# Tagged Questions

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

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### A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups. This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: $\mathcal{L}(H\subset G)$ ...
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### Character fields and Clifford's theorem

Let $G$ be a finite group with normal subgroup $N$. Let $\chi$ be an irreducible complex character of $G$. Then Clifford's Theorem says that $\mathrm{res}^{G}_{N}\chi = e(\eta_1 + \cdots + \eta_r)$ ...
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### Action of longest element of Weyl group on zero weight space

Let: $G$ be a real semisimple Lie group; $\rho$ be an irreducible representation of $G$ on a finite-dimensional real vector space; $A$ be a "Cartan subspace" of $G$ (a Lie subalgebra which is ...
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### Sum of Young symmetrisers of a given shape

Preliminaries and notation: Let $n\in \mathbb{Z}_{>0}$ and $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_s)\vdash n$ be a partition. Given a Young diagram of shape $\lambda$, we can associate it ...
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### Spectral decomposition on GL(n)

If $\Delta_1, \ldots, \Delta_{n-1}$ are a basis of the ring of commuting bi-$SL(n,R)$-invariant differential operators, $L_0^2=L_0^2(SL(n,Z)\backslash SL(n,R))$ is the space of cuspidal automorphic ...
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### Must normalizing field outer automorphisms “divide” the dimension?

Imprecise question: To get a normalizing field outer automorphism of order $r$, must we multiply the dimension by $r$? Precise hypothesis: Let $p\geqslant 5$ be a prime, let $q$ be a power of $p$ and ...
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### “Nice” basis for highest-weight irreducible module of a simple Lie algebra

Let $\mathfrak{g}$ be a simple complex Lie algebra, $\mathfrak{h}$ a Cartan subalgebra, $\Phi \subset \mathfrak{h}^*$ the associated root system, $\Sigma = \{\sigma_i : i\in I\}$ a basis of simple ...
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### Corepresentations of Tensor Products of Hopf Algebras

Given two cosemisimple Hopf algebras $H,G$ over ${\mathbb C}$, denote their usual (not braided) tensor product by $G \otimes H$. What conditions do we need to impose on the Hopf algebras to ensure ...
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### Reference request: proofs of the theorems in the paper “On the representation of the group GL(n, K) where K is a local field”

In the paper On the representation of the group $GL(n, K)$ where $K$ is a local field by Gelfand and Kazhdan, it is said that the proofs of the theorems in the paper are published in some other papers....
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### inductive construction of unipotent radicals

Consider a directed coxeter diagram $\vec{\Gamma}$, i.e. a finite graph where each edge is decorated with one of the integer weights $\big\{3,4,6\big\}$ and those edges with weights $4$ or $6$ are ...
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### Irreducible representations containing simple actions of $\mathrm{SL}(2,\mathbb{C})$

Let $G$ be a complex semisimple Lie group and let $\rho: G \longrightarrow \mathrm{SL}(n,\mathbb{C})$ be a faithful irreducible representation of $G$ with $n \geq 3$. Suppose that $G$ contains a copy ...
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### Reference for nonlinearity of covers of $\operatorname{SL}(2,\mathbb R)$

It is known that no nontrivial connected cover of $\operatorname{SL}(2,\mathbb R)$ admits a faithful finite dimensional linear representation (see, for example, page 143 in Fulton-Harris and Exercise ...
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I have posted this question on mathstack echange but did not get any answer. It mam trying my luck here. The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}... 0answers 114 views ### Singular Locus of a Schubert variety I am trying to compute the singular locus of the schubert variety$X_w$in$G_{2,7}$where$w=(4,7) \in I_{2,7}$. Following the notation in the book "The Grassmannian Variety: Geometric and ... 0answers 79 views ### Bounding Schur polynomials of a particular shape Consider Schur polynomials$s_\lambda$with$\lambda = (2m, m, m, \ldots, m, 0)$and$\ell(\lambda) = n$(that is,$\lambda$has$n$rows). Here$m \gg n$, which, for the sake of concreteness, let's ... 1answer 278 views ### Uncle of Witt algebra A Witt algebra W is an infinite-dimensional Lie-algebra defined by the generator relations: W:$[L_{j},L_{k}]:=(j-k)\cdot L_{j+k}$And my first thought was: What about the analogous algebra defined by ... 2answers 423 views ### When can a finite subgroup of$GL(2n,\mathbb{R})$be viewed as a subgroup of$GL(n,\mathbb{C})$? A finite group acting on a complex vector space of dimension$n$can be seen as acting on a real vector space of dimension$2n$just by forgetting the complex structure of the space. My question is, ... 1answer 180 views ### Bott-Samelson construction of a perfect Morse function on G/T An undergraduate student of mine is interested in writing a "senior" thesis on the topology of Lie groups. Let$G$be a simply connected compact Lie group and$T$a maximal torus. One way of ... 1answer 225 views ### Symmetrization for hyperalgebras in positive characteristic Let$G$be an algebraic group over an algebraically closed field$k$of arbitrary characteristic and let$U$be the hyperalgebra of$G$. Recall that$U$is defined as the subspace of the full linear ... 1answer 149 views ### A representation of Spin(9,1) Let$Spin(9,1)$denote the universal (double) cover of$SO(9,1)$.$Spin(9,1)$acts linearly on$\mathbb{R}^{16}$(see e.g. p.29 here https://arxiv.org/pdf/math/0105155v4.pdf ). Consider the induced ... 1answer 67 views ### Decomposition into irreducible components of a representation of$Spin(9)$It is well known that the group$Spin(9)$acts linearly on the vector space$\mathbb{R}^{16}$(see for example "Spinors and calibrations" by R. Harvey). Consider the induced representation of$Spin(9)...
Here is a definition which I invented and which I would like to understand better. Let $A$ be a complex affine algebraic group. Let $X \in \mathfrak g$ be an element in its Lie algebra. We say ...
### $k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?
For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set \$\...