Cyclic Symmetry of the Scaled Simplex

This notebook was written by Nathan Williams.  It implements large portions of the paper “Cyclic Symmetry of the Scaled Simplex” by Hugh Thomas and Nathan Williams (http://arxiv.org/abs/1207.5240).

Introduction

Definition.  Let consist of the alcoves contained in the m-fold dilation of the fundamental alcove of the type affine hyperplane arrangement.

Example. The 16 alcoves of are those alcoves contained within the thick black lines. As the fundamental alcove has a cyclic symmetry of order (k+1), so does . Let be the set of words of length (k+1) on Z/mZ with sum (m-1)  (mod  m), with the order (k+1) cyclic action given by rotation. As the orbit structure of is easily understood, we determine the orbit structure of with the following theorem.

Theorem. There is an equivariant bijection from under its cyclic action to under rotation.

Example. Each alcove of is labeled with its corresponding word in .  There are 5 orbits of size 3 and a single orbit (the alcove in the center) of size 1.  : a poset of words x of length k on Z/mZ

We first convert to a purely combinatorial object.

Definition. Let be the poset of words x of length k on Z/mZ with the partial order induced by the following covering relations:
1) For a<m-1, ya=(a+1)y, where y is a string of length k-1 on the alphabet Z/mZ.
2) For b<a, yabz=ybaz, where y and z are two strings on the alphabet Z/mZ, with the total length of the two strings being (k-1).

Example. The posets , In:= Out=     Theorem. There is a poset isomorphism between and . The geometric symmetry of the underlying graph is realized by the cyclic action φ.

Proof (sketch). Alcoves of are naturally labeled by increasing affine permutations.  The action of the affine symmetric group on is mirrored on (k+1)-cores, which allows us to pass from to (k+1)-cores.  The abacus diagram of the resulting (k+1)-core is necessarily flush, which allows us to just record the number of runners in each column.  For more detail on this bijection, see the paper or the presentation at http://www.math.umn.edu/~will3089/docs/SimplexPresentation/simplex.htm.

Example. The posets ,  Theorem. The graph of has (k+1)-fold cyclic symmetry, realized by an explicit action φ.

Proof (sketch). We will define a cyclic action φ  of order (k+1) that is a graph isomorphism.  Given a word , form the extended word of length (k+1)m That is, for any 1≤i≤(k+1), the entries in positions i,(k+1)+i,2(k+1)+i,…(m-1)(k+1)+i are cyclically decreasing by 1. Let be defined by cyclically rotating left so that its leftmost 0 appears as its rightmost character. This induces an action φ on by restricting the resulting word to its first k letters.  Observe that φ is a cyclic action of order (k+1), since there are (k+1) zeros in .  It is now a tedious check to show that φ takes edges to edges. Note that φ is not a poset isomorphism: it reverses the orientation of some edges.

Example. The graphs of , In:= Out=     : Words of length (k+1) that sum to (m-1)

Definition. Let be the set of words on Z/mZ of length (k+1) with sum equal to (m-1)  (mod  m).

We let act by left rotation, sending to .

Theorem. There is an equivariant bijection w from under φ to under left rotation.

Proof (sketch). Take the first letter of each word from an orbit of under φ, and concatenate these letters into a single word, cyclically repeated to make the resulting word of length (k+1). (This trick is called a bijaction.)  By construction, φ maps to left rotation. A description of the (much more complicated) inverse is in the paper. is the set of all words of length 2 on {0,1,2}.   to --the set of all words of length 3 on {0,1,2} that sum to 2 mod 3.   It turns out that finding is equivalent to the following problem.

Dendrodistinctivity

Definition. Let be the set of all words of length (k+1) on Z/mZ.

Definition. Let and let be an (m-1)-tuple with entries in {0,1,2,…,k+1}. Define a partitioned word to be a partition of w into m connected blocks, where b specifies where the dividers are placed. We denote the set of all partitioned words on Z/mZ with w of length k+1 by .

Definition. Define an infinite complete m-ary tree by
1) The 0th rank consists of the empty word φ, partitioned as .
2) The children of a partitioned word (w,b) are the m words obtained by prepending -i  (mod  m) to . .  Theorem [Dendrodistinctivity].
Let be the infinite complete m-ary tree obtained from by replacing each partitioned word (w,b) with its underlying word w. Then each word of length k+1 on Z/mZ appears exactly once. .  The proof is technical (see sections 8 and 9 of the paper): it first constructs a partition that is an approximation to the desired partition.  This approximation is then adjusted to the desired partition.  Code In:=  : Words of length (k+1) that sum to (m-1)

In:= Dendrodistinctivity

In:= 