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Mathematics, Number Theory, math.NT,High Energy Physics - Theory, hep-th,Mathematics, Group Theory, math.GR
Abstract:
We discuss the Iwasawa-decomposition of a general matrix in SL($n$,
$\mathbb{Q}_p$) and SL($n$, $\mathbb{R}$). For SL($n$, $\mathbb{Q}_p$) we
define an algorithm for computing a complete Iwasawa-decomposition and give a
formula parameterizing the full family of decompositions. Furthermore, we prove
that the $p$-adic norms of the coordinates on the Cartan torus are unique
across all decompositions and give a closed formula for them which is proven
using induction. For the case SL($n$, $\mathbb{R}$), the decomposition is
unique and we give formulae for the complete decomposition which are also
proven inductively. Lastly we outline a method for deriving the norms of the
coordinates on the Cartan torus in the framework of representation theory. This
yields a simple formula valid globally which expresses these norms in terms of
the vector norms of generalized Pl\"ucker coordinates.
