**2**

votes

**1**answer

149 views

### Computing tangent spaces of resolutions to Slodowy slices

This question is about (a special case of) the varieties discussed here: Does the preimage of the Slodowy slice in $T^*G/P$ have a name?.
Let $G = SL_n(\mathbb{C}), \mathfrak{g} = ...

**4**

votes

**2**answers

204 views

### What is the map from nodes of the E8 diagram to conjugacy classes in the binary icosahedral group?

Let $G \subset \mathrm{SL}_2(\mathbf{C}^2)$ be a finite subgroup isomorphic to the binary icosahedral group. Let $Y$ be the minimal resolution of $\mathbf{C}^2/G$. The irreducible components of the ...

**5**

votes

**0**answers

67 views

### Contragredient via an involution?

So in reading about epsilon factors of pairs for $G=GL_n(F)$ over a local nonarchimedian field $F$, the following fact was used. If $\pi$ is an irreducible smooth (complex) $G$-representation, then ...

**1**

vote

**1**answer

125 views

### classification of irreducible finite dimensional representation of affine hecke algebra of type A

Let $H_{n}$ be the affine Hecke algebra with parameter q, where q is not root of unity.
The classification of irreducible finite dimensional representations has been given by Kazhdan-Lusztig in terms ...

**2**

votes

**4**answers

644 views

### Are there natural examples of non-symmetric Frobenius algebras?

Symmetric Frobenius algebras arise everywhere, but the non-symmetric variety seem difficult to come by. Are there any natural examples/constructions that produce non-symmetric Frobenius algebras in ...

**6**

votes

**1**answer

122 views

### About G-modules with good filtrations

Let $k$ be an algebraically closed field of positive characteristic, and let $G$ be a reductive algebraic group over $k$ (for instance a classical group).
Let $V$ be a (rational) $G$-module. We say ...

**9**

votes

**1**answer

164 views

### Fixed set of order p automorphism of Bruhat-Tits tree

I would like to know the structure of the fixed set of an order $p$ automorphism [Edit: induced by a matrix in $GL_2(K)$] on the Bruhat-Tits tree for a p-adic field $K$, specifically in the case where ...

**3**

votes

**2**answers

242 views

### Algebraic Groups, Modules, and Comodules

Background:
Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For
$$
\widehat{H} := \text{Alg}_k\{H; k\},
$$
we recall (see Abe Chapter 4 ...

**2**

votes

**1**answer

173 views

### R-linear representations of sl(2,C)

Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}_2{\mathbb C}$?
Equivalently, what ...

**1**

vote

**2**answers

253 views

### A semisimple group ring

Let $n \in \mathbb{N}$, $p$ a prime number, and $G$ a finite group of order coprime to $p$. Let $R = \mathbb{Z} /p^n \mathbb{Z}$ be the ring of integers mod $p^n$. Must $R[G]$ be semisimple?
As noted ...

**1**

vote

**1**answer

138 views

### A Simple Bijective Proof Of Stanley's Hook-Content Formula for Hook Shapes

this is my first post on math overflow so I hope it goes well. I believe I have a fairly simple bijective proof for Stanley's Hook-Content Formula in the case of hook shapes. I wanted to see if ...

**10**

votes

**2**answers

254 views

### Which finite nonabelian groups have all their quaternionic representations of degree one?

A finite group $G$ has a finite set of irreducible representations over the complex numbers. All of these representations are linear (that is, are maps in 1x1 complex matrices) if and only if $G$ is ...

**0**

votes

**0**answers

92 views

### Actions and representations of profinite groups

Let $p$ be a prime number, and denote by $\mathbb{Z}_p$ the additive profinite group of p-adic integers. Let $G$ be a finitely generated profinite group of order coprime to $p$, and $V = ...

**3**

votes

**0**answers

354 views

### An exact sequence which does not split

Let $X$ and $Y$ be indecomposable modules over a finite dimensional algebra and let $f \colon X \to Y$ be a non-zero morphism which is neither a monomorphism nor an epimorphism.
Suppose that it is ...

**3**

votes

**1**answer

152 views

### Integral representations of groups of small order

I have a problem in which it would be helpful to know about the integral representations of some groups of small order (probably of fairly low degree). From what I've gathered so far, cyclic groups of ...

**2**

votes

**1**answer

225 views

### Quiver representations

I'm wondering how to find indecomposable representations of a given quiver explicitely. In particular, I'm interested in the maximal indecomposable representation of $\mathbb{E}_8$(I'm working over ...

**4**

votes

**3**answers

165 views

### Questions on constructions of supercuspidal representations

To my knowledge, usually there are two ways to construct supercuspidal representations over p-adic fields. The first is via theory of types (for GL(n) and classical groups), notably by Bushnell, ...

**1**

vote

**0**answers

209 views

### Relationship between algebraic groups and Lie groups? [closed]

In the literature, e.g. in representation theory, there seems to be a passage from Lie groups to (linear) algebraic groups. It is clear, particularly over $\mathbb R$ and $\mathbb C$ that they are ...

