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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 …

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 …

I thought that "Mathematical aspects of quantum field theory" by Edson de Faria and Welington de Melo was nicely written. Summary from the Publisher: "Over the last century quantum field theory has m …

I guess the first product is expanded as $$ \prod_{k=1}^n(1+x+y_k)=(1+x)^n\prod_k(1+y_k(1+x)^{-1})=\sum_{k\geq 0}e_k(y_1,\ldots,y_n)(1+x)^{-k+n}.$$ For the other products you can write $$ \prod_{j=1}^ …

Not a complete answer, but note that \begin{align} (*) \;\;\;e_k =e_k(x_1,\ldots,x_n)=\sum x_{i_1}x_{i_2}\cdots x_{i_k}, \end{align} where the sum is over $1 \leq i_1 < i_2 < \ldots < i_k \leq n$. Mo …

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 …