**2**

votes

**0**answers

60 views

### Alternating elements in free graded-commutative algebras

It is classical that every alternating polynomial is (uniquely) the product of a symmetric polynomial with the Vandermonde polynomial, in particular the alternating polynomials are a free rank-one ...

**1**

vote

**0**answers

63 views

### Irreducible representations of quantum affine algebras

The finite dimensional simple modules of quantum affine algebras are parameterized by Drinfeld polynomials. Some other modules of quantum affine algebras are studied in the paper. Some infinite ...

**5**

votes

**1**answer

136 views

### Center of quantum affine algebras

Are there some references about the center of quantum affine algebras? I searched on google and only find the paper. In particular, what is the center of $U_q(\widehat{\mathfrak{sl}_2})$. Thank you ...

**3**

votes

**0**answers

80 views

### Quasi-split subgroup

Let $G$ be a split symplectic group over a number field $K$, and let $T \subset G$ be a maximal torus defined over $K$ (not necessarily a split torus). The long roots of $T$ form a ...

**2**

votes

**1**answer

81 views

### Why is the ker-hull-topology on $Irr(A)$ is the discrete topology?

Let $A$ be a C$^*$-algebra. Let $Irr(A)=\{[\pi]: \pi$ is an irreducible representation of A}, here is $\rho\in [\pi]$ if there is an unitary operator $V:H_{\pi}\to H_{\rho}$ such that ...

**5**

votes

**0**answers

93 views

### (Double) Crystal reflection operators on SSYTs

I am not that familiar with the language of crystals, but this is what I know:
Let $SSYT(\lambda, \mu)$ be the set of semi-standard Young tableaux with shape $\lambda$ and weight $\mu$.
There are ...

**5**

votes

**1**answer

154 views

### Regular functions on nilpotent orbits and their covers

Let $G$ be a complex semisimple algebraic group with Lie algebra $\mathfrak{g}$.
In 1989 McGovern described the structure (as $G$-module) of the ring of regular functions on a finite cover of the ...

**2**

votes

**0**answers

136 views

### Properties of Higman's group

The infinite group of Higman which has no finite quotient is given by the presentation (with 4 generators and 4 relations):
$$
G = \langle a_i, i \in \mathbb{Z}/4\mathbb{Z} \mid a_ia_{i+1 \,(\text{mod ...

**9**

votes

**0**answers

104 views

### Failure of surjectivity in Hotta-Springer specialization: examples for special unipotents?

Last weekend's workshop on Springer theory and its generalizations at UMass demonstrated how far the subject has expanded over four decades, but the original set-up for the Springer correspondence ...

**4**

votes

**1**answer

151 views

### Sign of 3j symbol (in view of interpolation)

Question
Is there a closed formula for the sign of a 3j symbol?
Context
I need to compute Wigner 3J symbols/Clebsch–Gordan coefficients,
$$
\begin{pmatrix}
\ell_1 &\ell_2 &\ell_3\\
...

**10**

votes

**1**answer

133 views

### Growth of dimension of fixed spaces in $GL_n(\mathbb{Q}_p)$-representations

Let $\pi$ be a generic irreducible admissible representation of $GL_n(L)$, where $L$ is a $p$-adic field, $R$ is its ring of integers, and $\mathfrak{p}$ is its prime ideal. The conductor of $\pi$ ...

**4**

votes

**1**answer

101 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

260 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?
$$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Z}_{l})$$

**4**

votes

**0**answers

72 views

### positions of polyhedrons with vertices on the unit sphere

Let $S^2$ be the unit $2$-sphere canonically embedded in $\mathbb{R}^3$. Let $P$ be a polyhedron whose all vertices are in $S^2$. Let $\text{Iso}(S^2)$ be the isometry group of $S^2$ and ...

**5**

votes

**2**answers

168 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

103 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

212 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

186 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

187 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

83 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 ...

**6**

votes

**0**answers

109 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

151 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 ...

**6**

votes

**1**answer

121 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

201 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

328 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

166 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

98 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

130 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

136 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

249 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

209 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

34 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

239 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

138 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

61 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 ...

**1**

vote

**1**answer

134 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

236 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

125 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

75 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

409 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

209 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

109 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

150 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 ...

**2**

votes

**0**answers

69 views

### actions of the hyperoctahedral group

I am looking for actions (i.e., permutation representations) of the hyperoctahedral group $H_n$ studied in the literature, i.e., homomorphisms from $H_n$ to $\mathfrak S_X$, the set of bijections from ...

**1**

vote

**2**answers

193 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

156 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

115 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

178 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

247 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

65 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 ...