# Sage Days 65June 8 – June 12, 2015A mathematics and computer programming workshop at...

## Loyola University Chicago

IES Building (#38), Rooms 123 & 124.

Main Focci

• Development of Sage code in support of MV-polytopes and affine crystals.
• Development of Sage code in support of combinatorial Hopf algebras.
• Tutorials designed to get newcomers to Sage as up to speed as possible in a week!

Tentative Schedule

 Mon Tue Wed Thu Fri 9:30 Coffee Coffee Coffee Coffee Coffee 9:45 10:00 open 10:15 10:30 10:45 11:00 Project Intros Tutorial: Thiruvathukal+Albert Tutorial: Tingley+Peters Tutorial: Lauve Tutorial: open 11:15 11:30 Tutorial: Doty 11:45 12:00 Lunch Lunch / Free Afternoon Lunch Final Progress Reports 12:15 Lunch 12:30 12:45 13:00 13:15 13:30 13:45 14:00 14:15 14:30 14:45 15:00 Coffee Coffee 15:15 15:30 Coffee Small groups (coding/tutorials) Small groups (coding/tutorials) 15:45 16:00 Small groups (coding/tutorials) 16:15 16:30 16:45 17:00 Progress Reports Progress Reports 17:15 17:30 Progress Reports 17:45 18:00 18:15 18:30 18:45 19:00

(Participant-Supplied) Goals for the Week

• Develop code for Hopf monoids in species
• Learn how to use SAGE in my classroom
• Resume coding basic algebraic structure for KLR-algebras, quantum shuffle algebras, etc
• Start a wiki for combinatorial Hopf algebras, in the format of FindStat
• Crystals of tableaux for the Lie superalgebra gl(m|n)
• Improve NC-Gröbner basis calculations, implement dual Quasi-Schur basis #18447
• Non-commutative version of Faugere's F5 algorithm in Sage
• Code test for satisfaction of $$A_\infty$$-algebra relations

