Instructions for taking notes while watching the linked videos by Federico Ardila.
Instructions for taking notes while reading the assigned pages:
- Correct any typos you see
- Read slowly and carefully - verify all the given examples on your own
- Write down vocab words and concepts that you’ve never seen before or have forgotten.
- If the words or concepts are explained in this textbook, write down the definitions/explanations. Otherwise, look up all these vocab words/concepts (on the internet or your algebra textbook) and attempt to understand them.
- Bring the questions about these new concepts to the meeting
Instructions for exercises:
- Be ready to present to me
- You can use any sources (book, online, or people - please credit your sources)
- Hand-written is fine.
Coxeter groups Part 1
Section 1.1 + examples
Take notes while watching videos:
Take notes while reading Bjorner Brenti:
- All of Section 1.1 (pages 1-4): Definition and basic facts about Coxeter systems
- Selected examples from Sec 1.2:
- Example 1.2.3 (page 4) type A Coxeter group (optional: review symmetric group facts and notations from your Abstract Algebra class)
- Example 1.2.6 (top of page 6) infinite path
- Example 1.2.7 (most of page 6) Dihedral groups (if needed, review Dihedral group facts from Abstract Algebra class)
Prepare to present to me:
- Bjorner Brenti Ch 1, exercises 1 and 2 (page 22)
- One of the examples or theorems from the videos and book reading
Coxeter groups Part 2
Take notes while watching videos:
Take notes while reading Bjorner Brenti:
- Selected examples from Sec 1.2:
- Example 1.2.4 (page 5, top) type B/C and D Coxeter groups
- Example 1.2.5 (page 5, bottom) type affine A Coxeter groups
- (addendum) Section 1.3 Write a Coxeter system as a copy of a signed permutation group.
Prepare to present to me:
- (addendum) Warm-up exercises for Bjorner Brenti Ch 1, exercise 3 (page 22).
- What is the number of elements in the alternating subgroup for S5 (that is, the group of even permutations on 5 elements).
- Write the presentation for the Coxeter group of type H3. The Coxeter diagram is given in Appendix A1 (page 297).
- Let k = the order of the type H3 Coxeter group. Using the presentation for H3, write down some positive integers which divide k.
- Bjorner Brenti Ch 1, exercise 3 (page 22), except the last part.
- One of the examples or theorem from above videos or book.
- Compute some of the elements of D12 (the Dihedral group of order 12) as signed permutations. Write the signed permutations in cycle notation.
Research Part 1
Lecture notes from Mar del Plata 2016, Cluster algebra from surfaces
Take notes on:
- Section 3.1 Snake graphs, pg 23-25
(Optional: If there is extra time, read Section 3.2 Band graphs)
- Chapter 2 Cluster algebras of surface type pg 17-22 (you can ignore the self-folded trianlges for now)
Homework exercises after taking notes on section 3.1:
- Write down the sign functions of all the snake graphs with at most 4 tiles in Figure 3.2
- Draw all the snake graphs with 5 tiles. How many are there? Write down the sign functions of some of them.
- Represent the sign function by the signs of the south edge (of the first tile) and the interior edges and the north/east edge (of the last tile). Let the minus (-) sign be assigned to the south edge.
- a. Draw the snake graph corresponding to sign function (-,-,-,-,-,-) and (-,+,+,+,+,+)
- b. Draw the snake graph corresponding to sign function (-,+,-,+,-,+) and (+,-,+,-,+,-)
- c. Draw the snake graph corresponding to sign function (-,+,+,-,-,-) and (-,-,-,+,+,+)
Homework exercises after taking notes on chapter 2:
- draw all the triangulations of the pentagon and draw the edges connecting each one.
- draw all the triangulations of the hexagon (there should be 14 triangulations) and draw the edges connecting each one. This forms a 3-dimensional polyhedron. Redraw this polyhedron so that it looks nice and symmetric. Count all the faces. How many of the faces have 3 sides, 4 sides, 5 sides, 6 sides?
- draw just several of the triangulations of the heptagon (there are 42 total, so you don’t need to draw all), and draw all the edges connecting them.
(No presentations for this part)
Research Part 2
Lecture notes from Mar del Plata 2016, Cluster algebra from surfaces
Take notes on:
- Section 3.3 From snake graphs to surfaces and 3.4 Labeled snake graphs from surfaces, pg 26-28
- Section 3.5 Perfect matchings, height and weight and 3.6 Expansion formula, pg 28-30
- Section 3.7 Examples pg 30-31
Homework exercises
- Pick one of your triangulations of the heptagon, call it T1. Draw two arcs (call them gamma 1 and gamma 2) of the heptagon which cross T1. Draw snake graphs corresponding to gamma 1 and gamma 2, call them G1 and G2.
