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I have a question about proving that a module is non-rigid using Auslander-Reiten quiver. Suppose that we have some algebra $A$ and the components of its Auslander-Reiten quivers are tubes like the fo …

This is described in equation (5.2) in the paper which follows from the paper: Quantum affine algebras and affine Hecke algebras.

I am trying to figure out what should the following definition correspond to in the setting of quantum affine algebra: $$ X \circ Y = Ind_{\beta, \gamma}^{\beta+\gamma} X \boxtimes Y \quad (1) $$ This …

In Lusztig's paper Canonical bases arising from quantized enveloping algebras, the formula (a) in Section 9.4 on page 483, there is a formula \begin{align} \tilde{\gamma}_c' = \sum_{c' \le c} p_{c'}^c …

For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices: \begin{align} \left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\ -x^T & C & -A^ …

There is a symmetric group action on the power set of positive roots in type A. The action is defined as follows. Denote by $\alpha_1, \ldots, \alpha_n$ be the set of simple roots in a root system. In …

In Cluster algebra III by Berenstein-Fomin-Zelevinsky, Theorem 2.10, for any pair of reduced words $(u,v)$, they constructed an initial seed for the cluster algebra $\mathbb{C}[B^{u,v}]$, where $B^{u, …

In the paper A categorification of Grassmannian cluster algebras, an algebra $B_{k,n}$ is defined as follows. Denote by $C=(C_0, C_1)$ the circular graph with vertex set $C_0=\mathbb{Z}_{n}$ clockwise …

Let $R$ be an Artin algebra and let $0 \to A \to B \to C \to 0$ be an Auslander-Reiten sequence of finitely generated left $R$-modules. Is it always true that the projective cover of $B$ equals to the …

Grassmannian cluster categories are studied in A categorification of Grassmannian cluster algebras and Cluster categories from Grassmannians and root combinatorics. The category $CM(B_{k,n})$ of Cohen …

Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker rel …

Let $F$ be a p-adic field and $GL_n(F)$ the general linear group over $F$. The irreducible complex finite length smooth representations are parametrized by multi-segements in the paper. A multi-segmen …

Let $V$ be a vector space over $\mathbb{R}$. An element $s \in GL(V)$ is a reflection if $H_s:=\ker(s-1)$ is a hyperplane and $s^2=1$. The eigenvalues of a reflection $s$ are $1, -1$. Every reflection …

Extremal vectors are defined in Kashiwara's paper. The definition is as follows. Simple reflections in the Weyl group of $\mathfrak{g}$ acts on the crystal basis of integrable $U_q(\mathfrak{g})$-mo …

Are there some good references about representations of classical groups? What are the fundamental representations of classical groups of type $B, D$? I would like to know explicit formulas of the a …