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

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2
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0answers
80 views

An equality of discriminant and resultant divisors

Let $\Phi$ be the root system of a split group $G$ over a field $k$. The differentials $d\alpha$ of the roots define a polynomial called the discriminant $$\prod_{\alpha\in\Phi}d\alpha$$ on $\mathfrak ...
4
votes
0answers
80 views

Fell topology vs. convergence of matrix coefficients

My question is partially inspired by the following discussion: Topology on the Unitary Dual Let me remind/explain how the Fell topology is defined (at least I recall the definition which I saw): let ...
4
votes
0answers
142 views

Can we drop commutativity assumption?

Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...
3
votes
0answers
51 views

Anosov representations and boundaries of (harmonic) maps

Let $\Sigma_g$ be a closed hyperbolic surface and $\rho\colon\pi_1\Sigma_g\to G$ an Anosov representation into a suitable Lie group. By definition of Anosovness, one has a $\rho$-equivariant ...
5
votes
1answer
211 views

Constructing a simple $A$-module

Let $n \ge 2$, and let $A$ be the (unital and associative, but noncommutative) $\mathbb{C}$-algebra with generators $x_1, \dots, x_n$ and relations $x_ix_j + x_j x_i = 2\delta_{ij}$. What is a ...
6
votes
1answer
194 views

Does an element in the center of universal enveloping algebra becomes a scalar in irreducible representations?

I'm asking a question about Lie group representation. Let $G$ be a Lie group, not necessarily connected. Let $\Omega$ be an element in the center of the universal enveloping algebra $U(\mathfrak{g})$ ...
4
votes
1answer
210 views

$Ext$-algebra generated by $Hom$ and $Ext^1$ as $A_\infty$-algebra?

In [Keller: A-infinity algebras in representation theory, Proposition 1(b)], Keller states that for an associative algebra the $Ext$-algebra of the simples is generated by $Ext^1(S,S)$ as an ...
6
votes
0answers
99 views

Rings that are $K_0$ of finite groups

Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of ...
5
votes
0answers
110 views

Indecomposable representations of a wreath product

If $G$ is a finite group, we know the irreducible representations of $G ≀ S_n$ (over $\mathbb Q$) are classified by partitions of $n$ 'decorated' by an irrep of $G$. I'm wondering to what extent the ...
-1
votes
0answers
42 views

Representation ring of circle group over complex field [migrated]

Can someone please describe how to find a representation algebra of circle group over complex field ? I am reading " representation theory of compact Lie group" chapter 3 section 7. It will be great ...
2
votes
0answers
56 views

Element of Grothendieck group is eigenvector of operator [migrated]

Let $K_\mathbb{C}(G)$ be the Grothendieck group (over $\mathbb{C}$) of finite dimensional representations of a finite group $G$. Associated with any such representation $V$, there is a linear ...
0
votes
0answers
108 views

What is the natural action of $U(\mathfrak{g})$ on $\mathbb{C}[G]$? [migrated]

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. What is the natural action of $U(\mathfrak{g})$ on $\mathbb{C}[G]$? It seems that the natural action comes from the following. We have a ...
0
votes
0answers
46 views

Equivalence classes of pairs linear transformations

Consider the set of 4-tuples: $$S_{(x, y), k} = \{ (a_i, b_i, a_j, b_j) : \|a_ixb_i - a_jyb_j\|_F^2 = k \}$$ for $a \in GL(m, \mathbb{R})$, $b \in GL(n, \mathbb{R})$, $x, y \in \mathbb{R}^{m \times ...
12
votes
1answer
267 views

Applications of Lubotzky's linearity theorem?

Lubotzky's theorem is a necessary and sufficient set of conditions for a finitely generated discrete group to be linear, i.e. isomorphic to a subgroup of $GL_n(K)$, where $K$ is a field of ...
2
votes
1answer
196 views

Any representation is a sub representation of direct sum of regular representation

I need a reference for the following statement: Let G be a linear algebraic group over algebraically closed field k. Let V be a finite dimensional G-module. The V is sub representation of k[G]^n for ...
8
votes
1answer
318 views

Three involutions on the set of 6-box Young diagrams

The set of $n$-box Young diagrams classifies both conjugacy classes in $S_n$ and equivalence classes of irreducible representations of $S_n$. There is an outer automorphism of $S_6$, of order 2. ...
3
votes
1answer
124 views

