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# Classification of the T-avoiding permutations and generalizations to other Coxeter groups

A permutation of a set of objects is simply a rearrangement of those objects. If we have n of objects, then a permutation can be represented as a function from 1,2,,n to 1,2,,n. We say that a permutation w has property T if there exists i such that either w(i) is greater than w(i + 1), w(i + 2) or w(i + 2) is less than w(i), w(i + 1). A permutation w is T-avoiding if neither w nor its inverse have property T. We will present a classification of the T-avoiding permutations in the symmetric group, which is a Coxeter group of type A. In addition, we will discuss generalizations to other Coxeter groups and classify the T-avoiding elements in Coxeter groups of types B, which can be thought of as the group that rearranges coins and flips them over.

This poster was presented by my undergraduate research students Joseph Cormier, Zachariah Goldenberg, and Jessica Kelly (Plymouth State University) on April 27, 2012 at the 2012 PSU Research Symposium at Plymouth State University. April 27, 2012

## Transcript

1. Classiﬁcation of the T-avoiding permutations
and generalizations to other Coxeter groups
J. Cormier, Z. Goldenberg, J. Kelly, Plymouth State University
s4
s4
s3
s1
s2
s5
Symmetric Group
The symmetric group Sn
is the collection of bijections from {1, 2, . . . , n} to
{1, 2, . . . , n} where the operation is function composition (left ← right). Each
element of Sn
is called a permutation.
Cycle Notation
One way of representing permutations is via cycle notation.
Example
If σ = (1 3 5 2), then σ(1) = 3, σ(3) = 5, σ(5) = 2, σ(2) = 1, σ(4) = 4.
String Diagrams
A 2nd way of representing permutations is via string diagrams. Given a permutation
σ, there are many ways to draw associated string diagrams. Conventions:
1. no more than 2 strings cross each other at given point,
2. strings are drawn so as to minimize crossings.
Example
The following string diagram corresponds
to the permutation σ = (1 3 5 2).
Theorem
Sn
is generated by the adjacent 2-cycles: s1
= (1 2), s2
= (2 3), . . . , sn−1
= (n−1 n).
Relations of Symmetric Group
Sn
satisﬁes the following relations:
1. s2
i
= 1 for all i (2-cycles have order 2)
2. short braid relations: si
sj
= sj
si
, for |i − j| ≥ 2
3. long braid relations: si
sj
si
= sj
si
sj
, for |i − j| = 1.
Reduced Expressions
If sx1
sx2
· · · sxm
is an expression for σ ∈ Sn
and m is minimal, then we say that the
expression is reduced. By Matsumoto’s Theorem, any two reduced expressions for
σ ∈ Sn
diﬀer by a sequence of braid relations.
Example
Consider σ = s2
s1
s2
s3
s1
s2
∈ S4
. We see that
s2
s1
s2
s3
s1
s2
= s1
s2
s1
s3
s1
s2
= s1
s2
s1
s1
s3
s2
= s1
s2
s3
s2.
The original expression is not reduced, but it turns out that last expression is. The
only reduced expressions for σ are: s1
s2
s3
s2, s1
s3
s2
s3, s3
s1
s2
s3
.
Heaps
A 3rd way of representing permutations is via heaps. Fix a reduced expression
sx1
sx2
· · · sxm
for σ ∈ Sn
. Loosely speaking, the heap for this expression is a set
of lattice points (called nodes) in N × N, one for each sxi
such that:
1. The node corresponding to sxi
has vertical component equal to n + 1 − xi
,
2. If i < j and sxi
and sxj
do not commute, then sxi
is left of sxj
.
Example
There are two distinct heaps for the reduced expressions from previous example:
s3
s2
s1
s2 and s3
s2
s1
s3
Correspondence Between String Diagrams & Heaps
There is a 1-1 correspondence between
string diagrams and heaps. In the absence
of a node, the string “bounces.” s4
s4
s3
s1
s2
Property T
A permutation σ has Property T iﬀ there exists i such that
1. σ(i) > σ(i + 1), σ(i + 2),
i
i + 1
i + 2
or
2. σ(i + 2) < σ(i), σ(i + 1).
i
i + 1
i + 2
Example
If σ = (1 3 5 2), then σ and σ−1 have
Property T in 1 and 2 spots, respectively.
T-avoiding
σ is T-avoiding iﬀ both σ and σ−1 do not have Property T .
Example
The permutation σ = (1 2)(4 5) is T-avoiding.
s4
s1
Theorem (Cormier, Ernst, Goldenberg, Kelly, Malbon)
σ is T-avoiding iﬀ σ is a product of disjoint adjacent 2-cycles (iﬀ heap consists of a
single column iﬀ string diagram consists of “bars” & “X’s”).
Sketch of Proof
Fix a reduced expression for σ and consider its heap. The reverse implication of the
theorem is trivial. For the forward direction, consider the contrapositive.
The Easy Case
The easy case occurs when a node in the 2nd column in on either side is ”blocked”
by at most one node in the 1st column. There are 4 possibilities:
blocked 1x
i
i + 1
i + 2
i
i + 1
i + 2
blocked 1x
i
i + 1
i + 2
i
i + 1
i + 2
The Hard Case
i
i + 1
i + 2
blocked 2x
apply braids up & right
By applying a sequence
of long braid relations,
you can convert a heap
in the hard case to a
heap in the easy case.
Coxeter Groups
A Coxeter group consists of a group W together with a generating set S consisting
of elements of order 2 with presentation
W = S : s2 = 1, (st)m(s,t) = 1 ,
where m(s, t) ≥ 2 for s = t. Since s and t are elements of order 2, the relation
(st)m(s,t) = 1 can be rewritten as
m(s, t) = 2 =⇒ st = ts short braid relations
m(s, t) = 3 =⇒ sts = tst
m(s, t) = 4 =⇒ stst = tsts
.
.
.

long braid relations
Coxeter groups of Types A and B
The symmetric group Sn+1
with the adjacent 2-cycles as a generating set is a Coxeter
group of type An
. Coxeter groups of type Bn
(n ≥ 2) having generating set S =
{s1, s2, · · · , sn
} and deﬁning relations:
1. s2
i
= 1 for all i,
2. si
sj
= sj
si
if |i − j| > 1,
3. si
sj
si
= sj
si
sj
if |i − j| = 1 and 1 < i, j ≤ n,
4. s1
s2
s1
s2
= s2
s1
s2
s1
.
Generalization of Property T to Coxeter Groups
Let (W , S) be a Coxeter group and let w ∈ W . Then w has Property T iﬀ w has a
reduced expression of the form stu or uts, where m(s, t) ≥ 3 and u ∈ W . In terms
of heaps, w is T-avoiding iﬀ no heap for w has the property that a node in the 2nd
column in on either side is ”blocked” by at most one node in outer column.
Theorem (Cormier, Ernst, Goldenberg, Kelly, Malbon)
In types A and B, w ∈ W is T-avoiding iﬀ w is a product of commuting generators.
Concluding Remarks
1. Our advisor has classiﬁed the T-avoiding elements in type C, which consists of
more than just products of commuting generators (e.g.,“sandwich stacks”).
2. Tyson Gern (University of Colorado) has classiﬁed the T-avoiding elements in type
D. Again, classiﬁcation is more complicated than just products of commuting
generators.
3. Our advisor is currently working with a group of students on the classiﬁcation in
type F.
Joint work with J. Cormier, Z. Goldenberg, J. Kelly, and C. Malbon. Research conducted under the guidance of D.C. Ernst, Plymouth State University Typeset using L
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