September 7: Neha Siddiqui (LUC). On a Three-Letter 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 Burrows-Wheeler transform, minima of binary quadratic forms, positive generation of the free group \(F_2\), and on and on. In this talk, we define three-letter Christoffel words using a tiling of the plane by hexagons. We investigate what elements of the two-letter theory carry over to the new context—such as lower-upper conjugation, palindromicity, unique (Christoffel) factorizations, and Christoffel morphisms—as well as differences between the two.
(This ongoing work started as part of a LUROP research project this past summer, joint with Aaron Lauve.)

September 23: Stuart Martin (Cambridge). Young-Jucys-Murphy 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 multiplicity-free 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 finite-dimensional semisimple algebras, formulae for computing primitive idempotents follow from well-known Frobenius-Schur 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 YJM-sequence 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 so-called 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 multiplicity-free 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 multiplicity-free family, then one can solve this problem by computing the eigenspectrum of a sequence of Jucys-Murphy operators on the simple representations. This extends some well-knowncombinatorics associated with symmetric groups to other, more exotic, multiplicity-free 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 finite-dimensional 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 Iyama-Reading-Reiten-Thomas, the latter of which included a representation-theoretic 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 representation-theoretic interpretations of the join-irreducible torsion classes, and the so-called 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 non-standard ribbon element, which then allows one to consider a larger topological category.