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I don't really understand what you are asking for. Root systems are geometric objects associated to finite reflection groups. Try reading Jim Humphrey's "Reflection Groups and Coxeter Groups" for a ve …

A complete analysis of this is given in the paper by Orellana-Ram. Actually, they consider the action of the affine braid group on $M\otimes V^{\otimes n}$, but you can recover your case by taking $M$ …

Look at the paper Quantum Affine Algebras at Roots of Unity of Chari and Pressley (published as https://doi.org/10.1090/S1088-4165-97-00030-7).

There is a recent work of Weiqiang Wang and his student Jinkui Wan on spin-invariant theory. It appears to be the first of its kind.

What you are trying to do is done in complete detail in Leclerc's paper "Dual canonical bases, quantum shuffles and q-characters" (MR) based on the paper "Standard Lyndon bases of Lie algebras and en …

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 …

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 …

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.