The Algebra, Combinatorics, Geometry, and Topology (ACGT) Seminar meets on Tuesdays at 12:45-2:00pm in Room 164 of the Adel Mathematics Building. If you are interested/willing to give a talk, please contact Dana C. Ernst, ACGT Coordinator.

Note that talks are listed in reverse chronological order.

**Date:** 12/8

**Speaker:** Dana C. Ernst (NAU)

**Abstract:** Recall that the symmetric group $S_n$ is generated by the adjacent 2-cycles $(1,2),(2,3),\ldots, (n-1,n)$. That is, every element in $S_n$ can be written as a word using the alphabet consisting of the adjacent 2-cycles. It is important to note that there are potentially many different ways to express a given permutation as a product of adjacent 2-cycles. If we express a permutation as a product of adjacent 2-cycles in the most efficient way possible, then we call the expression a **reduced expression**. There may be many different reduced expressions for a given permutation, but all of them can be written in terms of the same number of adjacent 2-cycles occurring in the product (called the **length**).

We say that two reduced expressions are **commutation equivalent** if we can obtain one from the other by only commuting disjoint adjacent 2-cycles (no need to apply any **braid moves**). A **commutation class** of a permutation is the subset of all its reduced expressions that can be obtained from one another by commuting disjoint cycles. For example, there are 11 reduced expressions for $(1,3,5,4)$ that split into 2 commutation classes consisting of 7 and 4 reduced expressions, respectively. The **longest element** in $S_{n}$ is the (unique) element having maximal length. The number of reduced expressions for the longest element is known. However, the answer to the following question, originally posed by Richard Stanley, is unknown:

How many commutation classes does the longest element in the symmetric group have?

In $S_{4}$, the longest element is $(1,4)(2,3)$. In this case, it turns out that there are 8 commutation classes.

Dustin Story (NAU undergrad) and I (with some recent help from Nandor Sieben) have been playing with the problem given above. In particular, we have most of a proof for a bound on the number of commutation classes. In this talk, I will describe the original problem together with a few problems to which it is equivalent and outline the proof for our bound.

**Dates:** 11/17, 12/1

**Speaker:** Dana C. Ernst (NAU)

**Abstract:** The Sylver Coinage Game is a game in which 2 players, *A* and *B*, alternately name positive integers that are not the sum of nonnegative multiples of previously named integers. This seemingly innocent looking game is the subject of one of John Conway’s open problems with monetary rewards. In particular, the question that Conway asks is: “If player *A* names 16, and both players play optimally thereafter, then who wins?” I’m currently mentoring an undergraduate research project with 4 students (Joni Hazelman, Parker Montfort, Robert Voinescu, and Ryan Wood) aimed at exploring a simplified version of the Sylver Coinage game. In the simplified version of the game, a fixed positive integer $n\geq 3$ is agreed upon in advance. Then 2 players, *A* and *B*, alternately name positive integers from the set $\{1,2,\ldots,n\}$ that are not the sum of nonnegative multiples of previously named numbers among $\{1,2,\ldots,n\}$. The person who is forced to name 1 is the loser! One goal is to determine who wins under optimal play for given values of $n$. Moreover, we want to compute the Nim-values for the simplified game. We have some preliminary results, but now we are stuck. I’ll introduce the game, tell you what we currently know, and show you where we are stuck.

**Dates:** 10/27, 11/3, 11/10

**Speaker:** Jim Swift (NAU)

**Abstract:** Given a graph, each vertex represents an identical dynamical system (a cell), and the edges represent coupling between the cells. Some invariant subspaces can be predicted from the automorphism group of the graph, but additional “anomalous” invariant subspaces are present in many cases. This series of talks describes the combinatorial and algebraic understanding of the invariant subspaces.

**Dates:** 9/8, 9/15, 9/22, 9/29, 10/6, 10/13, 10/20

**Speaker:** Michael Falk (NAU)