**1**

vote

**0**answers

8 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 $F \in G_k$ be a Frobenius element, so that $G^F$ is a finite group of Lie type.
I'll start with a somewhat vague question and make my ...

**1**

vote

**0**answers

56 views

### The Irreducible Representations of the Sekine Quantum Groups

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

**7**

votes

**2**answers

221 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 ...

**3**

votes

**0**answers

51 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 ...

**0**

votes

**0**answers

64 views

### Showing a permutation module is reducible [on hold]

CROSS POST
For a permutation module $V$, which is a permutation module if it has the basis $B = \{v_1,...,v_n\}$ such that the matrix of every $g \in G$ with respect to this basis is a permutation ...

**7**

votes

**1**answer

141 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 ...

**2**

votes

**1**answer

120 views

### Sampling from random totally unimodular matrices of a particular type?

Is there a way to parametrize totally unimodular $(3n+2)\times(2n+2)$ matrices of form
$$\begin{bmatrix}
\pm1 & \pm1 & 0 & 0 &\dots & 0 & 0 & 0 & 0\\
A_{2n} & ...

**5**

votes

**0**answers

193 views

### Character tables of the p-core of the binary modular congruence group of p-power level

Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See The characters of binary modular congruence group, Bulletin of the
American Mathematical Society. 79 (1973), no. 4.), ...

**6**

votes

**1**answer

342 views

### In general, are 'Young symmetrisers' given by Littlewood-Richardson 'Orthogonal projection Operators'?

Consider $V^{\otimes n}$ where $V$ is vector space and the representation of GL(V) acting in the usual way. Now if I consider tensor products or plethysms of irreducible spaces, this is not in general ...

**0**

votes

**1**answer

72 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.

**4**

votes

**0**answers

88 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 ...

**7**

votes

**2**answers

493 views

### embedding of local tempered representation into cuspidal automorphic representation

Let v be a finite place of a number field F. Let $\pi_{v}$ be an irreducible tempered representation of $ GL_{n}(F_v)$. Is it true that we can find some irreducible cuspidal automorphic representation ...

**3**

votes

**1**answer

177 views

### Reference request - Jacquet module and asymptotic of matrix coefficients

Hello,
I would like to know some nice references about the relation between asymptotics of matrix coefficients of representations of reductive groups over local fields, and the pairing between the ...

**3**

votes

**0**answers

115 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 $U_q(sl_n)$ ...

**8**

votes

**2**answers

196 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 ...

**3**

votes

**1**answer

699 views

### Special automorphisms of extraspecial groups

Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...

**2**

votes

**1**answer

284 views

### Two questions about Whittaker functions

I am watching the video: Modeling p-adic Whittaker functions, Part I. I have two questions about Whittaker functions in the video.
From 33:00 to 37:00, it is said that after changing of variables, ...

**7**

votes

**1**answer

224 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 ...

**5**

votes

**1**answer

478 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: $\mathcal{L}(H\subset G)$ ...

**8**

votes

**0**answers

70 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 \ ...

**1**

vote

**0**answers

244 views

### quantization for integral coadjoint orbits

I am looking for a referrence for proof of following statement.
Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ be its Lie algebra and $\alpha_k$ be simple roots of $\mathfrak{g}$ and $\mu_0$ ...

**6**

votes

**1**answer

223 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?

**6**

votes

**1**answer

497 views

### A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...

**2**

votes

**0**answers

29 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 ...

**3**

votes

**0**answers

138 views

### Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$

Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...

**8**

votes

**0**answers

353 views

### On cyclic homology of Ginzburg's DG algebra

Let $(Q,W)$ be a quiver with potential, and $D$ be Ginzburg's DG algebra associated to it (as explained in http://arxiv.org/abs/math/0612139 and other places), so that it is a 3-Calabi-Yau algebra. I ...

**7**

votes

**1**answer

124 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 ...

**1**

vote

**1**answer

123 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 ${\mathfrak sl}_n$. ...

**0**

votes

**1**answer

53 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 ...

**4**

votes

**0**answers

192 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 ...

**4**

votes

**0**answers

117 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 ...

**21**

votes

**7**answers

4k views

### Why are the characters of the symmetric group integer-valued?

I remember one of my professors mentioning this fact during a class I took a while back, but when I searched my notes (and my textbook) I couldn't find any mention of it, let alone the proof.
My best ...

**1**

vote

**1**answer

67 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 ...

**1**

vote

**2**answers

178 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 ...

**6**

votes

**1**answer

202 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 ...

**18**

votes

**1**answer

273 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, ...

**37**

votes

**11**answers

5k views

### Why is the exterior algebra so ubiquitous?

The exterior algebra of a vector space V seems to appear all over the place, such as in
the definition of the cross product and determinant,
the description of the Grassmannian as a variety,
the ...

**13**

votes

**1**answer

351 views

### Hyperbolic polynomials and group representations

Recall that a homogeneous polynomial $P\in{\mathbb R}[X_1,\ldots,X_d]$ of degree $n$ is hyperbolic in the direction of a vector $V\ne0$ if for every vector $W$, the univariate polynomial $t\mapsto ...

**5**

votes

**1**answer

108 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 ...

**2**

votes

**1**answer

71 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 ...

**5**

votes

**6**answers

1k views

### Geometric proof of Robinson-Schensted-Knuth correspondence?

Famous Robinson Schensted Knuth correspondence gives a correspondence between the matrices with non-negative integer entries and pair of semi standard tableaux. The proof that I have seen is highly ...

**6**

votes

**2**answers

239 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).
...

**4**

votes

**1**answer

222 views

### An equivalence of derived categories by Happel-Reiten-Smalø

I have a problem in understanding the proof of a theorem by Happel-Reiten-Smalø. The original reference is this article
http://arxiv.org/abs/0911.4473
.
I write down the text of the theorem and a ...

**2**

votes

**0**answers

67 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

**1**answer

2k views

### Irreducible representations of Heisenberg algebra

I need some references for irreducible representations of the Heisenberg algebra with three generators or the category of its finite length modules.
Here, the Heisenberg algebra with three generators ...

**1**

vote

**1**answer

114 views

### How does an element $T\left(z\right)$ act on a $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)\left[\left[z\right]\right]$-module?

Context
Let $V$ be a 2-dimensional evaluation representation of the quantum loop algebra $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)$ with $a=q$. Also, for $m\in\mathbb{Z}$, the ...

**1**

vote

**1**answer

84 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

170 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 ...

**4**

votes

**2**answers

302 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 ...

**2**

votes

**0**answers

61 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 ...