Morphism \(\colon\) Tits Monoid \(\to\) Birkhoff Monoid

  • Find the \(4^{th}\) lecture notes linked: Lecture Notes 4

  • Previously, we had defined a product operation the set of faces determined by the sign sequence multiplication as formulated by Tits.
  • Now, we see the Geometric interpretation of the Tits product \(FG\) of two faces \(F\) and \(G\) as the Tits projection of \(G\) onto \(F\). Stand at \(F\) and move towards \(G\), the face that you find yourself walking along is the face \(FG\) given by the Tits Product. Original paper by Tits actually defines the product geometrically. In fact the partial order on poset of faces is determined completely by the Tits product.
  • Define Birkhoff Monoid on the set of flats. The product here is simply induced from the join opertaion \(\vee\).
  • Check Properties: A lot of nice properties can be listed. We primarily focus on the support map being a morphism at the monoid level. Rank \(0\) face in the Tits Monoid gets mapped to the minimal flat in the Birkhoff monoid. It is in fact a homomorphism
\[s(FG) = s(F) \vee s(G)\]
  • Arrangements under a flat and over a flat are exactly what you think they should be. We have, as usual another Isomorphism Theorem (what’s the count now?)
\[\Sigma [\mathcal{A}]_F \cong \Sigma [\mathcal{A}]_G, \quad K/F \mapsto GK/G\]