# Tag Archives: algebraic group

## 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 $(-1)^\sigma$ […]