Center of an irreducible representation over $\mathbb{Q}$

Let $G$ be a finite group, $\rho\colon G \rightarrow \mathrm{GL}_n(\mathbb{Q})$ its irreducible representation, and $D$ the division algebra of $G$-endomorphisms of $\mathbb{Q}^n$. The division ...
26
votes
2answers
1k views

How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the ...
4
votes
1answer
78 views

Relations among Young symmetrizers of non-standard tableaux

For any Young tableau, one can form the Young symmetrizer. I'm naturally interested in young symmetrizers coming from standard tableaux, but I'm forced to look at Young symmetrizers of non-standard ...
3
votes
1answer
87 views

Classification of finite abelian hypergroups and table algebras

Update: Originally, I formulated this question for finite abelian hypergroups, but in a discussion with Geoff Robinson below I realized that the abelian hypergroups defined below are equivalent to ...
3
votes
1answer
119 views

Faithful linear representation of a nilpotent Lie algebra

Let \begin{align} \mathfrak{g} = Span_{\mathbb{C}}\{ e_1, e_2, e_3, e_4, e_5: \text{ non-zero brackets are } [e_1, e_i]=e_{i+1}, i=2,3,4, [e_2, e_3]=e_5 \} \end{align} be a $5$-dimensional Lie ...
4
votes
1answer
279 views

“set of all irreducible representations of a group”, set-theoretic issues [closed]

I am working on a problem related to representations of the Weil group of a local field $\mathcal{W}_F$. In many articles one introduces the set $\hat{\mathcal{W}}_F$ of all equivalence classes of ...
1
vote
1answer
73 views

Normalized invariant form on a Kac-Moody Algebra

For a symmetrizable Kac-Moody Algebra, we can define a normalized invariant form that performs the same role as the Killing form in the finite dimensional case. My question is, do these forms ...
0
votes
0answers
36 views

Relation between $\Gamma$-percuspidal parabolic subgroups and split parabolic subgroups of real semisimple Lie groups

Let $G$ be a reductive algebraic group defined over $\mathbb{Q}$. Let $\Gamma$ be a lattice in $\mathcal{G}:= G(\mathbb{R})$. I am interested in knowing under what conditions on either of ...
9
votes
2answers
488 views

Schur's Lemma for Hilbert spaces

Let $H$ be a complex Hilbert space and let a group $G$ act on $H$ such that there are no invariant closed subspaces besides $H$ and $(0)$. Let $D$ be the ring of bounded operators which commute with ...
0
votes
0answers
111 views

Irreducible representations of $S_n$ inside the ring of symmetric polynomials

I will describe two ways to associate irreducible representations of $S_n$ with polynomials inside the ring of symmetric polynomials and I want to know if there is any connection between the two. ...
0
votes
1answer
63 views

Highest weight formulas for quadratic Casimir and dimension for the simply laced Lie algebras

Intro (tldr-ish): In the meantime, in the literature I dug up the formulae not only for the dimension D of a $G_2$ module, but also its quadratic Casimir C2 (eigenvalue). After some playing, I ...
1
vote
0answers
103 views

Classification of Automorphism set of a Regular graph

Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation ...
2
votes
1answer
104 views

Projectivity of torsion-free modules over integral group rings

Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$. If we assume ...
16
votes
1answer
330 views

What is the homomorphism between the third exterior and third symmetric power of the adjoint representation of a simple Lie algebra?

Let $\mathfrak{g}$ be the adjoint representation of a simple Lie algebra (which is not of type $A$). Then the space of intertwiners between the third exterior power of $\mathfrak{g}$ and the third ...
1
vote
1answer
181 views

Is it possible to classify the indecomposable representations of the wild quiver with one vertex and two arrows using infinite sets of parameters?

I have read several other questions about wild representation type but may I ask..what actually can be done and what have been done (such as partial results) about classifying indecomposable ...
8
votes
1answer
501 views

Which finite groups have no irreducible representations other than characters?

A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...
4
votes
3answers
235 views

Generalization of a theorem of Burnside to non-compact groups

The following two theorems are often attributed to Burnside: Theorem Let $G$ and $H$ be compact groups. Then the irreducible representations of $G\times H$ are precisely the representations ...
11
votes
4answers
421 views

The conjugacy classes of diagonalizable $2 \times 2$ matrices can be identified with their eigenvalues, what about pairs?

