**4**

votes

**1**answer

106 views

### Invariant regular cones in Lie group representation

I am following Analysis and Geometry on Complex Homogeneous Spaces by Faraut et al. I'll set up all of what I need and then ask my questions.
Let $G$ be a connected semi-simple non-compact real Lie ...

**1**

vote

**1**answer

272 views

### Why is every l-adic Galois representation conjugate to one over the l-adic integers? [closed]

Why is every l-adic Galois representation
$$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Q}_{l})$$
conjugate to one over the l-adic integers?
$$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Z}_{l})$$

**5**

votes

**2**answers

178 views

### Lefschetz Principle for semisimplicity

I think I can prove the following using the compactness of first order logic and I am wondering what a purely algebraic proof would look like.
Let $R$ be a unital ring (not necessarily ...

**4**

votes

**1**answer

105 views

### references for faithful orthogonal (or unitary) representation of symmetric groups

Let $S_n$ be the symmetric group of $n$ points. I want to find references (or proofs) for the following statement (1).
(1). There does not exist any faithful orthogonal representation
$$
...

**7**

votes

**2**answers

223 views

### Schur polynomial, change of variable

Let $k=(k_1,k_2,k_3,k_4)\in \mathbb{N}^4$ and let $s_k(x_1,x_2,x_3,x_4)$ be the Schur polynomial on $GL_4$.
Question 1: If I replace $x_3$ with $x_1$ and $x_4$ with $x_2$, can $s_k(x_1,x_2,x_1,x_2)$ ...

**9**

votes

**0**answers

205 views

### Calculation-free proof of the Weyl Integral formula for U(n)

The Weyl integral formula states that if f is a class function on U(n), T is the torus of diagonal matrices in U(n), and dU(n) and dT are the standard Haar measures on U(n) and T, then
$$\int_{U(n)} ...

**5**

votes

**1**answer

210 views

### Plethysm of $S^3(S^2V)$ as $\mathfrak{sl}_3(\mathbb{C})$-module

I have asked this question in MSE before, but have not got any answer. So here I am asking it again with some more detail.
I believe that the following sequence of ...

**1**

vote

**0**answers

85 views

### What are the E7(7) invariants in the adjoint representation?

Take a real vector space $R$ transforming in the adjoint representation of
the ${\rm E}_7(7)$ Lie group as $R \rightarrow G R G^{-1}$. One can define
invariants using traces of products of $R$ as ...

**7**

votes

**1**answer

188 views

### Operads and the Stable Module Category

I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too.
Let $k$ be a field and $R$ a $k$-algebra. The stable ...

**7**

votes

**1**answer

166 views

### Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations

Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.
I'll start with a somewhat vague question and make my question more specific further down:
How do ...

**7**

votes

**1**answer

130 views

### Equality of codimension under Lusztig-Spaltenstein induction

Pardon if this is well known. Suppose I have a (say complex) connected reductive group $G$ with the $\tilde{\Delta}=\Delta\cup\{\alpha_0\}$ being the simple roots plus the negative highest root. Any ...

**6**

votes

**1**answer

212 views

### The Irreducible Representations of the Sekine Quantum Groups

Here Y. Sekine introduces a one-parameter family of finite quantum groups of dimension $2n^2$. Let $n\geq 3$ be fixed and $\zeta=e^{2\pi i/n}$. Set
...

**8**

votes

**2**answers

340 views

### Confusion about Subcategories of Category $\mathcal{O}$

So, in learning about category $\mathcal{O}$ representations of a semisimple Lie algebra $\mathfrak{g}$, I've come across two natural kinds of subcategories, and I think I'm confused about their ...

**7**

votes

**1**answer

174 views

### Groups without property (T) but all finite quotients are expanders

What is an example of a group $G$ which
1- is finitely generated by $S$,
2- does not have property (T),
3- admits infinitely many finite quotients which do not factor through an homomorphism $G ...

**4**

votes

**0**answers

107 views

### Exotic 2-adic lifts of mod $2$ Steinberg idempotent

Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of
upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices.
The (conjugate) Steinberg idempotent is defined to be ...

**0**

votes

**2**answers

141 views

### Is the restricted root system of a simple real Lie group irreducible?

As the title asks, is the restricted root system of a simple real Lie group irreducible?
I believe this is true but I need a reference to cite.

**3**

votes

**0**answers

138 views

### Deformation and Representations

Let $\widetilde{U_q(sl_n)}$ denote a deformation of the algebra $U_q(sl_n)$. In particular, $\widetilde{U_q(sl_n)}$ is defined by the same generators and relations and $*$-operations as $U_q(sl_n)$ ...

**6**

votes

**1**answer

264 views

### Do representations of real semisimple algebraic group have to be algebraic?

If $G$ is the real points of a semisimple algebraic group and $\rho:G\to GL(n,\mathbb R)$ is continuous representation. Is $\rho$ an algebraic morphism?

