Cyclic Symmetry of the Scaled Simplex

Nathan Williams

University of Minnesota

July 11, 2012

Joint with Hugh Thomas

A2 Root System

Root Systems

A collection of vectors invariant under reflections like this slide.

(among other conditions...
and yes, I have projected A2 into 2 dimensions.)

why is this type different from all other types?
A2 Root System

A2 Root System

Weyl Group for A2

Weyl Groups

Generated by reflections in the roots.
The reflections break the space into chambers.
We can label the chambers by elements of the group.

In A2, labels are permutations of length 3 (=2+1).
Weyl Group for A2
(also known as the symmetric group S3)
Weyl Group for A2
(also known as the symmetric group S3)
Weyl Group for A2
(also known as the symmetric group S3)
Weyl Group for A2
(also known as the symmetric group S3)
Weyl Group for A2
(also known as the symmetric group S3)
We'll work with inverses

Inverses

The difference between acting on the left or the right.

In type A, this is the difference between acting on positions and entries.
Inverses

Inverses

Inverses

Inverses

Affine Symmetric Group

Affine Permutations

Are represented by a window

...-2 -1 0 1 2 3 4 5 6...
Affine Symmetric Group

Affine Symmetric Group

Acting on Affine Permutations

What happens in the window is mirrored throughout.

...-2 -1 0 1 2 3 4 5 6...
Affine Symmetric Group

Affine Symmetric Group
Affine Symmetric Group
Affine Symmetric Group
Affine Symmetric Group
Affine Symmetric Group
Affine Symmetric Group
Affine Symmetric Group
Affine Symmetric Group
Affine Symmetric Group
Affine Symmetric Group
4-Fold Dilation of the Fundamental Alcove in Type A2

Since dimension is 2, contains 42 alcoves.

Has a cyclic symmetry of order 3 (=2+1).

To understand the orbit structure of the alcoves,
we label the alcoves with words
so that the cyclic action on alcoves becomes rotation
on words.

4-Fold Dilation of the Fundamental Alcove in Type A2

Alcoves are labeled by words

• of length 3,
• with letters taken from {0,1,2},
• and with sum 3 mod 4.
(So there are 42 of them)
Rotation gives a cyclic symmetry of order 3.

Theorem (H. Thomas, W.):
There is an equivariant bijection between the alcoves of the m-fold dilation of the fundamental alcove in Ak and words

• of length k+1,
• with letters taken from {0,1,...,m-1},
• and with sum (m-1) mod m.

Affine Symmetric Group
Affine Symmetric Group
Abacus
Cores
Cores
A (k+1)-core is a(n integer) partition with no hook-length a multiple of (k+1).

(This is a 3-core.)
Cores
A (k+1)-core is a(n integer) partition with no hook-length a multiple of (k+1).

(This is a 3-core.)
Cores
A (k+1)-core is a(n integer) partition with no hook-length a multiple of (k+1).

(This is a 3-core. And also a 6-core.)
Cores
The affine symmetric group Ak acts on (k+1)-cores (just like with affine permutations).

Cores
The affine symmetric group Ak acts on (k+1)-cores.

By s1
Cores
The affine symmetric group Ak acts on (k+1)-cores.

By s1
Cores
The affine symmetric group Ak acts on (k+1)-cores.

By s1
Cores
The affine symmetric group Ak acts on (k+1)-cores.

By s2
Cores
The affine symmetric group Ak acts on (k+1)-cores.

By s2
Cores
The affine symmetric group Ak acts on (k+1)-cores.

By s2
Cores
The affine symmetric group Ak acts on (k+1)-cores.

By s0
Cores
The affine symmetric group Ak acts on (k+1)-cores.

By s0
Cores
The affine symmetric group Ak acts on (k+1)-cores.

By s0
Cores
Cores
Cores
Cores
Cores
Cores
Cores
Cores
Cores
Cores
Abacus

An abacus diagram represents a alcove
iff its columns are flush.

(It encodes the boundary of a core.)

Abacus
Abacus Sum
Boundary Path Words

We want to understand the cyclic action...

...But these words don't encode the cyclic symmetry by rotation.
So we'll force them to do so by using their action against them.

We'll pick off one piece of information from each word in an orbit to make new words
with the desired property.

I call this a bijaction.

Thank You.

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