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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 …

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 …

Let $A$ be an algebra over some field $k$. Let $K_P(A)$ be the Grothendieck group of the category of projective $A$-modules and $K_F(A)$ the category of finite dimensional $A$-modules. I've been told …

I guess I should plug Sasha Kleshchev's book "Linear and Projective Representations of Symmetric Groups". Chapter 1 reconstructs the representation theory of symmetric groups from the Jucys-Murphy ele …

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 $M$ …

Let $V$ be the vector representation of $GL_n(\mathbb{C})$, and let $d\leq n$. You want to see the multiplicities of a given irreducible $S_d$ module in $V^{\otimes d}$ in terms of the dimension of …

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 …

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 …

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 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 …

Ben, my paper on the Shapovalov form does give a generating series for the entries of a Gram matrix in Corollary 3.4, and those entries are evidently positive. It is not very hard to deduce a q-versi …

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.

For $\mathfrak{gl}_{\infty}$ the answer is easy, you realize the Fock space as a direct limit of polynomial representations of finite $\mathfrak{gl}_n$ modules. You can read about the construction her …

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 $D(b)=0$. …

Sasha Kleshchev's book "Linear and Projective Representations of Symmetric Groups" is the reference I'd suggest. Chapter 1 contains the connection with Young's lattice, and the subsequent chapters dev …