**8**

votes

**2**answers

219 views

### Characters of cuspidal representations

Let $\pi$ be an irreducible cuspidal representation of a semi-simple $p$-adic group $G$. It is well-known that the character of $\pi$ is concentrated in the set of compact elements in $G$.
What is ...

**2**

votes

**0**answers

38 views

### Sum rules for Clebsch-Gordan series

Suppose just for example $A\bigotimes{B}=2C+D^++E^-$ with irreps $A...E$.
You have a dimension sum rule ($2*d_C+d_D+d_E=...$) and a Dynkin index sum rule ($2*i_C+i_D+i_E=...$). If $A=B$, you get ...

**7**

votes

**1**answer

257 views

### What is $\hat{A}=\{[\pi]:\pi$ is a irreducible representation of $A$} ( $A$ is a $C^*$-algebra)?

Let $A=\{f:[0,1]\to M_2(\mathbb{C}): $f continuous and $ f(0)=\begin{pmatrix} f_{11}(0) & 0 \\ 0 & f_{22}(0) \end{pmatrix}\}$ be a $C^*$-algebra with pointwise multiplication, involutions and ...

**7**

votes

**1**answer

159 views

### Intertwiners and Clebsch-Gordan coefficients

Consider two unitary irreducible representations on vector spaces $V_1$ and $V_2$ of a Lie group $G$. For $G$ is compact and $V_1$ and $V_2$ finite dimensional there is a unique decomposition of $V_1 ...

**0**

votes

**1**answer

65 views

### An irreducible Lie algebra module decomposition over a subalgebra

My question in the most simple form:
Let $\mathfrak{g}=\mathfrak{g}_1\oplus \mathfrak{g}_2$ be a direct sum of simple finite-dimensional Lie algebras over $\mathbb{C}$ and let $M$ be a ...

**2**

votes

**1**answer

148 views

### Is it possible to describe the action of the Weyl group on the cohomology of the fibers of the Grothendieck-Springer resolution?

I am confused about the following: can one describe the action of the Weyl group on the cohomology of each fiber of the Grothendieck-Springer resolution? I only need the case of ${\mathfrak sl}_n$. ...

**4**

votes

**0**answers

264 views

### Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let $G$ be a finite group,let $Rep(G)$ denote the category of finite dimensional representations over $\mathbb{C}$. Let $V,W$ be representations of $G$ in ...

**5**

votes

**0**answers

126 views

### Relative invariants of $P\oplus P^*$

Let $P$ be a $\mathrm{GL}(V)$-module, and assume that the decomposition of $P$ into irreducible submodules is known. By a relative invariant of the module $P\oplus P^*$, I mean a homogeneous nonzero ...

**1**

vote

**1**answer

79 views

### Graded category O for for rational Cherednik algebras, but at t=0

The paper [1] introduced the category $\mathcal{O}$ for rational Cherednik algebras $H_{t,c}(W)$. This construction is tailored for the $t=1$ case (equivalently, the $t\neq 0$ case). The general setup ...

**20**

votes

**1**answer

475 views

### Bounding Schur symmetric polynomials on the unit circle

Recall the Schur polynomial in $n$ variables, indexed by the partition $\lambda$, with $\ell(\lambda) \leq n$, is given by
\begin{equation}
s_\lambda(x_1,\ldots, x_n) = a_{\lambda + \delta}(x_1, ...

**4**

votes

**1**answer

217 views

### canonical action of symmetric groups on orthogonal groups

There is a canonical faithful orthogonal representation of the symmetric group $S_{n+1}$, for $n\geq 1$:
$$
S_{n+1}\to O(n)
$$
given as follows.
(1). I regard $O(n)$ as the isometry group of the unit ...

**3**

votes

**1**answer

112 views

### symmetric group of regular polyhedrons

Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example,
$\Delta^2$:
$\Delta^3$:
Let ...

**5**

votes

**1**answer

151 views

### A natural Lascoux-Schützenberger involutions on plane partitions

The Lascoux-Schützenberger involutions, $s_i$, that permute the weight of semi-standard Young tableaux are fairly known.
They satisfy some nice Coxeter relations, for example, if $v$ and $w$ are ...

**11**

votes

**2**answers

387 views

### actions of the hyperoctahedral group

I am looking for actions (i.e., permutation representations) of the hyperoctahedral group $H_n$ (also known as the group of signed permutations) studied in the literature, i.e., homomorphisms from ...

**1**

vote

**2**answers

202 views

### Generic irreducibility of parabolic induction

In J.Bernstein's notes: REPRESENTATION OF P-ADIC GROUPS, he remarked the following result(see P.88):
Let $G$ be a reductive group defined over nonarchimedean local field $F$, $P$ parabolic subgroup of ...

