Tits Monoid

  • The \(3^{rd}\) lecture notes are linked below: Lecture Notes 3

  • Since there were a few discussions on the number-theoretic Möbius function and Riemann Zeta function in relation to those defined in class on Incidence Algebras, such as if we take the poset of natural numbers in which order is defined by divisor relation, will the Möbius function be the one defined in Basic Number theory. The answer is Yes! Hence Dirichlet series was introduced along with a couple of classic exercises and the resulting Möbius function was motivated. Similar arguments show that Zeta function acts as exponential whereas Möbius function acts as logarithm.

  • Isomorphisms of arrangements were defined: gisomorphisms and cisomorphisms. Cisomorphisms are of more practical use to us.

  • Essentialization along with quotient spaces were defined and examples of mod-ing out by equivalence relations was given. Quotient spaces as dual of subspaces in linear algebra.

  • Some classifications for Arrangements of small rank were discussed and it was soon apparent that its a difficult task.

  • Flats and Support Map were defined. Cones analogously defined if in the case of faces, arbitrary intersection had been allowed. Flats are easier to study than faces as their posets form and don’t form a lattice respectively. This is because joins may not exist in the poset of faces (for instance, chambers don’t have any join). Although joins always exist (meets exist and upper bounds exist) in the poset of flats, they can be very complicated. Join is not simply the union or the subspace spanned by the two elements. Join of an element of small rank may be of very high rank.

  • Finally, we reached the definition of the Tits Monoid after an introduction to sign sequences (canonical encoding of faces) and product of two faces. Associativity of the product and existence of the central face as the identity element makes the set of all faces a Monoid which is called the Tits monoid.

  • My friend suggests that we are going towards a structure theory for Hopf Algebras (called by physicists as Quantum Groups) using structures mainly arising from algebraic combinatorics (arrangements, operads, buildings). As of yet monoids (as most other topics) have only been concretely demonstrated by examples and the abstract stuff is yet to come. Hopefully, we are not too far away from putting Category theory into use.

Some references for Category theory recommended by me and a friend are:

For my friends in engineering, I personally also suggest :