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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 …

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 …

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 …

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 …

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 …

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 …

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).

As this is a representation theory question, the connection to affine Hecke algebras deserves a few more words. The (degenerate) affine Hecke algebra, $H_d$ is isomorphic as a vector space to $\mathbb …

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 …

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 …

This is not a complete answer, but does give some information. In addition to having an embedding of the Iwahori-Hecke algebra into the affine Hecke algebra, $\iota:H_n^{\mathrm{fin}}\hookrightarrow …

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 $M$ …

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 …

Ben, I have heard the same thing, but I have never seen an example. After thinking about it a bit, I came up with the following 'heuristic' reason why the structure constants should be positive for ha …

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 …