# Algebra and Combinatorics Seminar, Fall 2016

### Wednesdays, 3:45–4:45. IES 111

Webpages from previous semesters

Organizers:  Aaron Lauve, Peter Tingley

 September 7 *4:00* Neha Siddiqui (LUC) On a Three-Letter Generalization of Christoffel Words 14 No talk 23 *Friday* Stuart Martin Young-Jucys-Murphy 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 on-campus 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 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.