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 
Steve Doty 
Canonical idempotents in multiplicityfree families of algebras 
November  
2  
9 


16 
Andrew Carroll (DePaul) 
Canonical Join Representations for Torsion Classes 
23 
No talk (thankgiving) 

30 
Peter Tingley (Loyola) 
A minus sign that used to annoy me but now I know why it is there.i 
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.

September 30: Frederick Goodman (Iowa). Nice bases for Brauer's centralizer algebras
In 1937, Richard Brauer defined certain diagrammatic algebras in order to describe the centralizer
algebras for the complex orthogonal or symplectic groups acting on tensor space. The abstract Brauer
diagram algebras and their representation theory can be understood using the framework of cellular
algebras. The actual Brauer centralizer algebras, acting on tensor space, are quotients of
specializations of the diagram algebras, and, somewhat paradoxically, the cellular structure of the
abstract diagram algebras seemed to have no implications for the structure or representation theory of
the concrete centralizer algebras, as cellularity does not generally pass to quotient algebras.
We have resolved this paradox by producing cellular bases of the diagram algebras that split into a
basis of the kernel of the representation on tensor space, and a a piece that survives as a cellular
basis of the concrete centralizer algebras. This can be viewed as an integral version of the 2nd
fundamental theorem of invariant theory for the orthogonal and symplectic groups. It also has the
consequence that those irreducible representations of diagram algebras (with integer loop parameter) that
factor through the representation on tensor space, are actually defined over the integers (as are all
irreducible representations of the symmetric groups in characteristic zero).

October 26: Steve Doty (Loyola). Canonical idempotents in multiplicityfree families of algebras
(Joint work with Aaron Lauve and George Seelinger). I will discuss an elementary problem: how to compute a complete set of primitive orthogonal idempotents in a split semisimple finite dimensional algebra. If the algebra in question fits into a multiplicityfree family, then one can solve this problem by computing the eigenspectrum of a sequence of JucysMurphy operators on the simple representations. This extends some wellknowncombinatorics associated with symmetric groups to other, more exotic, multiplicityfree families, of which there are many examples in the recent literature. This approach is based on the new approach to the representations of symmetric groups as explained in two papers of Vershik and Okounkov (1996, 2004). I will outline the results (many of which are easy extensions of previous known results scattered across the literature) and discuss how they apply to symmetric group algebras and Brauer algebras.

November 16: Andrew Caroll (dePaul). Canonical Join Representations for Torsion Classes
The lattice structure of the poset of torsion classes over a finitedimensional associative algebra has recently seen a resurgence of attention. Beautiful connections between these posets and the combinatorics of the weak order on associated Coxeter groups have been explored by Mizuno, and IyamaReadingReitenThomas, the latter of which included a representationtheoretic description of the objects known as shards, which appear in the combinatorics of reflecting hyperplane arrangements for said Coxeter groups. In this talk, I will review some of these connections, and describe new work that extends the aforementioned connections. In particular, we give representationtheoretic interpretations of the joinirreducible torsion classes, and the socalled canonical join representations of elements. Background notions from both representation theory and combinatorics will be provided. This is based on joint work with Emily Barnard, Gordana Todorov, and Shijie Zhu.

November 30: Peter Tingley (Loyola). A minus sign that used to annoy me but now I know why it is there
We consider two well known constructions of link invariants. One uses skein theory: you resolve each crossing of the link as a linear combination of things that don't cross, until you eventually get a linear combination of links with no crossings, which you turn into a polynomial. The other uses quantum groups: you construct a functor from a topological category to some category of representations in such a way that (directed framed) links get sent to endomorphisms of the trivial representation, which are just rational functions. Certain instances of these two constructions give rise to essentially the same invariants, but when one carefully matches them there is a minus sign that seems out of place. We discuss exactly how the constructions match up in the case of the Jones polynomial, and where the minus sign comes from. On the quantum group side, one is led to use a nonstandard ribbon element, which then allows one to consider a larger topological category.
