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Bernšteĭn, I. N.; Gelʹfand, I. M.; Gelʹfand, S. I. Differential operators on the base affine space and a study of $\mathfrak{g}$-modules. Lie groups and their representations (Proc. Summer School, Bol …

The representation theory of the Clifford algebra you are asking about can be understood using the theory of associative superalgebras. In this context, the irreducible $\mathbb{Z}_2$-graded represent …

I assume you mean to decompose $V$ into orbits since $V$ is only a $G$-set. I'm also going to guess that the action of $\alpha\in G=\mathbb{F}_p$ on $V$ is $\alpha.f(x)=f(x+\alpha)$. In this case, eve …

I would think that this result would go all the way back to Frobenius. Anyway, the proof seems easy enough: Let $V$ be the trivial $\Gamma$-module. We want to show that $\dim H_\Gamma=\dim\mathrm{Hom …

Let $m$ be maximal such that $D^ma\neq 0$. Then, $$\Phi(a)=\frac{b}{(D\epsilon)^m}$$ where $$ b=\sum_{k=0}^m\frac{(-1)^k}{k!}(D^ka)(\epsilon)^k(D\epsilon)^{m-k}. $$ Now you need to show that $D(b)=0$. …

Irreducible representations of $S_n$ are absolutely irreducible, meaning that they remain irreducible after extension of scalars. Therefore, if $V$ is irreducible as an $\mathbb{R} S_n$-module and $\p …

I assume you are talking about representations of $S_n$ over $\mathbb{C}$. In this case, $\mathbb{C}S_n$ is left semi-simple by Maschke's Theorem, so every representation of $S_n$ is a direct sum of i …

Couple of quick observations for $\alpha_i(x)=x^{(d_i)}$ ($x^{(d)}=x^d/d!$ as usual). Note that if $d_i>d_{i+1}$, for all $i$, then $G(x_1,\ldots,x_n)=0$ unless $d_i-d_{i+1}=1$. In particular, if th …

This is done in Orellana-Ram, `Affine braids, Markov Traces and the category O'. The answer is essentially the same as for the degenerate affine Hecke algebra.

I've been teaching some elementary representation theory to undergraduates, and want to provide applications of Maschke's theorem to complex group algebras to present in class. In particular, I'd like …

Then the answer is yes, finite dimensional irreducible $G_0$-modules are projective in the category of finite dimensional modules. Note that $G_0=G_0'\oplus Z$, where $G_0'$ is semisimple and $Z$ is …

Since you only care about the completely reducible case, I'll assume $K=\mathbb{C}$. The easiest way that I know of to construct irreducible $S_n$ representations is a special case of the constructio …

Using the notion of a graph with compatible automorphism, Lusztig constructs all symmetrizable Cartan data (i.e. Cartan matrices $A$ for which there is a diagonal matrix $D=\mathrm{diag}(d_1,\ldots,d_ …

The degenerate affine Hecke algebra $H(k)$ over a field $F$ is isomorphic (as a vector space) to the tensor product $$ H(k)=^{\mathrm{v.s.}} FS_k \otimes F[x_1,\ldots,x_k] $$ where $FS_k$ is the group …

Given a reduced expression for the longest word $w_0$ in the Weyl group of $\mathfrak{g}=\mathfrak{n}^+\oplus\mathfrak{h}\oplus{n}^-$, one obtains a convex ordering on the set of positive roots, $\be …