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

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2
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78 views

Geometry of Rogers-Ramanujan continued fraction

I'd like to understand the underlying geometry of the Rogers-Ramanujan continued fraction from the point of view of integrable systems (ideally Toda type theories). The generating function $R(z) = ...
0
votes
2answers
127 views

How to write $\mathbb{C}[G/U_-]=\oplus_{\lambda} V_{\lambda}$ explicitly?

Let $G=GL_n$ and $U_-$ the set of all lower unipotent triangular matrices. Then by Gauss Decomposition, we have $G = U_-B$, where $B$ is the set of all upper triangular matrices. The group $U_-$ acts ...
3
votes
0answers
88 views

Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...
3
votes
0answers
50 views

Vanishing of power of nilpotent operator $\mathrm{ad} \, \;e$ in different characteritics

This question needs some background: (1) In his influential 1959 paper here, Kostant studied the adjoint representation of a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (which can be ...
9
votes
0answers
123 views

Definition of discrete spectrum and continuous and basic properties

I apologize if this is too basic for MO. I have an embarrassing admission to make: I don't know the actual definition of the discrete/continuous spectrum of a reductive group $G/\mathbb{Q}$ (in the ...
2
votes
0answers
117 views

For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]: If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free ...
-3
votes
0answers
155 views

Maps from $S^3$ to $S^3$ [on hold]

As a physicist, I apologize for imprecise language. I am interested in maps from $S^3$ to $S^3$ (identical to the group $SU(2)$). Since $S^3$ is threedimensional, there is some similarity to maps ...
10
votes
2answers
375 views

From Weyl groups to Weyl groupoids?

I'm trying to find a framework where the choices in the classical construction of a root system of a semi-simple lie algebra are not needed. Let $\mathfrak{g}$ be a semisimple lie algebra. ...
0
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0answers
95 views

Rank of a locally free $\mathbb Z[G]$-module

This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free. ...
2
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0answers
68 views

Dimension of curvature invariants

EDIT: Let $V$ be a Euclidean space and let $O(V)$ denotes its orthogonal group. Let $K(V)\subset Sym^2(\wedge^2(V))$ denote the subspace of curvature tensors, i.e. the subspace of elements satisfying ...
19
votes
1answer
366 views

Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For ...
9
votes
2answers
264 views

Representation viewpoint on Chern Weil (cohomology computations done with rep theory?)

Let $G$ be a compact lie group. Chern-Weil theory tells us that there's a homomorphism: $$H^{*}(BG;\mathbb{R}) \to (Sym^{\bullet} \mathfrak{g^*})^G$$ Which in our case is an isomorphism since $G$ ...
2
votes
0answers
69 views

Primitive ideal space and unitary dual of a [SIN] group - when are they Hausdorff?

Recall that a locally compact group $G$ is said to be an $[FC]^-$ group, if each conjugacy class in $G$ has a compact closure; an $[SIN]$ group, if each neighborhood of the identity includes a ...
0
votes
0answers
29 views

Integrable modules and comodules

Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Do we have the following result: $V$ is an integrable $U(\mathfrak{g})$-module if and only if $V$ is a $\mathbb{C}[G]$-comodule? ...
5
votes
0answers
200 views

Deligne's theorem on the characterisation of Tannakian categories

I am trying to understand the proof of the Theorem 7.1 from Catégories Tannakiennes by Pierre Deligne https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf. Essentially, it is ...
5
votes
3answers
286 views

Character Values for Alternating Groups of degree $\geq 7$

Is it true that in each row and column of the character table of alternating groups with degree $\geq 7$ there are at most two complex values? Any reference will be highly appreciated.
9
votes
1answer
281 views

A careful roadtrip from locally symmetric spaces to algebra

I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning ...
1
vote
0answers
120 views

Deformation of a Hopf algebra

A deformation of a Hopf algebra is defined as follows. On page 171 of the book a guide to quantum groups, Remark 2, it is said that Any deformation of a Hopf algebra $A$ as a bialgebra is ...
5
votes
2answers
179 views

