Notes and Sage worksheets from our meetings can be found below.
| Date | Summary |
| May.11. | We introduced symmetric groups and studied relations between basic transpositions using wiring diagrams. We then defined free groups, group presentations and the Coxeter presentation of symmetric groups. Notes |
| May.13. | We generalized symmetric groups to Coxeter groups and talked about the groups of type B. We introduced reduced words and reduced word graphs, talked about Matsumoto's theorem, and defined fully commutative (FC) elements. Notes |
| May.15. | We defined heaps of words and FC elements, Stembridge's FC criterion for full commutativity, and a simpler specialization of the criterion in type A. Notes |
| May.18. | We introduced Sage and wrote an FC test to check if a word is the reduced word of an FC element, in type A. We also wrote code to produce all reduced words of all FC elements in a Coxeter group of type A. Sage Worksheet |
| May.21. | We talked about descents and descent tests for FC elements, as well as their connection to heaps. We described how to obtain the Cartier–Foata canonical forms of FC elements. Notes |
| May.25. | We shared our code for finding the descents and Cartier–Foata forms of FC elements. We also discussed a method for enumuerating FC elements of a Coxeter group by increasing length. Sage Worksheet |
| May.27. | We wrote code for enumerating all FC elements in FC-finite groups, and verified the results against Stembridge's results. We also talked about documenting our code to match Sage conventions. Sage Worksheet |
| May.29. | We learned about coset decompositions of Coxeter groups with respect to parabolic subgroups. Then we talked about star operations and started discussing the star operation orbits of FC elements (the orbits often turn out to be Kazhdan–Lusztig cells!). Notes |
| Jun.01. | We talked about star reducibility of FC elements and of Coxeter groups, with an emphasis on visualizing star operations via heaps. Notes |
| Jun.03. | We shared our code for various computations related to star operations, including computation of orbits of star operations. Sage Worksheet |
| Jun.05. | More discussion on orbits of star operations. We also talked about streamlining our code and migrating to Github. Our Github Repository |
| Jun.08. | We defined group algebras, group rings, and Hecke algebras of Coxeter systems. We talked about the standard basis of Hecke algebras, and discussed how we can define homomorphisms from objects given by generators and relations using universal properties. Notes |
| Jun.10. | We introduced the three realizations of the Temperley–Lieb (TL) algebra in type A: as a quotient of the Hecke algebra, as an algebra given by generators and relations, and as a diagram algebra. Notes |
| Jun.12. | We briefly reviewed relevant current code from Sage and Fokko du Cloux's Coxeter3 on Hecke algebras and diagram algebras, and introduced other types of TL algebras (namely, types B,D,E,H and affine C) with diagrammatic realizations. Notes |
| Jun.15. | We studied current Sage code for diagram algebras and discussed a plan to implement generalized TL diagrams, complete with decorations. |
| Jun.17. | We discussed a first version of our code for generalized TL diagrams. Using the paper Decorated tangles and canonical basis by Green, we also discussed how to find the canonical basis and diagram associated to each FC element for the TL algebra of type H. |
| Jun.19. | We coded Python classes for decorated tangles and generalized TL algebras that take advantage of tools such as CombinatorialFreeModules and AlgebraWithBasis from Sage. The code can draw the diagram representations of elements in the TL algebras of types A, B, and H. We also introduced Kazhdan–Lusztig cells and the Mathas–Lusztig (ML) involutions. Code for TL diagrams; Notes |
| Jun.22. | We discussed how to right justify the reduced word of an FC element w in types B and H based on Green's paper Decorated tangles and canonical basis. Here a right justified word of w will allow us to associate a single TL diagram to w which turns out to be a canonical basis element in a certain sense. We also started to describe our strategy for computing the effect of the ML involutions on FC elements. Notes |
| Jun.24. | We talked about ideals in Hecke algebras and how to use them to significantly simplify the computations of both KL cells with only FC elements and the restriction of ML involutions on FC elements. We also discussed how to compute products of KL basis elements of Hecke algebras in a way faster than the standard tools currently offered in Sage. Notes |
| Jun.26. | We further discussed how the product computations mentioned last time can help optimize the computation of both FC cells and ML involutions in type B. Code for cells and ML involutions in type B; Code for fast Hecke algebras computations |
| Jun.29 | We tried to find a good algorithm for right justifying a reduced word of a fully commutative element w in types B and H, and we found some algorithms based on coset decompositions which proved wrong in technical places. We also discussed how to code an important reduction step in producing the canonical diagram for w from a right justified word (this step reduces a linear combination of several diagrams to a single diagram according to certain local reduction rules). Code for the diagram reductions |
| Jul.01. | We designed two correct, slightly different algorithms for right justifying an FC element and obtaining the corresponding caonical basis element in the TL algebras of types B and H. We also have plots of pairs of TL diagrams corresponding to pairs of element (w, \lambda(w)) where w is an FC element in type B and \lambda is the left ML involution. Code for right justification and canonical bases in types B and H; Notes; Plots of pairs of TL diagrams related by ML involutions |
| Jul.03. | We demonstrated how to use the key functions we developed, such as functions for computing KL cells, canonical bases of TL algebras, and plotting ML involutions, on a local installation of Sage. We also briefly summarized all our results throughout the program. (Update: These results were later turned into two SageTrac tickets and a third set of Python code usable in Sage; they were also used to verify most of the enumeration results in this paper.) Notes |