coordinates. There is an induced action on ...

**17**

votes

**2**answers

789 views

### What is modular representation theory for groups good for?

I am an absolute beginner in modular representation theory of finite groups. I know some things in representation theory in characteristic zero. My questions are regarding the main goals of this part ...

**3**

votes

**1**answer

203 views

### Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$

I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over ...

**5**

votes

**1**answer

438 views

### exceptional cases in Kazhdan-Lusztig

The Kazhdan-Lusztig story doesn't apply to the four exceptional cases $(E_6)_1$, $(E_7)_1$, $(E_8)_1$, $(E_8)_2$ (see this earlier question of mine).
What's special about those cases?

**1**

vote

**0**answers

79 views

### Is specht module the intersection of two induced modules?

I heard someone said( maybe Okonov) that specht module is the intersection of two induced modules, but I do not know why.The details of my question is as follows.
Let $\lambda\vdash n$ be a partiton, ...

**4**

votes

**0**answers

134 views

### Number of Irreducible Representations of $U_q(n)$ of Dimension $n$?

For quantum group $U_q(n)$, is it true that it has exactly two non-isomorphic irreducible corepresentations with dimension $n$, and that one is the dual of the other? I know result is in the Chapter ...

**1**

vote

**0**answers

81 views

### representation of $SO(p)\times SO(q)$ with $p,q$ odd

Assume $p,q$ odd. We denote by $\sigma_p$ the standard representation of $SO(p)$, that is the representation of $SO(p)$ acting on $\mathbf{R}^p$ as matrix. So is $\sigma_q$.
Take $K=SO(p)\times ...

**5**

votes

**2**answers

221 views

### Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$

This question follows up a question I asked on math.SE. This is a refinement and a reference request.
For what groups $G$ does there exist a $Z(G)$-extension of $\operatorname{Aut}G$ (call it ...

**5**

votes

**1**answer

193 views

### Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$

Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel.
Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and ...

**1**

vote

**2**answers

78 views

### Non-degenerate characters of the unitriangular group $U$

I made a previous post which was unclear and mistaken in fundamental aspects, so that it was actually more worthy making this new post than actually editing the previous one.
I'm studying the ...

**2**

votes

**1**answer

191 views

### highest weight representations inside tensor product

Let $G$ be a semisimple simply connected group over an algebraically closed field $k$ of characteristic zero, $B$ a Borel and $T$ a maximal torus.
Let $\lambda,\mu,\nu$ be dominant characters of $T$.
...

**4**

votes

**1**answer

153 views

### Is the restriction of a representation semisimple?

Let $F$ be local field of characteristic zero and $\pi$ be a irreducible admissible representation of $GL_n(F)$.
Let us consider its restriction to $GL_{n-1}(F)$. Then I want to know whether ...

**1**

vote

**0**answers

84 views

### Compatibility of two definitions of Koszul dual

Let $k$ be a field and $A$ a nonnegatively graded ring over $k$. Assume $A_0 = k.$ We have a bigrading on $\operatorname{Ext}(k,k)$ (one corresponding to homological degree, one corresponding to the ...

**8**

votes

**1**answer

322 views

### Gabriel's theorem over a commutative ring

Is Gabriel's theorem on the indecomposables of representations of quivers of finite type true over a commutative ring, i.e. not necessarily a field?

**0**

votes

**0**answers

58 views

### adjoint quotient and points in DVRs

Let $G$ be a connected reductive group over an algebraically closed field $k$, $T$ a maximal torus and $W$ its Weyl group.
We have a Steinberg map $\chi:G\rightarrow \mathfrak{C}:=T/W$ if we have a ...

**1**

vote

**2**answers

107 views

### irreducibility of certain subspaces of the permutation group in quantum mechanics

Let $P_j$, $j = 1, \dotsc, N!$ be a set of unitary operators constituting a representation of the symmetric group $S_N$, acting in a sub-Hilbert space $V_0 \subseteq H$ (of a separable Hilbert space ...

**1**

vote

**0**answers

118 views

### Young symmetrizers question

Let $\lambda$ be a partition of $n$, and let $T$ be the standard tableau associated to $\lambda$ (write the Young diagram of $\lambda$ down and fill in the boxes with $1$ through $n$ left to right, ...

**2**

votes

**1**answer

87 views

### Rational Points of a Quotient of a Reductive Group by a Parabolic Subgroup

Let $G$ be a reductive group and let $P$ be a parabolic subgroup of $G$ all defined over $\mathbb{Z}$.
Also, let $F$ be a number field, is it true (and if so, please provide a reference) that
$$ ...

**3**

votes

**1**answer

165 views

### Are norm-continuous representations smooth?

Let $G$ be a real Lie group and $A$ a unital Banach algebra. Let us call a map $\varphi:G\to A$ a (norm-)continuous representation, if it is continuous
$$
x_i\to x\quad\Longrightarrow\quad ...

**3**

votes

**1**answer

160 views

### How to think about the simple reflection s_0 in the affine Weyl group?

Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...