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T-avoiding for type A
New section:
- Definition and examples for Property T
- Definition and examples for T-avoiding
- Lemma: If product of commuting generators, then T-avoiding.
- Question: Are there others?
- Definition and example for "bad"
- Theorem: T-avoiding iff product of commuting generators (i.e., no bad elements).
- Theorem: If Property T, then "hard" converts to "easy
Section 2.2
Just for general knowledge and to help me understand the way the paper is defining equivalence classes in regards to commutation classes. It defines = (but squiggly) as by taking the reflexive transitive closure of ~ which we have defined as the relation between words (w'~w if they differ by a single commutation).
The definition I found for reflexive transitive closure is here
Given R is a relation, reflexive transitive closure of R is the smallest relation S on A such that
RโS;
S is reflexive; and
S is transitive.
In the context is this extending ~ so that things that differ by multiple commutations are in the same commutation class. For example in A_3 the element 13231=13213=31231=31213. They are all in the same commutation class but 13231 and 31213 differ by two commutations?
Insert missing figures in T-avoiding in type A section
Question
Since we are claiming in the hard case that \sigma is not fully commutative, do we need to have a case showing that sigma being fully commutative has Property T?
Add a bib file
Write Intro
Do last-ish.
Type B needs an intro
Fiddle with overlaying strings on heaps
Check out my students' old slides that use pstricks.
Fix code to draw "sideways" heaps
Currently, the heap TikZ code draws heaps so that top corresponds to left. We want left column of heap to correspond to left descent set.
Write abstract
Do this last.
Retool heap examples to type A
At least temporarily, let's stick with examples in type A.
At least for now, comment out FC stuff
I don't think we need FCness.
Retool old heap styles
Create "Type A stuff" section
This section should include:
- Connection to symmetric group
- String diagrams
- Examples of string diagrams
- Connection between string diagram diagrams and heaps (in type A)
What is a saturated chain in reference to a saturated heap?
It sounds like from what I looked up a saturated chain is where in an representation of a group element we build everything in between. IE a saturated chain between 2 and 5 would be 3,4. Is this correct?
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