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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 $\hat{H}_m=\hat{H}_m(q)$ be the Iwahori-Hecke algebra of $GL_m$, see for example, Section 2. The simple $\hat{H}_m$-modules are parametrized by Zelevinsky's multi-segments, See Section 2.2 of the …

The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, \ldo …

Let $G = GL_n$. By algebraic Peter-Weyl theorem, we have $$ \mathbb{C}[G] = \bigoplus_{\lambda} V_{\lambda} \otimes V_{\lambda}^*, $$ where $\lambda$'s are dominant weights. Let $U^-$ be the unipote …

I found two versions of definitions of orthosymplectic supergroups. It seems that they are not equivalent. I don't know which version of the definition is standard. The first version of the definitio …

In the paper, the orthosymplectic supergroup $OSP(2p|n)$ is defined as follows. Let $A = A_0 \oplus A_1$ be a supercommutative superalgebra, where elements in $A_0$ are even and elements in $A_1$ are …

Dihedral groups are Coxeter groups of type $I_m$, $m \geq 3$. The Coxeter matrix of $I_m$ is \begin{align} \left( \begin{matrix} 1 & m \\ m & 1 \end{matrix} \right). \end{align} When $m=3,4,6$, $I_m$ …

I am reading the lecture notes and trying to understand dressing actions. Let $G$ be a Poisson-Lie group and $G^*$ its dual Poisson-Lie group. In the lecture notes above, Proposition 5.22 on page 80 …

Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Do we have the following result: $V$ is an integrable $U(\mathfrak{g})$-module if and only if $V$ is a $\mathbb{C}[G]$-comodule? T …

I have a question about decomposing elements in $SL_2$ as a pair of elements in $SL_2^*$. Here $SL_2^*$ is the dual Poisson Lie group of $SL_2$ which is defined as follows. Let $G$ be a Poisson-Lie …

Let $\mathfrak{g}$ be a semisimple Lie algebra and $I$ its vertices of Dynkin diagram. The weight $\rho$ is defined by $\rho = \sum_{i \in I} \omega_i = \frac{1}{2} \sum_{\alpha \in \Phi^+} \alpha$, w …

Let $(\mathcal{G}, \mathcal{G}^*, \delta)$ be a Lie bialgebra. Suppose that the structure constant on $\mathcal{G}^*$ and $\mathcal{G}$ are \begin{align} & [t^a, t^b]_* = C_c^{ab} t_c, \\ & [t_a, t_b] …

Suppose that $G, H$ are Lie groups and $\mathfrak{g}$ the Lie algebra of $G$. Suppose that there is a Lie group action $G \times H \to H$. Is there a natural $\mathfrak{g}$ action on $C^{\infty}(H)$? …

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

Let $G=GL_n$ and $U_-$ the set of all lower unipotent triangular matrices. Then by Gauss Decomposition, we have $G = U_-B$, where $B$ is the set of all upper triangular matrices. The group $U_-$ acts …