dcernst / t-avoiding Goto Github PK

New section:

1. Definition and examples for Property T
2. Definition and examples for T-avoiding
3. Lemma: If product of commuting generators, then T-avoiding.
4. Question: Are there others?
5. Definition and example for "bad"
6. Theorem: T-avoiding iff product of commuting generators (i.e., no bad elements).
7. Theorem: If Property T, then "hard" converts to "easy

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
R⊆S;
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?

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?

Do last-ish.

Check out my students' old slides that use pstricks.

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.

Do this last.

At least temporarily, let's stick with examples in type A.

I don't think we need FCness.

This section should include:

1. Connection to symmetric group
2. String diagrams
3. Examples of string diagrams
4. Connection between string diagram diagrams and heaps (in type A)

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?