# Tag Archives: algebraic group

## A positive characteristic theory for polar representations?

Let $G$ be a split reductive algebraic group over a field $k$ of characteristic zero and $\mathfrak{g}$ it's Lie algebra. If $T\subset G$ is a maximal torus with Lie algebra $\mathfrak{t}$ and Weyl group $W$, then there is a well-known isomorphism of algebras $$k[\gfr]^G\xrightarrow{\sim}k[\mathfrak{t}]^W.$$ This is called the Chevalley restriction theorem. There are many ways […]

## Guess The Algebraic Group

Suppose one day you run into the following algebraic group, defined on $\mathbb{Z}$-algebras $R$ by $$G(R) = \left\{ \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{11} & -a_{14} & -a_{13} \\ a_{13} & -a_{14} & a_{11} & -a_{12} \\ a_{14} & -a_{13} & -a_{12} & a_{11} \end{pmatrix} \in\mathrm{GL}_4(R) \right\}$$ Can you […]

## Determinants, Permutations and the Lie Algebra of SL(n)

Here is an old classic from linear algebra: given an $n\times n$ matrix $A = (a_{ij})$, the determinant of $A$ can be calculated using the permuation formula for the determinant: $\det(A) = \sum_{\sigma\in S_n} (-1)^\sigma a_{1\sigma(1)}\cdots a_{n\sigma(n)}$. Here $S_n$ denotes the permutation group on $n$ symbols and $(-1)^\sigma$ […]