Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7,076 questions
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Representation of the group of automorphisms on the holomorphic forms
Let $X$ be a compact Riemann surface and $G = Aut(X)$ be its group of automorphisms (biholomorphisms between $X$ and $X$). It is known that $G$ acts on the space $Harm(X)$ of all harmonic forms and ...
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Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)
Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature?
There are two sets of partition polynomials, not in the OEIS, that serve as the ...
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The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)
Original question (without additional information from Wendy):
Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way:
Taking the E8 as {128,...
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Criteria for a character being a root of unity [closed]
Let G be a finite group , g$\in$G and $\chi$ be a character of G. If |$\chi(g)$|=1 then show that $\chi(g)$ is a root of unity. $\\$
Hint: Let |G|=n and consider E=$\mathbb{Q}(\zeta_{n})$ where $\...
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In Algebraic Compact Quantum Groups, is an Irreducible Corepresentation equivalent to its Conjugate?
A quantum group $A$ here is an algebraic compact quantum group --- a Hopf*-algebra with a Haar State. Here $\hat{A}$ is the set of linear functionals $\{\mathcal{F(a)}:a\in A\}$ of the form $\mathcal{...
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Orthogonality of irreducible and non-isomorphic representations [closed]
Let V and W be any two subspaces of $(\mathbb{C}^d)^{\otimes n}$ such that there exists two irreducible and non-isomorphic representations $\rho_V: G \to GL(V)$ and $\rho_W: G \to GL(W)$. Does this ...
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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, \...
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irreducible subgroup of SL(n,R)
Suppose a subgroup of SL(n,R) is irreducible; i.e. R^n contains no proper invariant real subspaces except {0}. Then is it irreducible as a subgroup of SL(n,C)? i.e. Does C^n contain no proper ...
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Killing form that is not diagonalizable?
An example Lie algebra $L$ with non-diagonalizable Killing form would have to be non-semisimple, and the Killing form complex. (Otherwise diagonalizability is obvious.) I tried with a few $L$, but (in ...
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Pseudo commutativity
Operator commutativity is the basis for things like homomorphisms and linearity, e.g., $f(x+y) = f(x) + f(y)$.
Is there any meaning or development on a more general nature of this property? E.g., $f(x+...
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Representation of Lie algebra $\operatorname{SE}(2)$
When I read the paper Universal approximations of invariant maps by neural networks of Dmitry Yarotsky, it happens on page 36 that he used some concepts about the representation of Lie algebra of the ...
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irreducible Classical Lie algebras [closed]
which submodule of FG-module of a lie algebra $L$ will be determined I want to check that how we can find out a classical lie algebra like $D_4$ and $E_6$ are irreducible?
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Subgroup of the semidirect product of two subgroups with coprime orders [closed]
It is well known that if $\gcd (|H|,|K|)=1$ then all subgroups
of $H\times K$ are of the form $H^{\prime }\times K^{\prime }$ such that $H^{\prime}$ is a subgroup of $H$ and $K^{\prime}$ is a subgroup ...
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proj of an Algebra [closed]
Let $\mathbb Z_2= \langle\sigma\rangle$ act on $\mathbb C^6$ by $(x_1,x_2,x_3,x_4,x_5,x_6)=-(x_6,x_5,x_3,x_4,x_2,x_1)$. Then what is $\operatorname{Proj}\left(\left(\frac{\mathbb C[x_1,x_2,x_3,x_4,x_5,...
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infinite left degrees
I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by
CLAUDIA CHAIO AND SHIPING LIU. This is a part of the paper:
Definition: Let $f: X \rightarrow Y$ be an irreducible morphism ...
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How to show the following properties of $Coker(d^{-n-1})$?
Let $A$ be a k-algebra,where k is a fixed field. We denote by $\mathfrak{D}^b(A-mod)$ the bounded derived A-module category. A complex $Z^{\bullet}=(Z^i,d^i) \in \mathfrak{D}^b(A-mod)$ such that all $...
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representation theory and finite order automorphisms
Let $\sigma$ a finite-order automorphism of a finite-dimensional complex simple Lie algebra $g$. Denote the order of sigma by $k$ and fix a $k^{th}$ root of unity $\omega$.
It is well know that $k=1,2 ...
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sl(2)-modules... [closed]
I'm trying to learn some Lie algebra without much knowledge of representation theory. While being asked to prove some things about an sl(2)-module, why can one assume that the module is irreducible ...
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no classification of nilpotent lie groups
there is no classification of (simply connected) nilpotent lie groups, but I am tempted to try to generalize the construction of the Heisenberg group. For an upper triangular matrix:
$$ \left(
\...
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Does a group representation being transitive on a basis imply irreducibility?
Let $G$ be an infinite discrete group and $\pi$ a representation of $G$ on the Hilbert space $H$.
Suppose that the group representation is transitive on an orthonormal basis $B = \{e_j\}_{j=1}^{\infty}...
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Existence of orientable finite volume complete cusp hyperbolic 3-manifolds $\mathbb{H}^3 / \Gamma$, where $\Gamma$ has no parabolic generators?
Let $K$ be a hyperbolic knot, i.e., $S^3 - K$ is an orientable finite volume cusp hyperbolic 3 manifold. Let $M=S^3 - K$ then $M= \mathbb{H}^3/\Gamma$, where $\Gamma$ (Kleinian group) is discret ...
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Is there any Lefschetz-like principle for representations of finite groups?
Representation theory (at least the origin of this terminology) aims to exhibit a model (a represetative) in the group of matrices for an abstract group which is known by only its group law. So ...
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On Haar measure and Spherical measure [closed]
Let $d$-dimensional complex sphere be
$$\{(c_1,\cdots,c_{d})\sum_{i=1}^{d} |c_i|^2=1.\}$$
We can define the Haar measure on this sphere by regarding the unitary group $U(d)$.
We can regard the $d$-...
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Orthogonality of invariant subspaces for restricted representations [closed]
Let $G$ be a finite group and $H_1$ and $H_2$ are two proper subgroups of $G$. Also, let $\rho:G \rightarrow \mathbb{C}^m \times \mathbb{C}^m$ be an irreducible non-trivial representation of $G$. Let $...
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Borromean rings, Condorcet's paradox and Quantum chromodynamics [closed]
In https://plus.google.com/108432079989441783124/posts/LHewqvcj5Xo T. Abderrahman explains what Borromean rings are. As I noticed in a comment, the underlying order structure is the same as in ...
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How to find matrix representations of a boolean algebra? [closed]
Given a boolean algebra with a finite number of elements {a, b, c, ...}, and the usual operations: $\cup, \cap, \neg$.
How to find matrix representations of the elements such that:
boolean $\cup$ ...
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