Anne Schilling, UC Davisquestions in representation theory and mathematical physics by combinatorial...

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Crystal story Programming Wishlist

Why Sage?Features we have and want in combinatorics in Sage

Anne Schilling, UC Davis

IMA, Minnesota, August 21, 2017

Eclipse

http://www.vogella.com/tutorials/Python/article.html

http://www.eweek.com/development/eclipse-behind-the-name

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Why Sage?Features we have and want in combinatorics in Sage

Anne Schilling, UC Davis

IMA, Minnesota, August 21, 2017

Eclipse

http://www.vogella.com/tutorials/Python/article.html

http://www.eweek.com/development/eclipse-behind-the-name

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A Story

One of my passions are

crystal bases which provide a combinatorial tool to studyalgebraic/geometric structures such as

quantum groups

affine Schubert calculus

symmetric functions

representation theory

Combinatorics lends itself to computational analysis!

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A Story



quantum groups


symmetric functions



2/9


A Story



quantum groups


symmetric functions



2/9


A Story



quantum groups


symmetric functions



2/9


A Story



quantum groups


symmetric functions



2/9


A Story



quantum groups


symmetric functions



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Crystals

B( ) B( )

1

3

2

1

2

1 22

1 23

1 13

1 12

2 23

2 33

1 33

1 32

2

2

2

21

1

1

1

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Lonely Programming ...

Programmed crystals in Mathematica

Programmed what I needed right then for research

No tests or documentation

Could not reuse my own code a few weeks later (forgot how itworked ...)

Kept writing similar code over and over

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Sage Days 7 at IPAM in 2008

with Nicolas Thiery started Dan Bumpporting crystal code to Sage uses crystals in number theory

William Stein Franco Saliola Mike Hansen Mike Zabrocki

https://wiki.sagemath.org/days7

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What can Sage do?� �sage: B = crystals.Tableaux(['A',2],shape=[2,1])sage: u = B.highest_weight_vector(); u[[1, 1], [2]]sage: b = u.f(1); b[[1, 2], [2]]sage: type(b)<class 'sage.combinat.crystals.tensor_product.CrystalOfTableaux_with_category.element_class'>sage: u.weight()(2, 1, 0)sage: b.weight()(1, 2, 0)� �

But the following does not work ...� �sage: b.shape()-------------------------------------------------------------AttributeError:'CrystalOfTableaux_with_category.element_class' objecthas no attribute 'shape'sage: b.to_tableau().shape()[2, 1]� �

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Implementation of a crystal� �class HighestWeightCrystalOfTypeA(UniqueRepresentation, Parent):

def __init__(self, n = 3):Parent.__init__(self, category = ClassicalCrystals())self.n = nself._cartan_type = CartanType(['A',n])self.module_generators = [ self(1) ]

def _repr_(self):return "Highest weight crystal of type A_%s

of highest weight omega_1"%(self.n)

class Element(ElementWrapper):

def e(self, i):if self.value == i+1:

return self.parent()(self.value-1)else:

return None

def f(self, i):if self.value == i:

return self.parent()(self.value+1)else:

return None� �

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Moral of the Story ...End/beginning of the Story ...

Semester long program at ICERM onAutomorphic Forms, Combinatorial Representation Theoryand Multiple Dirichlet Series, Spring 2013

Thematic Tutorial: Lie Methods and Related Combinatorics inSage

World ScientificWorld Scientificwww.worldscientific.com9876 hc

ISBN 978-981-4733-43-4

BumpSchilling

CRYSTAL BASES Representations and Combinatorics

CRYSTAL BASES

Daniel Bump

Anne Schilling


This unique book provides the first introduction to

crystal base theory from the combinatorial point of

view. Crystal base theory was developed by Kashiwara

and Lusztig from the perspective of quantum groups.

Its power comes from the fact that it addresses many

questions in representation theory and mathematical

physics by combinatorial means. This book approaches

the subject directly from combinatorics, building

crystals through local axioms (based on ideas by

Stembridge) and virtual crystals. It also emphasizes

parallels between the representation theory of the

symmetric and general linear groups and phenomena

in combinatorics. The combinatorial approach is

linked to representation theory through the analysis

of Demazure crystals. The relationship of crystals to

tropical geometry is also explained.

k -Schur functions and affine Schubert calculus

http://icerm.brown.edu/sp-s13


http://www.sagemath.org/doc/thematic_tutorials/lie.html


http://www.worldscientific.com/worldscibooks/10.1142/9876

https://arxiv.org/abs/1301.3569

8/9


Moral of the Story ...End/beginning of the Story ...

Semester long program at ICERM onAutomorphic Forms, Combinatorial Representation Theoryand Multiple Dirichlet Series, Spring 2013

Thematic Tutorial: Lie Methods and Related Combinatorics inSage

World ScientificWorld Scientificwww.worldscientific.com9876 hc

ISBN 978-981-4733-43-4

BumpSchilling


CRYSTAL BASES

Daniel Bump

Anne Schilling


This unique book provides the first introduction to

crystal base theory from the combinatorial point of

view. Crystal base theory was developed by Kashiwara

and Lusztig from the perspective of quantum groups.

Its power comes from the fact that it addresses many

questions in representation theory and mathematical

physics by combinatorial means. This book approaches

the subject directly from combinatorics, building

crystals through local axioms (based on ideas by

Stembridge) and virtual crystals. It also emphasizes

parallels between the representation theory of the

symmetric and general linear groups and phenomena

in combinatorics. The combinatorial approach is

linked to representation theory through the analysis

of Demazure crystals. The relationship of crystals to

tropical geometry is also explained.

k -Schur functions and affine Schubert calculus





http://www.worldscientific.com/worldscibooks/10.1142/9876

https://arxiv.org/abs/1301.3569

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Wishlist

trac #23560 Benkart-Kang-Kashiwara crystals for super Liealgebras

trac #22921 Shifted tableaux, Krazkiewicz insertion, Haimanmixed insertion

trac #22922 Faster implementation of LLT polynomials

Karnofsky–Rhodes and McCammond expansion of a rootedgraph

your wish?

https://trac.sagemath.org/ticket/23560



9/9


Wishlist

trac #23560 Benkart-Kang-Kashiwara crystals for super Liealgebras

trac #22921 Shifted tableaux, Krazkiewicz insertion, Haimanmixed insertion

trac #22922 Faster implementation of LLT polynomials

Karnofsky–Rhodes and McCammond expansion of a rootedgraph

your wish?




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Anne Schilling, UC Davisquestions in representation theory and mathematical physics by combinatorial...

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