“Diagonalizing” Littlewood-Richardson coefficients

Let's consider the Littlewood-Richardson coefficients $c^{\lambda}_{\mu \nu}$ so that \begin{equation} V_\mu \otimes V_\nu = \bigoplus_\lambda V_\lambda^{\oplus c^{\lambda}_{\mu \nu}} \end{equation} ...
2
votes
0answers
47 views

Order of metaplectic operator

I have a weak background on this subject. Suppose $S$ be a $2m \times 2m$ symplectic matrix of order $n$. Suppose $W_S$ be the corresponding metaplectic operator on $\mathcal{S}(\mathbb{R}^m)$, the ...
6
votes
3answers
126 views

Sum of the dimensions of the rational irreducible representations of $S_k \times S_j$

For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge ...
6
votes
1answer
86 views

U(n)-submodules of SO(2n)-modules

Let $\Gamma_{(\lambda_1, \dots, \lambda_{n})}$ denote an irreducible $SO(2n)$-module with highest weight $(\lambda_1, \dots, \lambda_n)$ and let more specifically $X = \Gamma_{(2\lambda, \dots, 0)}$ ...
1
vote
0answers
204 views

How to check that an ideal of $\mathbb{C}[GL_n]$ is a coideal or not?

Let $I$ be an ideal of $\mathbb{C}[GL_n]$. Are there effective methods or software to check whether $I$ is a coideal or not? Thank you very much. For example, let I be the ideal of $\mathbb{C}[GL_3]$ ...
1
vote
0answers
70 views

Notation clash between a representation and spectral radius

I am currently writing a paper where I need talk both about a representation of a semisimple Lie group (usually denoted by $\rho$), and about spectral radii of linear maps (also usually denoted by ...
0
votes
0answers
76 views

representation theory in Magma

I have a submodule named sub of a representation named rep in Magma defined in the following way : sub:= Submodules(rep)[4]. I search for its generators using: Generators(sub), but I am not getting ...
13
votes
0answers
304 views

Does the category of G-spectra know G?

I was recently in the situation of having access to the category of $G$-modules (for some group $G$ which I had forgotten), as just a category, i.e. no monoidal structure, together with the forgetful ...
3
votes
1answer
115 views

How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic?

Let $X:=x^3$, $Y:=x^2y$, $Z:=xy^2$ and $W:=y^3$ be the 4 independent generators of $S^3\mathbb{C}^2$, and observe that the kernel of the natural epimorphism (total symmetrisation) $$ ...
4
votes
0answers
155 views

positivity of semicanonical basis

Given a quiver $Q$, there is an algebra isomorphism from $U\frak{n}$ to $\mathcal{M}\subset \sum_{v}\text{Const}(\Lambda_v)^{G_v}$ by Lusztig's construction. Fro each $Z\in \text{Irr}(\Lambda_v)$. ...
5
votes
1answer
106 views

Relation between unipotent cuspidal representations and cuspidal local systems

This could well be a question for reading suggestion. Hope it's not too bad and thanks a lot. So the question is as in the title. What are the relations between the notion of unipotent cuspidal ...
1
vote
1answer
75 views

Jordan algebra of $3 \times 3$ quaternionic hermitian matrices

Let $\mathbb H = \mathbf H \otimes_{\mathbf R} \mathbf C$ be the tensor product of the quaternions with $\mathbf C$, and let $\mathcal J_3(\mathbb H)$ denote the set of $\mathbb H$-hermitian $3 \times ...
3
votes
1answer
89 views

Unclear asymmetry in Lie-algebra module structure on space of linear transformations Hom(V,W)

Let $L$ be a (finite dimensional) Lie-algebra. Let $V, W$ be finite-dimensional vector spaces. If $V,\; W$ are in addition $L$-modules (see, e.g., 6.1 in Humphreys Introduction to Lie Algebras), then ...
30
votes
1answer
814 views

Roadmap to Geometric Representation Theory (leading to Langlands)?

