Questions tagged [representation-theory]
For questions about representations or any of the tools used to classify and analyze them. A representation linearizes a group, ring, or other object by mapping it to some set of linear transformations. A common goal of representation theory is classifying all representations of some type. Representation theory is a broad field, so questions not including the word "representation" may be appropriate.
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Serre's Representation Theory exercise 7.3(d)
I am trying to solve Exercise 7.3(d) in Serre's Linear Representation of Finite Groups. I have solved all other parts. The important points of where I am stuck at boils down to the following facts. (I ...
- 93
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Mapping between vectors in irreducible Sp-representation
Let $V$ be the standard Sp-representation with symplectic basis $\{ a_i, b_i \}$. I believe the vector $(b_1 \wedge b_2) \otimes (b_1 \wedge b_2 \wedge a_3 \wedge a_4)$ lies inside the irreducible ...
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Does the formal character determine the representation?
Suppose $V,W$ are two finite-dimensional representations of a Lie algebra $\mathfrak{g}$.
Is it true that if their formal characters coincide, $$\mathrm{ch}_V=\mathrm{ch}_W ,$$ then the ...
- 1,562
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"Linear independency" of Lie Brackets
I was watching this eigenchris video. At 21:49, he says:
$$[g_i, g_j]=\sum_k {f_{ij}}^{k}g_k$$
for $\mathfrak{so}(3)$.
Does this mean $[g_i, g_j]$ and $g_i, g_j$ can be linear independent? What about ...
- 113
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Complexification of complex Lie algebras like $\mathfrak{su}(2)$
I'm reading Brian Hall's book on Lie theory.
He defines the complexification $V_{\mathbb{C}}$ of a real vector space $V$ as the linear combinations $v_1+iv_2$, with $v_1,v_2\in V$. Next, he proceeds ...
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Prime ideals dividing the Artin conductor
Let $L/K$ be a Galois extension of number fields, and let $(\phi,V)$ be a representation of $\operatorname{Gal}(L/K)$. Let $\mathfrak{f}_\phi$ be the Artin conductor of this representation, which is ...
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Real Commutant Algebra of a Set of Matrices
Suppose I have a collection of $N\times N$ real, symmetric matrices $R_1, R_2, \dots$ and I want to find their orthogonal commutant---that is, the group of real, orthogonal matrices that commute with ...
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Number of left ideals in the simple components of groups algebras
Let $G$ be a finite group and $K$ a field with characteristic zero. Suppose $G$ has $m$ irreducible $K$-representation $W_i$ with character $\chi_i$. $KG$ is semisimple algebra, and
$$KG=KGe_1 \times \...
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Generating a conjugate representation of an irreducible self-conjugate representation of $S_n$
Suppose we have a complex matrix representation $\Gamma_{ij}^\sigma \in \mathbb{C}^{d \times d}$ of dimension $d$ for the permutations $\sigma$ of the group $S_n$ of permutations of $n$ objects.
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Projective representations that are compatible with a group automorphism
We know that projective representations of a group $G$, $\rho(g) \rho(h) = e^{i\omega(g,h)} \rho(gh)$, is classified by the second group cohomology $H^2(G,U(1))$. Let us now specify a group ...
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Let $a$ be the reflection of the plane $\mathbb{R}^2$ over the bisector of the odd quadrants [closed]
Let $a$ be the reflection of the plane $\mathbb{R}^2$ over the bisector of the odd quadrants (line with equation $y = x$), and let $b$ be the reflection of the same plane over the bisector of the even ...
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The action of $SL(n,\Bbb{R})$ on the tangent space of $SL(n,\Bbb{R})/SO(n)$.
SETUP. It is a standard result that $\text{GL}(n,\Bbb{R})/O(n)$ is isomorphic to the set $P'$ of positive definite $n\times n$ matrices, as manifolds: the basic idea is that $\text{GL}(n,\Bbb{R})$ ...
- 3,337
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Working with character tables
I am currently a bit stumped by an old exam question, which gives a character table and wants me to deduce properties of the group:
What is the order of $g$?
Show that $g \notin C_G(G)$
Show there ...
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Are linear representations of finite groups in one-to-one correspondence with modules over the group algebra?
Suppose $G$ is a finite group, and $V$ is a complex vector space. It is often said that linear representations $\rho:G\to\DeclareMathOperator{\GL}{GL}\GL(V)$ correspond to modules over $\mathbb C[G]$. ...
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What is the intuition for representations of the symmetric group?
What is the (physical) intuition for representations of the symmetric group? In particular, matrix coefficients of the Fourier coefficients corresponding to a representation.
For the cyclic group $C_n$...
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