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There is a paper: Representations of the hyperoctahedral groups by L. Geissinger and D. Kinch.

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 …

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 …

I am trying to understand the notation $\rho^{\vee}(-1)$. Let $T$ be a maximal torus of a semi-simple algebraic group $G$ and $\mathbb{G}_m$ the multiplicative group. I think that $\rho^{\vee}$ is a m …

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 …

The answer of this problem is given in the paper by Chari and Pressley (the Corollary on Page 272).

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 …

I am reading Kashiwara's paper GLOBAL CRYSTAL BASES OF QUANTUM GROUPS. On page 462, the action of $\tilde{e}_i$ on an integrable $U_q(g)$-module $M$ is defined as follows. For $u \in \ker e_i \cap M_{ …

The actions are defined in the Kashiwara's paper: On crystal bases of the Q-analogue of universal enveloping algebras in (2.2.5). Let $M$ be an integrable $U_q(\mathfrak{g})$-module. Then \begin{ali …

Let $\mathfrak{g}$ be a semisimple Lie algebra and $U_q(\mathfrak{g})$ the corresponding quantum group. Are there some general method to compute the upper global basis (dual canonical basis) of an irr …

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 …

Littelmann path is a combinatorial tool to compute multiplicity. I have some questions about the definition of Littelmann path. It is said that a Littelmann path is a piecewise-linear mapping $$\pi …

Are there any definitions of quivers for algebras which are not basic or unital? I am reading the book Elements of the representation theory of associative algebras: volume one. The ordinary quivers a …

If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, indecomposable projective modules, indecomposable injective modules of $A$. If we o …

Let $V, W$ be two vector spaces. We have $\Lambda^2(V \otimes W) \cong (\Lambda^2 V \otimes S^2 W) \oplus (S^2 V \otimes \Lambda^2 W)$. I am trying to find similar results for $\Lambda^3(V \otimes W)$ …