# Seminar Archive

Semester 1 programme, Session 2023-24

**Thursday 14th September 2023**: Jack Allsop (Monash University)

**Row-Hamiltonian Latin squares and perfect 1-factorisations**

Abstract: A Latin square of order n is an n x n matrix of n symbols, such that each symbol occurs exactly once in each row and column. Let L be a Latin square of order n. Each pair of distinct rows of L forms a 2-line permutation. If this permutation is a single n-cycle, for any choice of rows, then L is called row-Hamiltonian. Each Latin square has six conjugate Latin squares, obtained by uniformly permuting the coordinates of its (row, column, symbol) triples. Let \nu(L) denote the number of conjugates of L which are row-Hamiltonian. It is known that \nu(L) \in \{0, 2, 4, 6\} and for each m \in \{0, 2, 6\} there are known infinite families of Latin squares with \nu = m. We construct the first known infinite family of Latin squares with \nu=4.

A 1-factorisation of a graph is a partition of its edge set into 1-factors. A 1-factorisation is perfect if the union of edges in any pair of its 1-factors forms a Hamiltonian cycle. A perfect 1-factorisation of the complete bipartite graph K_{n, n} is equivalent to a row-Hamiltonian Latin square of order n. Our family of Latin squares with \nu=4 allows us to build the eighth known infinite family of perfect 1-factorisations of complete bipartite graphs.

**Thursday 28th September 2023**: Rosemary Bailey (St Andrews)

**Designs on strongly regular graphs**

Abstract: I will talk about experiments where each treatment is applied to some of the vertices of a strongly regular graph. The corresponding Bose-Mesner algebra has three common eigenspaces. I assume that these are also the eigenspaces of the variance-covariance matrix of the responses on the vertices.

There are two (conflicting!) desirable properties of the design that do not depend on the eigenvalues of this matrix. I will describe both of these in the context of two familiar strongly regular graphs.

**Thursday 12th October 2023**: Scott Harper (St Andrews)

**Bases and generating sets for finite groups**

**Thursday 26th October 2023**: Bromlyn Cameron (St Andrews)

**Transition chains and Thompson’s groups**

Abstract: In the 1960s R. Thompson defined a family of three finitely presented groups of homeomorphisms of the Cantor set. Two of these groups, T and V, were the first known examples of finitely presented infinite simple groups. We are interested in the other group – Thompson’s group F – and more specifically groups of PL_+(I) homeomorphisms which do not contain an isomorphic copy of F. In this talk we will introduce transition chains and discuss their place in the study of F-less groups of homeomorphisms.

**Thursday 9th November 2023**: Florent Hivert (University Paris-Saclay / Orsay)

**Diagrammatic for Okada monoid and algebra**

Abstract: It is well known that the Young lattice is the Bratelli diagram of the symmetric groups expressing how irreducible representations restrict from S_N to S_{N-1}. In 1975, Stanley discovered a similar lattice called the Young-Fibonacci lattice which was realized as the Bratelli diagram of a family of algebras by Okada in 1994. In this talk we will present a combinatorial model for the Okada algebra and the associated monoid using a labeled version of the arc diagrams of the Jones monoid and the Temperley-Lieb algebra. We prove that the cardinality of the Okada monoid and dimension of the Okada algebra is n! in full generality (it was only proven in the semisimple case by Okada). In particular, we interpret the natural bijection between permutations and labeled arc diagrams as an instance of the Robinson-Schensted-Fomin correspondence associated to the Young-Fibonacci lattice. This has a lots of algebraic consequences: the aperiodicity of the Okada monoid and its Green relations as well as the cellularity of the Okada algebra. (Joint work with Jeanne Scott.)

**Thursday 23rd November 2023**: Reinis Cirpons (St Andrews)

**Computing in Free Bands**