I believe there has been at least one question similar to this one and yet I still think this particular question deserves to have a thread of its own. I'm becoming increasingly fascinated by stuff ...
5
votes
1answer
337 views

On the symmetric group of 2^n elements

Consider the set $ X_1^n=\{1,2,...,2^n\} $. Then define $ X_2^n $ to be the set of two element subsets of $ X_1^n $. I will construct $ X_i $ by induction on $ i $. $ X_i^n $ is the set of two element ...
1
vote
1answer
125 views

Orbital integral for matrix coefficients

I am currently aiming at estimating orbital integrals. Maybe surprizingly, I hope for some help in the compact case (ramified places), in proving the usual formula $$O_\gamma(f) = \int_G ...
6
votes
0answers
83 views

Modular double of elliptic quantum group

By studying dynamical quantum Yang-Baxter equations and corresponding $RLL$ relations, Felder defined an elliptic version of quantum group $E_{\tau, \eta}(sl_2)$, which can be understood as ...
-2
votes
0answers
30 views

Quick question on Pauli matrices and u(2) [migrated]

The wiki page for Pauli matrices states "Together with the 2 × 2 identity matrix I (sometimes written as σ0), the Pauli matrices form an orthogonal basis, in the sense of Hilbert–Schmidt, for the ...
3
votes
1answer
82 views

Vector fields, diffeomorphism subgroups and lie group actions

Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization: Let $\{X_j\} \in Vect(M)$ be a ...
2
votes
0answers
52 views

Action of orthogonal group on the free Lie algebra

This question is somewhat related and inspired by this post of professor Montgomery. The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of ...
4
votes
1answer
180 views

Does there exist finite dimensional irreducible representation of Euclidean or Poincare group in which translation and rotation both act nontrivially?

Does there exist any finite dimensional irreducible rep. of Euclidean or Poincare group in which translation and rotation both act nontrivially? Let me firstly clarify my question. For example, we ...
11
votes
3answers
542 views

Philosophy behind cohomological representations

For a given real reductive Lie group $G$, we have the notion of a representation being cohomological using the Lie algebra cohomology. In particular we know that the discrete series representations of ...
2
votes
0answers
138 views

Is there a name for this construction from two representations?

I've come across the following elementary construction in representation theory. Does anyone know if this has a name? Let $G$ be a group and let $V$ and $W$ be two representations. The tensor ...
-1
votes
1answer
107 views

Representations of the $3\times 3$ Heisenberg group [closed]

I am trying to understand how the Heisenberg group is defined because I would like to understand the (irreducible) representations. Following this article given a symplectic bilinear form $\langle, ...
4
votes
1answer
103 views

Cartan subspaces for general algebraic representations

So I feel like asking the following likely open-ended question: What good generalizations of the notion of Cartan subspace do we have? To be precise, let $G\curvearrowright V$ be an algebraic ...
7
votes
1answer
174 views

Hopfian modules

My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any ...
4
votes
1answer
222 views

Is the direct sum in Maschke's Theorem an orthogonal decomposition?

I am reading a paper on coding theory, and it uses a statement, which was claimed to be a reformulation of Maschke's Theorem. But I felt that was false... Let's say $\mathcal(V):=\mathcal{F}_2^n$ is ...
11
votes
2answers
325 views

Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix?

Let $A,B$ be two rational rotations: $$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\ -\frac{4}{5} & \frac{3}{5} & 0 \\ 0 & 0 & 1 \end{array}\right] ...
9
votes
0answers
195 views

Geometric Satake and Restriction

The Geometric Satake correspondence (due to Lusztig, Ginzburg, Mirkovic-Vilonen) relates perverse sheaves on the Loop Group $\hat{G}$ (with their convolution product) to the Representations of the ...
6
votes
2answers
220 views

What are the special parahoric subgroups in unitary groups?

Let $L$ be a $p$-adic field and let $L'/L$ be a quadratic extension. Let $U_{L'/L}(n)$ be a quasi-split unitary group of $n\times n$ matrices with entries in $L'$. I'm curious about what the special ...
30
votes
4answers
1k views

Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...
7
votes
1answer
164 views

How fine an invariant of a representation is its quotient singularity?

This is a refinement of a question asked on MSE. Let $G$ be a finite group and let $V$ be a finite-dimensional faithful complex representation of $G$. Consider $V$ as an affine complex variety. In ...