Organizers: Aaron Lauve, Peter Tingley
September  
7 *4:00* 
Neha Siddiqui (LUC)  On a ThreeLetter Generalization of Christoffel Words 
14 
No talk  
23 *Friday* 
Stuart Martin 
YoungJucysMurphy elements for Schur algebras 
28 
Emily Gunawan 
Cluster algebraic interpretation of infinite friezes 
30 *Friday* 
Fred Goodman 
Nice bases for Brauer's centralizer algebras 
October  
5 


12 
No talk (fall break) 

19 


26 


November  
2  
9 


16 


23 
No talk (thankgiving) 

30 

December  
7  
Directions 

IES (the institute for environmental sustainability) is located at the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL (map)
Parking is available oncampus for $7 in the Parking Garage (building P1 on the Lake shore campus map). To get to the Parking Garage, enter campus at the corner of West Sheridan Road and North Kenmore Avenue.
Abstracts 


September 7: Neha Siddiqui (LUC). On a ThreeLetter Generalization of Christoffel Words
A \((p,q)\)Christoffel word is a sequence of letters "a" and "b" that provides a representation of
a line segment in the plane of slope \(q/p\).
These words are connected to several areas of mathematics and science,
including Diophantine equations, computer graphics, the BurrowsWheeler transform, minima of binary
quadratic forms, positive generation of the free group \(F_2\), and on and on.
In this talk, we define threeletter Christoffel words using a tiling of the plane by hexagons.
We investigate what elements of the twoletter theory carry over to the new context—such as
lowerupper conjugation, palindromicity, unique (Christoffel) factorizations, and Christoffel
morphisms—as well as differences between the two. 
September 23: Stuart Martin (Cambridge). YoungJucysMurphy elements for Schur algebras (Joint work with Stacey Law.) We study the Schur algebra \(S(n,r)\) in characteristic 0 or \(>r\). We produce a multiplicityfree family involving Schur algebras which reflects the branching properties of general linear groups. The motivation is to "glue together" Schur algebras up to some degree. We construct such a family \(A^+( r)\) for each \(r\in\mathbb N\) and show it has branching structure similar to that for the polynomial simple modules of GL\({}_n(\mathbb C)\). Then we investigate YJM elements, primitive central idempotents and primitive orthogonal idempotents of \(A^+( r)\). Some of this is known: for finitedimensional semisimple algebras, formulae for computing primitive idempotents follow from wellknown FrobeniusSchur relations involving irreducible characters and entries in irreducible matrix representations (see papers of Neunhoffer/Scherotzke and of Doty/Lauve/Seelinger). The PCIs of the Schur algebras \(S(n,r)\) are described in terms of symmetric group character values and the Green basis of the Schur algebras (as per Geetha/Prasad). Finally we answer a question of Doty to find a YJMsequence for our family \(A^+( r)\). 
September 28: Emily Gunawan (Gustavus Adolphus College). Cluster algebraic interpretation of infinite friezes
Originally studied by Conway and Coveter, friezes appeared in various recreational mathematics publications in the 1970s. We construct a periodic infinite frieze using a class of peripheral elements of a cluster algebra of type D or affine A. We discover new symmetries and formulas relating the entries of this frieze and socalled bracelets. We also present a correspondence between Broline, Crowe and Isaacs's classical matching tuples (which is a generalization of Conway and Coxeter's work) and various recent combinatorial interpretations of elements of cluster algebras from surfaces. This is joint work with Gregg Musiker and Hannah Vogel.
