Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

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2
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176 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
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2answers
255 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
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1answer
142 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
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2answers
258 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 ...
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93 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 = ...
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356 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
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1answer
154 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
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1answer
229 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 ...
5
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3answers
171 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, ...
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212 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
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135 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
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1answer
263 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 ...
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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
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2answers
442 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
1answer
298 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
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2answers
185 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 ...
6
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1answer
225 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. ...
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2answers
566 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
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2answers
336 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
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0answers
69 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
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2answers
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 $G \cong A_5\times A_5$? ...
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0answers
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
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1answer
156 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
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313 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 $(SL_2(\mathbb{C}), PGL_2(O_F))$ ...
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94 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 ...
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78 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
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2answers
221 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
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2answers
313 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
1answer
105 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 ...
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91 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
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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
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2answers
184 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 ...
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141 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 ...
2
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120 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
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87 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
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1answer
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
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2answers
544 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
1answer
140 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 ...
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2answers
347 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 ...
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2answers
183 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
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138 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 ...
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2answers
633 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 ...
2
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1answer
151 views

“Quantum Littlewood-Richardson” Rule?

Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor ...
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128 views

Reductive Group Actions and Completion

Suppose that $A$ is a Noetherian (not necessarily commutative) $\mathbb{C}[h]$-algebra equipped with a rational action of an affine reductive group $G$, i.e., $A$ is a $\mathbb{C}[G]$-comodule and ...
4
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3answers
530 views

Action of a profinite group

Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod_{i \in I} \mathbb{Z}_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be ...
2
votes
1answer
85 views

Equivalence of star products on two differents Poisson algebras?

Let $A$, $B$ be two commutative and associative $\mathbb k$-algebras and let $A_\hbar:=A[[\hbar]]$, $B_\hbar:=B[[\hbar]]$ be the corresponding ring of formal series. Of sense [Deformation theory and ...
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2answers
179 views

Can $T$ act trivial in a repn of SL$_2(\mathbb{Z}_N)$?

I am confronted with the following problem: If $\rho : \text{SL}_2(\mathbb{Z}) \to \text{GL}_{\mathbb{C}}(V)$ is a finite dimensional representation such that $\text{ker}(\rho)$ contains the ...
8
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3answers
563 views

Tensor products of two irreducible representations of reductive groups and their inclusions

Let $G$ be a reductive group and $\lambda$, $\mu$ and $\nu$ be dominant weights of $G$. Denote by $V_\lambda$ the irreducible representation of $G$ of highest weight $\lambda$. It seems to be true ...
1
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1answer
253 views

The number of irreducible projective characters with associated factor set of any finite group

Is the number of a set of irreducible projective characters with associated factor set always strictly less than the number of the ordinary irreducible characters of a finite group G? If so, can you ...
5
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1answer
230 views

Learning representation theory of real reductive lie groups

I am interested in any sources that can be helpful for learning the representation theory of real reductive groups. I am currently reading Wallach book, but I feel that I don't understand the subject ...