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$\DeclareMathOperator{\SL}{\operatorname{SL}}$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 $N$ 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 …

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 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 …

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\ …

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) …

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 …

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 …

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 …

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 …

Let $R_n$ be the category of complex-valued smooth finite-length representations of the group $GL_n(F)$, where $F$ is a local field. By the result of Borel, the subcategory of $R_n$ consisting of repr …

Bernstein and Zelevinsky classifies the irreducible complex smooth representations of a general linear group over a local field in terms of cuspidal representations. The irreducible modules are parame …

Super Grassmannians are introduced by Manin, see for example. Elements in a grassmannian can be written as matrices, see for example. Can we write an element in a super Grassmannian as a pair of ma …

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 …

Let $I$ be an ideal of $\mathbb{C}[GL_n]$. Are there effective methods or software to check whether $I$ is a coideal or not? Thank you very much. For example, let I be the ideal of $\mathbb{C}[GL_3]$ …