**2**

votes

**1**answer

94 views

### Irreducible unitary representations of semidirect groups of a discrete abelian group by a discrete group

Recently in a paper we get the following result:
Let a discrete group $\Gamma$ act on a discrete abelian group $G$ by group automorphisms. Every irreducible unitary representation $\pi$ of ...

**1**

vote

**0**answers

147 views

### The representation theory for the fake Heisenberg groups over non-perfect local field

Let $K$ be a local field of characteristic $p$, where $p$ is a prime number greater than 2. In particular, $(x+y)^p=x^p+y^p$ for $x,y\in K$.
The fake Heisenberg group is defined to be
$$
...

**5**

votes

**1**answer

176 views

### What is Jantzen's formula for the determinant of the Shapovalov form in the case of generalized Verma modules?

The best reference I found is
[Kac, Kazhdan '79]
which extends the results of Shapovalov and Jantzen to the case of infinite dimensional Lie algebras.
Theorem 1 of this paper gives the Shapovalov ...

**1**

vote

**0**answers

51 views

### Relating the R-Transform in Free Probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolution of distribution functions, which plays well with the Fourier Transform.
In free (noncommutative) ...

**3**

votes

**1**answer

169 views

### Is every weight of an integrable highest weight module in the Tits cone?

Let $\mathfrak{g}$ be a Kac-Moody algebra with Cartan subalgebra $\mathfrak{h}$, Weyl group $W$, and simple roots and coroots $\alpha_i, \check{\alpha_i}, i \in I$, respectively. Let $L$ be an ...

**0**

votes

**0**answers

47 views

### Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action

I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...

**0**

votes

**1**answer

296 views

### On a claim of Zagier on extending a map to cocycle

Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$_2(\bf{ Z})$' (link, p. 6), studies the action of $\Gamma=PSL_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ ...

**2**

votes

**2**answers

169 views

### Permutation covering of a $G$-lattice

Let $G$ be a finite group.
By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$.
We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis ...

**1**

vote

**0**answers

57 views

### Distiguishing mutant knots

Can an invariant from a quantum Lie algebra ever distinguish mutant knots?
(Maybe if it is "chiral"...whatever that means :-)
(Note that Kauffman abstract tensors/skein equations CAN distinguish ...

**3**

votes

**0**answers

169 views

### Can the product of a simple and a non-simple indecomposable representation be semisimple?

Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but not irreducible (i.e., not ...

**2**

votes

**1**answer

180 views

### Understanding the Weyl Character Formula

Let $G$ be a compact (connected) Lie group with a maximal torus $T$. For each (analytically) integral weight $\lambda$ the Weyl character formula
$$\Theta_{\lambda}(H)=\frac{\sum_{w\in ...

**3**

votes

**2**answers

277 views

### Representations of complex semi-simple algebraic group “defined over $\mathbf{Z}$”?

If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over ...

**0**

votes

**2**answers

244 views

### Representation Theory of $U(N)$

(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...

**-1**

votes

**1**answer

69 views

### Using Magma to Find a Fixed Points Module [closed]

Let $G$ be a group and $H$ a subgroup. Suppose $M$ is a $kN_G(H)$-module ($k$ a field). Then the $H$-fixed points in $M$ denoted $M^H$ is a $kN_G(H)$-module. Is there a way to access this module in ...

**3**

votes

**1**answer

255 views

### Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?

Let $G$ be a group, $k$ a field and $T(n,k)\subset Gl(n,k)$,
the group of invertible upper triangular $n\times n$ matrices.
I know that if $\rho : G\rightarrow T(n,k)$ is faithful
(i.e. into) then ...

**3**

votes

**0**answers

86 views

### Approximating the norm of a finite dimensional representation on a Banach space by irreducible representations

Let $G$ be a compact group, let $X$ be a Banach space and let $\pi$ be a linear and isometric representation of $G$ on $X$ that is continuous with respect to the strong operator norm. For $v \in X$, ...

**1**

vote

**0**answers

114 views

### What is the definition of plethysm in the representation theory of permutation groups

Let $s_\lambda \circ s_\mu$ be a plethysm. Here let $\lambda, \mu$ be $m,n$ box Young diagrams.
I have seen the definition of plethysms in symmetric functions. I would like to understand the ...