For sake of simplicity, let's say that we live in $G = SL(2, \mathbb{C})$. Every conjugacy class of diagonalizable matrices $$[A] := \{gAg^{-1} \mid g \in G\}$$ can be identified with its set of ...
1
vote
1answer
115 views

Involutive automorphisms of a finite abelian p-group

First, let $A$ be a finitely generated free abelian group, and $s$ an automorphism of order $2$ of $A$. Set $G=\{1,s\}$. Then we know that $A$ is a sum of indecomposable $G$-lattices $A_i$, where ...
0
votes
0answers
41 views

Decomposition of polynomial ring as $S_n$-module [migrated]

I want to whether there is a containment relation between the $S_n$-modules $\mathbb{C}S_n$ and $\mathbb{C}[x_1,\ldots ,x_n]$. Is it true that $\mathbb{C}[x_1,\ldots ,x_n]$ contains an isomorphic copy ...
5
votes
2answers
228 views

Does the ring generated by the odd power sum symmetric functions have a name?

Let $\Lambda$ be the ring of symmetric functions and recall the power sum symmetric function $p_i = \sum x_1^i + x_2^i + \dots$ generate this ring. Let $\tilde\Lambda$ be the ring generated by the odd ...
2
votes
1answer
100 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} & ...
3
votes
1answer
145 views

The term $H^1(N,A)^{G/N}$ in the inflation-restriction exact sequence

[a repost from SE due to the lack of response] Given a group $G$, let $A$ be a $G$-module and let $N\trianglelefteq G$. If I understand it correctly, the superscript "G/N" in the third term of the ...
3
votes
0answers
119 views

What's the relationship between the different versions of the BBD decomposition theorem?

I have a few questions relating to the BBD decomposition theorem. I have come across the following two versions of the decomposition theorem. Version 1. Let $f : X \to Y$ be a proper map of ...
5
votes
2answers
248 views

1-dimensional representations of the affine Hecke algebra for $SL_2$

Kazhdan-Lusztig theory gives a correspondence between irreducibles of the affine Hecke algebra for a simply connected linear algebraic group $G$ and certain homological data extracted from the ...
5
votes
1answer
138 views

Do Iwahori-Hecke algebras come from cohomology classes?

Let $W$ be a Coxeter group. The Iwahori-Hecke algebra $H_q(W)$ is a deformation of $k W$. Question: is there some way to interpret the deformation $H_q(W)$ as a cohomology class? It doesn't ...
5
votes
0answers
174 views

Examples of Rankin-Selberg L-functions from Eisenstein series

I've been digging for awhile to not much success, so I figure I would try here: I am looking for some references which compute explicitly examples of Rankin-Selberg L-functions from the constant ...
4
votes
1answer
209 views

highest weight the half-sum of positive roots

Sorry if this one is already asked - couldnt find anything about it. If I take the irreducible representation of $GL_n$ whose highest weight is the half-sum $\rho$ of positive roots, it has ...
0
votes
1answer
211 views

Determinants of tensors [closed]

Consider a tensor of dimension $[d]\times[d]\times[d]$ which is symmetric with respect to every permutation of the indices. Are there any $\textbf{explicit}$ formulas for notions like determinant-like ...
4
votes
1answer
230 views

reference request: direct product of WOT-continuous unitary representations

In an article I'm revising, I spend some time giving a self-contained proof of the following result Let $G$ be a (Hausdorff) topological group and let $(\pi_i)$ be a family of unitary ...
3
votes
1answer
165 views

A representation of a finite group where every nonzero vector has a trivial stabilizer [duplicate]

What are the finite groups which admit a non-zero representation in char 0 where every non-zero vector has stabilizer equal to $\left<1\right>$? Cyclic groups of prime order is one obvious ...
1
vote
1answer
156 views

Mysterious central projections in the full group $C^*$-algebra

Let me quote the following theorem about the structure of $C^*(G)$ for property $T$ group (the reference is Higson and Roe "Analitycal K-homology"): Let $G$ be a property $T$ (discrete) ...
0
votes
0answers
56 views

Quadratic Casimirs of the E7 series

Of course a) you can compute the QCs for any member and irrep of the $E_7$ series ($E_7,H_{32},D_6$..., corresponding to $m=8,6,4$...) by standard liealgebraology and b) in general form via the Vogel ...
6
votes
1answer
270 views

Representations of the unit group in a ring of integers

Let $K/\mathbb{Q}$ be a finite extension of degree $d > 1$. Suppose that $\omega_1, \cdots, \omega_d$ is a basis for $K$ over $\mathbb{Q}$. Further, we assume that $\omega_1, \cdots, \omega_d \in ...