Abstracts

 Monday Franco Saliola Let's Start Using Sage! A whirlwind tour of what Sage can and cannot do (and why you should care). Stephen Doty Getting Started with the Sagemath Cloud Sagemath Cloud is a recent project to make Sage (and much more: e.g., Python, R, LaTeX, Terminal) available in any modern browser, without the need to install anything on the computer. This will be an introduction, with no prerequisites. Dinakar Muthiah MV polytopes in finite and affine type MV polytopes provide a model for highest weight crystals in finite and affine type. Interest in MV polytopes comes from the variety of different contexts in which they appear: MV cycles in the affine Grassmannian, irreducible components in preprojective varieties, character-support for KLR modules, and PBW bases. They also can be constructed purely combinatorially. I will focus on the combinatorics of MV polytopes and briefly mention the other contexts in which they appear. I will also discuss the MV polytope code that we have already written and explain some of the tasks that remain. Nantel Bergeron Homogeneous, Non-commutative Gröbner Bases Computing a non-commutative Gröbner basis takes an extremely long time. I will present the algorithm and indicate where it could be parallelized... Tuesday Anne Schilling Algebraic Combinatorics in Sage: How to use it, make it, and get it into Sage We will very briefly discuss the history of combinatorics in Sage and give some examples on how to use some features like crystals, permutations and words. We will then implement some new missing features together and see how to get them into Sage. Mark A. & George T. Code collaboration in SAGE and other open source projects We will have a brief introduction to the typical organizational structures and technologies used by large-scale open source projects and how one can contribute at various levels in each. This will be followed by a tutorial for working collaboratively on code to contribute directly to the SAGE environment. Mike Zabrocki How to program a combinatorial Hopf algebra (with bases) I will review the structure of the code for combinatorial Hopf algebras (symmetric functions/partitions, quasi-symmetric functions/compositions, non-commutative symmetric functions/compositions, symmetric functions in non-commuting variables/set partitions) that are already in Sage and explain how to create a new combinatorial Hopf algebra on another set of combinatorial objects. I will also point out the ongoing work on open tickets to implement other combinatorial Hopf algebras (packed words #15611, FQSym, WQSym, PQSym #13793, PBT/Loday-Ronco #13855) Wednesday Ben Salisbury Affine crystals in Sage I will give a brief overview of affine crystals (both irreducible highest weight affine crystals and affine Verma crytals) before discussing certain implementations of these crystals in Sage. I will also point to some current Sage work in this area as well as possible extensions beyond. Peter T. & Emily P. Linear Algebra in Sage We will lead a session on figuring out how to get sage to do something. This will mostly consist of participants working together to try and figure stuff out. That stuff will be from linear algebra and, if things go well, random matrix theory. Thursday Simon King An F5 algorithm for modules over path algebra quotients and the computation of Loewy layers The F5 algorithm is a signature based algorithm to compute Gröbner bases for modules over polynomial rings. The F5 signature allows to exploit commutativity relations in order to avoid redundant computations. When considering modules over path algebra quotients, one can instead exploit the quotient relations to avoid redundancies.   For my applications, it is important that Gröbner bases are actually not more than a by-product of the F5 algorithm. Indeed, the F5 signature provides additional information: If the quotient algebra is a basic algebra and if a negative degree monomial ordering is used, then the F5 signature allows to read off the Loewy layers of the module. Aaron Lauve Convolution Powers: step by step I share my personal story (I want to say "natural progression" but I'm sure it's nothing of the kind) from perceived gap in the Sage code for Hopf algebras to sage-trac ticket submission. George Seelinger TBA ... Jonathan Judge Root Multiplicities for Kac-Moody Algebras in Sage Root multiplicities are fundamental data in the structure theory of Kac-Moody algebras. We will give a brief survey on root multiplicities that highlights the differences between finite, affine, and indefinite types. Then we will describe the two main ways that these multiplicities are computed, namely Berman-Moody's formula and Peterson's recurrent formula. Lastly, we demonstrate an implementation of Peterson's recurrent formula in Sage. Friday open ...

Organizers

Funding

Limited travel and lodging support is available for early career researchers.
Deadline for requests: February 28 (sagedays@math.luc.edu).

Local Information

Location: Conference talks and coding sprint rooms will be in the IES Building (#38), Rooms 123 & 124, on the Lakeshore campus, near the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL.

Parking: Daily parking is available on-campus for \$7 in the Parking Garage (building P1 on the Lakeshore campus map). To get to the Parking Garage, enter campus at the corner of West Sheridan Road and North Kenmore Avenue. Overnight parking is also available (details).

Housing: A block of rooms is being held in San Francisco Hall, immediately adjacent to IES. (Register | Instructions: choose "Any Location" and use promotion code "sagedays") All rooms are Jack&Jill suites, which are two rooms with a shared bathroom. Attendees wishing to share their room to control costs should contact the organizers at sagedays@math.luc.edu. Alternatively, there are a number of reasonable hotel options in Evanston and the Chicago Loop that are a short drive or train-ride away. (Don't hesitate to ask the organizers for advice.)

Participants

• Darlyne Ann Addabbo (U Illinois)
• Mark V. Albert (Loyola Chicago)
• Nantel Bergeron (York U, Canada)
• Kevin Dilks (U Minnesota)
• Stephen Doty (Loyola Chicago)
• Mervin Fansler (tentative)
• Gabriel Feinberg (Haverford)
• Emily Gunawan (U Minnesota)
• Christine Haught (Loyola Chicago)
• Mee Seong Im (U Illinois)
• Jonathan Judge (U Connecticut)
• Wongeun Kim (CUNY)
• Simon King (FSU Jena, Germany)
• Michael Kratochvil (Loyola Chicago)
• Aaron Lauve (Loyola Chicago)
• Jake Levinson (U Michigan)
• Dinakar Muthiah (U Toronto)