**2**

votes

**0**answers

39 views

### Noncommutative fusion categories

Although noncommutativeness is almost a defining trait for fusion categories, offhand I recall only the extended Haagerup N-N (rank 8). It's a two minute computation to find that even a based ring ...

**0**

votes

**1**answer

125 views

### Clifford-Mackey theory, references

I am working on a problem related to the local Langlands correspondence and I am interested in certain smooth representations of locally profinite groups (in particular of the Weil group of a local ...

**2**

votes

**1**answer

137 views

### Ext groups in the equivariant derived category

I apologize in advance that this question is probably too basic for MO, but I reckoned I would not get an answer on Math.Stackexchange.
I am starting to learn about perverse sheaves, the ...

**0**

votes

**1**answer

74 views

### Non-graded representations over Lie superalgebra $\mathfrak{gl}(m,n)$

I have the following questions:
Let $m,n$ be positive integers. Consider representations over the general linear Lie super-algebra $\mathfrak{gl}(m,n)$. Namely, modules over the associative algebra ...

**27**

votes

**4**answers

936 views

### Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?

A $4\times 4$ symmetric matrix
$$
\left(
\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{22} & a_{23} & a_{24} \\
a_{13} & a_{23} & a_{33} & ...

**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

**1**answer

189 views

### Finite quotients of an infinite product of finite groups

Let $G$ be a finite group.
Consider the direct product $\Gamma=\prod_{i=1}^{\infty}G$ of (countably) infinitely many copies of $G$. For every finite set of numbers $\{i_1,\ldots,i_n\}$ we have the ...

**3**

votes

**0**answers

135 views

### The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$

It is commonly known that we have a chain of embeddings
$$SU(4)\subset Spin(7)\subset SO(8)$$
(there is more than one possible $Spin(7)$, just take one).
Which is the explicit analog for the Lie ...

**2**

votes

**2**answers

350 views

### Lifting projective Galois representation with condition

Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$.
A theorem of Tate tells us that we can always ...

**3**

votes

**1**answer

158 views

### Fusion categories: If infinity were an integer

Consider the following fusion categorie $F(i)$ with integer parameter $i$. Simple objects are $1,a,A,B$ (where $a$ and $A$ are conjugates). Nontrivial fusion rules are $a\bigotimes{a}=A$ (and ...

**10**

votes

**1**answer

319 views

### Embedding linear algebraic groups of a given dimension into a fixed $\mathrm{GL}_N$

Given $n$, can $n$-dimensional linear algebraic groups over $\mathbb{C}$ be embedded into $\mathrm{GL}(N,\mathbb{C})$ for a uniformly bounded $N$?
Thanks so much for your reply!

**4**

votes

**1**answer

274 views

### Irreducibility of the tensor product of two finite-dimensional irreducible group representations

Let $k$ be an algebraically closed field of characteristic 0, let $G$ be any group and $N\unlhd G$ a normal subgroup. Let $U$ be a finite-dimensional and irreducible $kG$-module, such that $U$ is also ...

**9**

votes

**3**answers

296 views

### Real and Quaternionic Representations according to Weights

According to this question, it is easy to know whether a representation is self dual or not: just check if the weight distribution in space is symmetric about the origin.
Now, for self dual ...

**5**

votes

**1**answer

202 views

### Do discrete groups with property $(T)$ have “modest” subgroup growth?

I saw it conjectured at http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0309.0317.ocr.pdf that "discrete subgroups with property $(T)$ may have modest subgroup growth." (Page 5, directly above ...

**2**

votes

**1**answer

212 views

### A computation about Whittaker functions and Eisenstein series

I have some questions about the computation of Eisenstein series and Whittaker functions in the book. The question is on page 29, Theorem 4.3.
My questions are in the following.
(1) I think that ...

**4**

votes

**0**answers

93 views

### Uniqueness of cohomological holomorphic discrete series representation

In Claus Sorenson's PhD thesis, he proves a theorem about level lifting of paramodular forms whose associated automorphic representation has component $\pi_{\infty}$ that is the "cohomological ...