**9**

votes

**0**answers

176 views

### Is an inclusion of finite groups with boolean lattice, linearly primitive?

Let $(H \subset G)$ be an inclusion of finite groups.
Definition: Let $W$ be a representation of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{H}:=\{w \in W \ \vert \ kw=w \ ...

**2**

votes

**2**answers

138 views

### Appropriate Recursion relations for Wigner 3j Symbols

I am attempting to code the Cosmic Microwave Lensed Temperature and Polarisation power spectra from first principles and have been told to code the relevant Wigner 3j symbols using recursion rather ...

**6**

votes

**1**answer

189 views

### Well-understood bases for Grothendieck groups of modular representation categories

Let $\mathfrak{g}$ be a semi-simple Lie algebra.
So in characteristic $0$, the Grothendieck group of a block of category $\mathcal{O}$ is given by the classes of the Verma modules. Unlike the ...

**6**

votes

**2**answers

254 views

### Is this algebra isomorphic to an incidence algebra?

This question is motivated by trying to establish a converse to Theorem 7.8 of our paper.
I have a finite poset $P$ with the following properties:
$P$ has binary meets (and hence a least element).
...

**2**

votes

**0**answers

70 views

### The role of the Vandermonde determinant in representations of affine Lie algebras

I am reading a paper 'Yangians and R-matrices' by Chari & Pressley (1990) and to classify representations for particular quantum groups, they define a "quantum Vandermonde determinant". They also ...

**4**

votes

**0**answers

82 views

### Smooth admissible representations, Hom, tensor and extension of scalars

(Remark: This has previously been posted on math.stackexchange, but I believe it might be suitable for this site as well. ...

**4**

votes

**2**answers

315 views

### Explicit description of the principal block of the symmetric group

Let $k$ be a field of prime characteristic $p$ and $\Sigma_n$ be the symmetric group.
If I have a concrete $k[\Sigma_n]$-module $M,$ how to compute the direct summand corresponding to the principal ...

**1**

vote

**1**answer

163 views

### On the definition of matrix coefficient

As far as I have known, for irreducible admissible representation $\pi$ of $p$-adic group $G$, the matrix coefficient is defined as follows:
For $v\in \pi$ and $w \in \pi ^\vee$, the contragredient ...

**7**

votes

**1**answer

147 views

### Littlewood-Richardson-Type Rule for Restriction from $S_{2n}$ to $S_{2(n-t)} \times (S_2 \wr S_t)$

It is well-known that the Littlewood-Richardson coefficient $c^{\nu}_{\lambda \mu}$ is the number of times the irreducible representation $V_\lambda \bigotimes V_\mu$ of the product of symmetric ...

**3**

votes

**1**answer

105 views

### Character degrees of extensions of 2^B_2(q^2)

The outer automorphism group of the Suzuki simple group
${}^2B_2 (2^{2m+1})$, $m \geq 1$ is cyclic of order $2m+1$ and
is generated by a field automorphism $\varphi$ of order $2m+1$.
For any ...

**4**

votes

**2**answers

169 views

### Questions about $\mathbb{C}[G/U^-]$ and $\mathbb{C}[B]$

Let $G = GL_n$. By algebraic Peter-Weyl theorem, we have
$$
\mathbb{C}[G] = \bigoplus_{\lambda} V_{\lambda} \otimes V_{\lambda}^*,
$$
where $\lambda$'s are dominant weights. Let $U^-$ be the ...

**7**

votes

**1**answer

237 views

### Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra

Definitions
Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a ...

**2**

votes

**0**answers

119 views

### Normal Subgroups of $UT_n(q)$

What is known about normal subgroups of $UT_n(q)$, the group of upper triangular matrices with entries in the finite field $\mathbb{F}_q$ and ones on the diagonal? Is there an interpretation of the ...

**2**

votes

**0**answers

89 views

### An equality of discriminant and resultant divisors

Let $\Phi$ be the root system of a split group $G$ over a field $k$. The differentials $d\alpha$ of the roots define a polynomial called the discriminant
$$\prod_{\alpha\in\Phi}d\alpha$$
on $\mathfrak ...

**9**

votes

**2**answers

311 views

### Fell topology vs. convergence of matrix coefficients

My question is partially inspired by the following discussion:
Topology on the Unitary Dual
Let me remind/explain how the Fell topology is defined (at least I recall the definition which I saw): let ...

**4**

votes

**0**answers

161 views

### Can we drop commutativity assumption?

Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...

**3**

votes

**1**answer

148 views

### Anosov representations and boundaries of (harmonic) maps

Let $\Sigma_g$ be a closed hyperbolic surface and $\rho\colon\pi_1\Sigma_g\to G$ an Anosov representation into a suitable Lie group. By definition of Anosovness, one has a $\rho$-equivariant ...