**2**

votes

**0**answers

134 views

### A decomposition of a representation via characters of a normal compact subgroup

This is connected to my question here. Let $K$ be a normal compact subgroup in a locally compact group $G$, $\widehat{K}$ the dual object for $K$, and $\mu_K$ the normed Haar measure on $K$ ...

**6**

votes

**1**answer

259 views

### What is the Grothendieck group of the category of $\mathbf{Z}_p[G]$-modules?

Let $G$ be a finite group. Let $\mathcal{O}$ be a suitably large finite extension of the $p$-adic integers, with residue field $\mathbf{F}_q$.
The Grothendieck group of the category of ...

**0**

votes

**0**answers

74 views

### “Multiplying” Clebsch-Gordan series

Assume you have a Lie algebra $G$ and a Clebsch-Gordan series $A\bigotimes{B}=C\bigoplus{D}\bigoplus...$
Assume you have a Lie algebra $g$ and a Clebsch-Gordan series ...

**8**

votes

**2**answers

438 views

### When is the conjugation character almost multiplicity free?

Let me give some motivation, which also explains how I arrived at the question. We may let the finite group $G$ act on itself by conjugation, and this makes the group ring into a $\mathbb{Z}G$-module ...

**5**

votes

**1**answer

290 views

### Is there a specific geometric meaning why fractional charges are allowed in SU(N) gauge theories?

So in the standard model of particle physics, there exist particles with fractional charge. What this means geometrically is as follows: We are given a smooth manifold with a principal $U(1)$ bundle ...

**2**

votes

**2**answers

183 views

### Minimal *-idempotents for the group algebra of the symmetric group

There is a well-known construction of minimal idempotents in the group algebra of the symmetric group $\mathbb C[S_n]$ using row symmetrizers and column antisymmetrizers. But these idempotents are ...

**5**

votes

**1**answer

190 views

### Condition on a Hopf operad for tensor product in the base categoy to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...

**16**

votes

**2**answers

557 views

### What's the relationship between these two isomorphisms involving G and T?

Let $G$ be a compact connected Lie group with maximal torus $T$ and Weyl group $W$. Recall the following two isomorphisms.
Isomorphism 1: $R(G) \cong R(T)^W$, where $R(-)$ denotes the representation ...

**4**

votes

**2**answers

333 views

### Jones polynomial of the concatenation of two braids

Let $\sigma_1$ and $\sigma_2$ be two braids with $n$-strings.
Are there any formulas relating $J_{\widehat{\sigma_1\sigma_2}}(q)$, $J_{\hat{\sigma_1}}(q)$, and $J_{\hat{\sigma_2}}(q)$?
Here, ...

**3**

votes

**0**answers

68 views

### Rational cohomology of $S$-arithmetic groups over function fields and Gauss-Bonnet

I have a question on the ranks of rational cohomology groups of
$S$-arithmetic groups over function fields. To fix the situation, $G$
is a simple Chevalley group of rank $r$, $k=\mathbb{F}_q$ a finite
...

**1**

vote

**2**answers

358 views

### on the extensions of $ A_5$ by $A_5$ [closed]

Let $G$ be a finite group such that $G$ has a normal subgroup $H$ and $H$ is isomorphic to the alternating group $A_5$. Also we know that $G/H \cong A_5$.
Can we say that $G \cong A_5\times A_5$?
...

**1**

vote

**0**answers

80 views

### G-Invariant Complete Intersection generated by G-representation

I have a smooth manifold $M$ with $G$-action and complete intersection of codimension $n$ given by ideal $I$ such that $gI=I$. I'm interested when I can choose a $n$-dimensional vector space of ...

**1**

vote

**1**answer

149 views

### Some question about polynomial representations of $GL(V)$

I'm sure this is something silly but I am trying to understand the following paper http://www.sciencedirect.com/science/article/pii/S0001870807001636# and something is not clear to me. The paper ...

**5**

votes

**3**answers

306 views

### Decomposition of $L^2(\Gamma \backslash G)$

Let $G$ be a semisimple Lie group, and $\Gamma$ be an lattice (arithmetic) - typical examples I am thinking about would be $(SL_2(\mathbb{R}), SL_2(\mathbb{Z})$, or $(SL_2(\mathbb{C}), PGL_2(O_F))$ ...

**1**

vote

**0**answers

84 views

### Constant terms of Eisenstein series and Gindikin-Karpelevich formula

Let $G$ be a split reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of G. Let $\chi$ be an unramified character of $M$ and $f_\chi$ be the spherical section of the ...

**3**

votes

**0**answers

77 views

### Orthogonal basis for the multilinear polynomials with zero “trace”

We say that a multilinear polynomial $P(x_1,\ldots,x_n)$ in $n$ commuting variables over $\mathbb{R}$ has zero trace if
$$ \frac{d}{dt} P(t,\ldots,t) = 0. $$
Equivalently,
$$ \left(\sum_{i=1}^n ...