**5**

votes

**0**answers

114 views

### Root-theoretic formulation of characteristic polynomial

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra of rank $n$ over $\mathbb{C}$. Let $G$ denote the corresponding simple simply connected algebraic group. By Chevalley's Theorem, ...

**4**

votes

**0**answers

82 views

### Polynomials invariant with respect to a nilpotent Lie algebra

Let $\mathfrak{u}$ be a nilpotent Lie algebra and let $\mathbb{C}[\mathfrak{u}]$ be the space of polynomials with the natural coadjoint action of $\mathfrak{u}$.
Can one describe ...

**2**

votes

**0**answers

96 views

### Classification of symplectic representations of quaternion division algebras

I would like to know the classification of representations of the form $\rho:B^{\times}\to Sp(V,F)$ or ($Gsp(V)$), where $B$ is a quaternion division algebra over a number field $F$ (or ...

**4**

votes

**0**answers

59 views

### Minimal rank of a permutation resolution of a $G$-lattice

Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...

**1**

vote

**0**answers

48 views

### General quantum highest-weights dimension formulas

The formulas hold modulo typos :-)
It is well known (tl;dr fun fact: not well enough for me, I forgot where I saw it so I guess-computed it from the data in the Hayashi paper; promptly I found it in ...

**0**

votes

**0**answers

21 views

### Decomposition of axial vector and vector representions of C$_{4v}$ group

Let $R$ be the orthogonal matrix corresponding to an operation in $O(3)$. If
R is a proper rotation, then both vectors $\vec{V}$ and axial vectors $\vec{A}$ are transformed in the same way,
$$ ...

**2**

votes

**1**answer

153 views

### The lower bound of a group with characters of special degrees

Is there any lower bound for the order of a group with an irreducible character of degree $p$, where $p$ is a prime.
Is there any similar result for $p^2$ or $p^3$ instead of $p$?
Thanks for your ...

**2**

votes

**2**answers

283 views

### Algebraic groups “generated” by a Lie algebra element

Here is a definition which I invented and which I would like to understand better.
Let $ A $ be a complex affine algebraic group. Let $ X \in \mathfrak g $ be an element in its Lie algebra. We say ...

**0**

votes

**0**answers

38 views

### Pseudo-braided fusion categories

A few definitions first, please replace with the standard terminology (and correct me if I confuse all the by-names of fusion categories :-)
I call a complex number $z$ pseudo-cyclotomic if $|z|=1$.
I ...

**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, ...

**8**

votes

**3**answers

748 views

### Catalan numbers as sums of squares of numbers in the rows of the Catalan triangle - is there a combinatorial explanation?

This question arose from an answer to my recent question How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, ... algebras?
What I need from that ...

**0**

votes

**0**answers

49 views

### Non-trivial summand in End(\rho)

Given a finite group representation $\rho:G\to GL_n(\mathbb C)$ one knows that the trivial representation $\mathbb 1$ is contained in $End(\rho)$.
Let $\rho'$ be the other summand, i.e., $\rho'$ is ...

**2**

votes

**0**answers

215 views

### References for 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'

Presently I am reading the 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'. I am finding it difficult for eg. the initial sections on $l$-adic geometric representation of finite ...

**3**

votes

**0**answers

89 views

### Localized at $p$ integral representations of finite elementary $p$-groups

Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...

**1**

vote

**0**answers

39 views

### Fusion category R-symbols diagonal even when multiplicity present?

Preface: Please bear my lousy ASCII :-)
I automatically thought that a twist move on a trivalent node -(/ = a* -( would be diagonal even in the case of multiplicity, for example: -1(/= a* -1( + 0* ...

**1**

vote

**0**answers

90 views

### Character sums over a fixed subset of skew tableaux

Let $f(\lambda)$ count the number standard young tableaux of shape $\lambda\vdash n$ and $\lambda=(\lambda_1,\cdots,\lambda_r)$. Let $\mu \vdash k$ be a partition for $k<n$. It is a consequence of ...

**5**

votes

**1**answer

292 views

### real representation of a product group

Let $G_1$ and $G_2$ be compact Lie groups. We know that each finite-dimensional complex irreducible representation of $G_1\times G_2$ is the tensor product of an irreducible representation of $G_1$ ...