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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