- Draw all the perfect matchings of the snake graphs G1 and G2. Follow the flip diagram (forming a lattice) on page 32. Another example of a flip diagram is Fig. 15 on page 29 of this paper Musiker-Schiffler-Williams: Bases for cluster algebras from surfaces - you can read this page and explanations related to the figure but don’t worry about understanding the entire paper.
- Use the perfect matchings above to compute the height and weight for the arcs gamma 1 and gamma 2.
- Draw two triangulations of an annulus with 2 marked points on the outer boundary and 2 marked points on the inner boundary. Pick one of them and call it T2. Draw two arcs (call them gamma 3 and gamma 4) of the heptagon which cross T2. Draw snake graphs corresponding to gamma 3 and gamma 4, call them G3 and G4.
- Draw all the perfect matchings of the snake graphs G3 and G4. Follow the flip diagrams on page 32.
- Use the perfect matchings above to compute the height and weight for the arcs gamma 3 and gamma 4.
(No presentations for this part)
Computing Part 1
Research Part 3
Tentative instruction:
Lecture notes from Mar del Plata 2016, Cluster algebra from surfaces
Take rough notes on Chapter 1 definition of cluster algebra. The most important parts are understanding the definition of clusters, seed, cluster variables, and being able to do quiver mutations. You don’t have to understand every single details and definitions.
Coxeter Group part 3
(Note: I should have assigned this along with Section 1.1 of Brojer Brenti because the free group concept was not covered in Abstract Algebra I)
Take notes while reading Sec 6.3 (A word on free groups) pg 215-220 of Dummit-Foote 3rd edition (I will print the pages for you).
Submit exercises 2, 4, 7.
Meeting Notes from meetings on Tues, Wed May 29-30, 2018.
- Started to put SageMath from
Git the Hard Way
- Explained poset, order ideals, order filters, continued fractions and their connection to the lattice of snake graph perfect matchings.
Meeting Notes from Wed June 20, 2018
Meeting (Skype) Notes from Saturday July 7, 2018
-
We looked at the lattice of order filters for the word w=10101
The subwords are:
empty,
1,
10,
11,
100,
101,
110,
111,
1001,
1010,
1011,
1101,
10101.
-
Fact: every order filter on the j-th row of the lattice has j elements. The number of elements of an order filter does correspond to the number of boxes that are shaded (the “height” when using the snake graph formula)
-
HW (easy): Draw the lattice of antichains.
For examples:
the order filter {2,4,5} corresponds to the antichain {2,5}.
the order filter {1,2,3,4,5} corresponds to the antichain {1,3,5}.
-
HW: Try finding a map between the subwords and the antichains.
-
HW (easy): The word 11001110 is long, so try shorter words. Pick word w=101110. Compute the continued fraction, draw the snake graph, draw the lattice of perfect matchings (include the heights when you draw), also the lattice of order filters/antichains.
-
HW: Look at figures 3-4 of COUNTING THE NUMBER OF NON-ZERO COEFFICIENTS IN ROWS OF GENERALIZED PASCAL TRIANGLES by LEROY, RIGO, AND STIPULANTI. Try to construct the trie in figures 3-4 for word w=10101 and w=101110. Do a ctrl-F for Figure 3 and Figure 4 to find where they explain the figure. Note: don’t need to understand everything not related to Figure 3 and 4.
-
HW: As you feel you need, take notes on the new definitions on poset theory:
- poset online book poset intro
- antichain of a poset Wikipedia Antichain
- ideal and filter Some sort of Wikipedia page
- There is a bijection between the set of antichains and the set of order filters. Do you see the bijection?
- Note: the above online references aren’t that great, but I couldn’t find better ones outside printed textbooks. You can also go to the back of your combinatorics textbook (the index) and see whether these words appear in your textbook.
Notes from (in person meeting) Tuesday, July 17, 2018
-
We looked at the lattice of snake graph matchings/order ideals/antichains. We noticed that the elements of the antichains are exactly the labels of the downward edges of the matching flips (in other words, adding one element of the order ideal to the lattice).
-
HW: Continue with COUNTING THE NUMBER OF NON-ZERO COEFFICIENTS IN ROWS OF GENERALIZED PASCAL TRIANGLES by LEROY, RIGO, AND STIPULANTI. Figure out the edge labels (where to put 1 and 0) rule for the trie. Label the vertices of the trie with the subwords.
-
HW: Try to figure out a logical way to put the the snake graph matchings/order filters/antichains as vertices of the trie.
Notes from (in person meeting) Wed, July 18, 2018
- HW Start an overleaf file. Share with me the edit link.
