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I am trying to understand Borel subgroups of $Sp(4,\mathbb{C})$. I think that the following is a Borel subgroup of $Sp(4, \mathbb{C})$: the subset of $Sp(4, \mathbb{C})$ of all lower triangular matric …

This question relates to the question and the question. Let $B^-$ be the Borel subgroup of $Sp(4)$ consisting of all lower triangular matrices. Let $X = \left(\begin{array}{cccc} x_{1,1} & 0 & 0 & 0\ …

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 …

Let $G$ be a reductive algebraic group and $I$ the set of vertices of the Dynkin diagram of $G$. Let $J \subset I$ and $P_J = BW_JB$ the parabolic subgroup of $G$ containing $B$, where $B$ is a Borel …

The Chevalley basis of $sp_4$ is generated by \begin{align} & e_1=\left(\begin{array}{cccc} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 \end{array}\right), \ e_2 = \left(\begin{arr …

The longest word in type $A_3$ Weyl group written as a matrix is \begin{align} w_0=s_1s_2s_1=\left(\begin{array}{cccc} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{array}\right) …

Let $W$ be the Weyl group of a simple algebraic group $G$. The Artin braid group $Br_{\mathfrak{g}}$ is generated by the $T_i$ , $i \in I$ such that for all $i, j \in I$, \begin{align} \underbrace{T …

Let $N$ be the maximal unipotent subgroup of $SL_k$. I think that the following is an additive basis of $\mathbb{C}[N]$: $$\{ e_T: T \text{ is a semi-standard Young tableau with at most $k-1$ rows and …

Let $\mathbb{G}_m$ be the multiplicative group and $T$ a maximal torus of a semisimple group. Let $X^*(T)=\{ \phi: T \to \mathbb{G}_m \}$ be the set of characters and $X_*(T)=\{ \phi^{\vee}: \mathbb{G …

What is the current status of representations of $GL_n(F)$ (and other algebraic groups)? When $F$ is a local field, the representations of $GL_n(F)$ are classified by Bernstein and Zelevinsky in ter …

$\DeclareMathOperator{\SL}{\operatorname{SL}}$Let $G=\SL_k$ be the special linear group, $U$ the unipotent subgroup consisting of all lower unipotent triangular matrices, $U^+$ the unipotent subgroup …

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 …

Let $G$ be a reductive algebraic group and $B$ a Borel subgroup of $G$. Let $T$ be a maximal torus of $G$ contained in $B$. The $B=UT=TU$ for some unipotent subgroup $U$ of $G$. We have Bruhat decompo …

Why we need to study representations of matrix groups? For example, the group $\operatorname{SL}_2(\mathbb F_q)$, where $\mathbb F_q$ is the field with $q$ elements, is studied by Drinfeld. I think th …