**3**

votes

**2**answers

220 views

### Which Weil group over a $p$-adic field?

For simplicity, call the Weil group of a local nonarchimedean field $F_v$ to be the following extension:
$$1\longrightarrow F^\times_v\longrightarrow W_{F_v}\longrightarrow\text{Gal}(F_v/\mathbb ...

**9**

votes

**2**answers

312 views

### Can the difference of non-conjugate pseudoreflections lie in the commutator subgroup?

Let $G$ be a finite group acting on a complex vector space $V$ by pseudoreflections (i.e. every element of $G$ is a product of elements which fix hyperplanes in $V$). I would like to understand the ...

**7**

votes

**1**answer

103 views

### One-Parameter Families of Indecomposable Representations of the Preprojective Algebra of type A5

Consider the preprojective algebra of type $A_n$. It is well known that this algebra is of finite representation type when $n<5$, of tame representation type when $n=5$, and of wild representation ...

**0**

votes

**0**answers

89 views

### Reps of a compact connected Lie group are equivalent iff they are equivalent as reps of a maximal torus

Let $G$ be a compact connected Lie group, $T$ a maximal torus in $G$ and $V$, $W$ finite-dimensional $G$-representations. Using characters and the fact that every element of $G$ can be conjugated into ...

**3**

votes

**0**answers

100 views

### Why “non-linear similarity” is the same as equivalence of representations for connected Lie groups?

Let $G$ be a compact Lie group and $V$ a finite-dimensional orthogonal $G$-representation. Write $S^V$ for the quotient $D(V)/S(V)$, where $D(V)$ and $S(V)$ are the unit disk and sphere in $V$, ...

**0**

votes

**2**answers

181 views

### Projective characters with corresponding factor set

The following is just a follow up to my previous question. I have a finite group $H$ with 14 ordinary characters. The Schur multiplier $M(H)\cong 2^2$. Hence the group $H$ will have 3 sets of ...

**4**

votes

**0**answers

139 views

### Is there a notion of “tame” representations of $GL_n(Z)$?

This is a followup to this question about the (left) noetherianity of the group ring of $GL_n(\mathbf{Z})$:
Does GL_n(Z) have a noetherian group ring?
Given that $\mathbf{Z}[GL_n(\mathbf{Z})]$ is ...

**3**

votes

**0**answers

119 views

### characterization of all periodic tiling of a simple set of Wang Tile

Consider a set of Wang Tile such that all the edges are either 1 or 0.... there are 16 elements in such a set.
Now, I wish to characterize all the periodic tilings of this set (better if they are ...

**3**

votes

**0**answers

81 views

### Shalika germ for local function field

I am wondering if there is a theorem of Shalika germ (as below) for local function field, for both the group version or the Lie algebra version, probably under assumption on the characteristic to be ...

**14**

votes

**1**answer

339 views

### Local-global principle for split extensions of Galois representations

I guess the following is well-known (and probably follows from Chebotarev's density theorem, but I'm not very comfortable with it):
Define some notation:
$K$ a global field,
$G$ the absolute Galois ...

**4**

votes

**2**answers

540 views

### Decomposing representations of finite groups

Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$.
Must there be ...

**5**

votes

**1**answer

137 views

### “Plucker” embedding of G/N, for reductive group G, affinization of quasiaffine varieties

I'll use "affinization" to describe the natural map of schemes $X \rightarrow \text{Spec}(\Gamma(X, \mathcal{O}_X))$. For quasi-affine varieties $X$ this is an open embedding.
Let $G$ be a reductive ...

**2**

votes

**2**answers

346 views

### Central idempotents from characters in Frobenius algebras (generalizing Lusztig arXiv:math/0208154v1 §19)

$\newcommand{\refone}{\textbf{(1)}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\tr}{\operatorname{Tr}}$ $\newcommand{\kk}{\mathbf{k}}$ Let $\kk$ be a field. Let $A$ be a $\kk$-algebra which ...

**0**

votes

**2**answers

179 views

### Matrix algebra as Clifford algebra

Many kinds of Clifford algebras have corresponding sub-algebras of matrix algebras in sense of isomorphism. Say, quaternion, spacetime algebra and also Dirac algebra. Generally, Clifford algebra has ...

**4**

votes

**0**answers

131 views

### Parahorics and their normalizers

Let $G$ be a reductive group over a local non-arch field $F$. For convenience let's assume $G$ has anisotropic center (or even that $G$ is semisimple if preferred). Let $x$ be any point in the ...

**5**

votes

**2**answers

628 views

### An application of Maschke's theorem

I've been teaching some elementary representation theory to undergraduates, and want to provide applications of Maschke's theorem to complex group algebras to present in class. In particular, I'd like ...