- Definition of the trie of subwords from [LRS] paper. Use an example like 101110 or something else. Refer to Fig 4. of [LRS] without drawing the picture - write “See [LRS, Fig 4]”.
- Following the [LRS] trie, define a trie of antichains (new definition). Use your previous example, and also the example from Fig 4 of [LRS], see trie of antichains for Fig 4. Explain how to build the trie of antichains and how to label the nodes (as antichains).
- Do another example where the nM (the number of the last block is not 1, for example [1,2,1,1,3]).
- Do last: Take notes and understand Cluster algebra and binary words slides. See source files on Overleaf
- In the overleaf file, define the maps you see on the slides.
- Try to prove that the maps are bijections (in Overleaf or handwritten).
- You can also try to come up with a nicer map.
- HW (do during the rest of July):
- Read just the introduction (usually a paragraph or two) at the beginning of each section from Sec 1.3, 1.4, 1.5, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 3.1, 3.2, 3.3, 3.4. Let me know which ones sound the most intriguing. I will pick just a few sections to focus on for the rest of the summer.
- Note: the chapter that may be relevant to the snake graph matchings is chapter 3. Originally I planned to propose a research question related to chapter 3 but the paper that talks about this (currently being written by other people) will not be out until after the summer, so we cannot work on this question right now.
Notes from (in person meeting) Wed, August 1, 2018
- HW: Continue Overleaf doc
- Write the examples. For now, do handwritten for the pictures. I will create one example with Tikz from your handwritten picture.
- Add bijection (between subwords and antichains) from Cluster algebra and binary words slides to Overleaf
- Try to prove bijection.
Notes from (in person meeting) Fri, August 3, 2018
-
Wrote a proof for surjection for antichain to subword map
- HW:
- Try to type up the injection proof on Overleaf
- Continue with definition of the trie of antichains. And connecting the our trie with the LRS trie of subwords.
- Draw diagrams in Tikz
-
If time, HW:Come up with your own definition of the map, or take notes of the bijection to the snake graph matchings from Cluster algebra and binary words slides
-
Later, will add instruction for some of Sections 1.4, 1.5, 2.1, 2.4, and 3.1 of BB.
- Try some exercises from Computing Part 1
Coxeter Group part 4 HW
- Take notes of just these definitions (from Sec 1.4):
- the left associated reflections and the right associated reflections TL and TR at the bottom of page 16.
- right descent set and left descent set at the top of page 17.
- Submit reading notes on Section 3.1. Try to look up notations that you haven’t seen before (in chapter 1 and 2). Write down questions.
- Write down the “chain property”, the exact statement of Proposition 3.1.2.
- Prove Prop 3.1.2(ii)
- Submit Exercise 1 (pg 84). Determine which of the x, y, z in Fig 3.3 is a product of the other two.
- Hint: Look up the generators and relation for H3 at the back of the BB textbook. In order to find a reduced expression for each of x, y, and z, you can color or label (part of) the Weak order poset with the generators. Then try all six possibilities to find the correct identity.
- Submit reading notes on just a few concepts (from Sec 3.2):
- Middle of page 70: definitions and notations of meet (greatest lower bound), complete meet-semilattice. Statement of Theorem 3.2.1
- Middle of page 71: joints (least upper bounds), lattice.
- Go to Figure 3.2 (weak order of S4) and Figure 3.3 (weak order of H3). For each, choose four different subsets and find the the meet and the join of each of your subsets.
- Extra:
- Write down the definition of interval and covering from Appendix A2.2 pg 300-301.
- Draw a small example illustrating Corollary 3.1.4 (pg 69), using either A2 (symmetric group of order 6 with 3 reflections) or A3 (symmetric group of order 24 with 6 reflections)
Notes from virtual meeting August 14
Notes from in person meeting Friday August 24
- Overleaf for Rachel:
- Clean up all the todo. Re-read some explanations and clean up/ add an English word at the beginning of each sentence.
Notes from in person meeting Tue Sept 4, 2018 and also Wed, Sept 19, 2018.
- Overleaf for Rachel:
- Just on paper (or on Overleaf if you want), describe a direct bijection between the subwords and the snake graph matchings. Follow the slides, or come up with another explanation that makes more sense to you.
- Continue to clean up (including describing LRS trie of subwords)
- Overleaf for Emily
- Continue to clean up the parts I copied and pasted from slides (sign function and snake graph)
- Make sure to add: In Example 7 (there is a todo yellow note), refer to the picture from slides 1011101100, but with numbers instead of letters. DONE.
- In fig of trie with red green circles, add the curves to indicate T2 T3 etc (maybe for the future).
LaTeX and Tikz resources:
See more algebra/SageMath resources
An assortment of intro to cluster algebras notes and papers: