Sage Days 30 Coding Sprint Projects

For the main SD 30 wiki page go here

Below a list of proposed projects.

Combinatorics

* Implement the full collection of d-complete posets and slant product of d-complete posets, and jeu de taquin for them. Implement jeu de taquin for increasing tableaux (for d-complete posets or something less general). For background, see this paper.

• (Hugh, Eric, Ryan, JP, Calin, Josh, Gaelan, Francis, Florent)

* Student project: iterator over LR-tableaux of fixed shape, as in Anders's talk (Hugh)

* Student project: implement double Schubert and double Grothendieck polynomials (Hugh)

* Finalize posets

• Christian, Franco, Nicolas, Neil, Carolina

* Set factories

• Jason, Florent, Travis, Nicolas, Anne, Franco

* Finalize Mike's permutation group patches

• Rob, Nicolas

* Get the reflection groups / Coxeter groups into a proper state

• Christian, (Nicolas, Anne ?)

* Implementation of energy function of crystals (suitable for an interested student!!!)

• Anne, Travis

* Implementation of cyclic tableaux

• Anne

* Quotient rings of rings of symmetric functions, examples of noncommutative Schur functions, etc.

• Jason, Anne, Chris

* Actions on combinatorial free modules

• Hugh, Anne, Franco, Nicolas, Florent, Chris

* Implementation of bijection between crystal paths and rigged configurations

• Travis, Anne

* Polyhedral representation of symmetries

• Aaron

* Partitions related to k-Schur functions

• Carolina, Jason

* (Something with poset partitions?)

• Holly

Number Theory

* Update IntegerVectors internal representation in Sage [l^p norms: norm(p='sage.rings.integer.Integer(2)')].

• Eva, Nicolas, Florent, Jordi

  Goal: Want to be able to enumerate n-tuples with l^p norm less than a given value.

  INPUT: n = dimension, p = norm to use, s = max size of l^p norm for vectors.

  OUTPUT: enumeration of vectors in ZZⁿ (or n-tuples) with l^p norm <= s.

  > Related stuff already in sage:

  • - cartesian_product_iterator (in sage.misc.mrange) enumerates n-tuples with entries in given lists (given per entry), so can be used to enumerate vectors in ZZⁿ with l^infinity norm <= s

    - Composition (in sage.combinat.composition) enumerates all vectors of length between 1 and n with nonnegative integer entries with l¹ norm <= s for a given value of s; relies on IntegerVectors?

    - norm in class sage.modules.free_module_element returns the l^p norm of a vector in ZZⁿ for given n and p >= 1, including p = infinity (p='sage.rings.integer.Integer(2)'), (p=2 is default)

* Update routines for enumerating lattice points in a polytope to help me find digit sets:

• -> One possible route: allow half-open polytopes, that include certain faces and exclude others

  -> Another possible route: allow vertices to be non-lattice points, so that I can shrink my polytope (which is a nice parallelopiped) by epsilon on the faces that I want to exclude

* Student project: Generate centered digit set for multidimensional radix representation

• Input: a dilation matrix A (nxn integer matrix, all of whose eigenvalues have modulus > 1). Output: a centered digit set D = {d_1, ..., d_r}; here r = |det A|. The centered digit set is a complete set of coset representatives of Zⁿ/A(Zⁿ), chosen to be the integer vectors contained in the set/polytope A((-1/2,1/2]ⁿ). Josh

* Student project: Generate minimum modulus digit set for multidimensional radix representation

• Input: a dilation matrix A. Output: a digit set D = {d_1, ..., d_r} consisting of the representatives of each coset of Zⁿ/A(Zⁿ) with minimum l² norm.

  - Method 1: Use algorithm for centered digit set D_0 to find an initial set of coset representatives. Find the smallest l² norm of any k in Zⁿ that is not in the initial digit set D_0. Search vectors k in Zⁿ of expanding l² norm: check the l² norm of the current representative for the coset that k is in, and replace that representative with k (in the digit list/set D) if k has smaller norm. Note that this is a finite search.

  - Method 2: Use LLL algorithm to find minimal basis for lattice A(Zⁿ). Then take D to be the set of Zⁿ lattice points (integer vectors) in the set/polytope A((-e_1/2,e_1/2] x ... x (-e_n/2,e_n/2]), where e_1, ..., e_n are the LLL basis vectors for A(Zⁿ).

* Student project: Visualizing attractors of iterated function systems and other fractal sets

• Review/search for graphics packages that are currently usable by Sage, as well as other open-source options for drawing fractals that are available. Find the best one for visualizing fractals, specifically fractals arising from iterated function systems, specifically iterated functions systems arising from multidimensional radix representations (sums of terms of the form Aⁿ d_n, where A is an nxn integer matrix all of whose eigenvalues have modulus > 1, and the d_n are "digit" vectors drawn from a complete set of coset representatives of Zⁿ/A(Zⁿ); for each digit d, we can define a function f_d = A^-1(x+d)). Good qualities in a graphics package include: ability to generate images from batches of examples automatically generated in Sage, ability to focus on areas of interest in the image and zoom in, ability to display both 2D and 3D images, and ability to rotate 3D images for different views. (Franco to advise)

* Potentially suitable as student project?: Implement Scheicher & Thuswaldner neighbor-finding algorithm

• Input: a dilation matrix A (nxn integer matrix, all of whose eigenvalues have modulus > 1) and digit set D = {d_1, ..., d_r} (set of integer vectors that comprise a complete set of coset representatives of Zⁿ/A(Zⁿ); here r = |det A|). Output: a list of integer vectors that give the neighbor translates of T(A,D), the set of "decimals" under the multidimensional radix representation with base A and digit set D. See Scheicher and Thuswaldner's paper.

Documentation

* Discussion thematic tutorials

• Jason, Anne, Franco, Rob, Nicolas, Florent, Jeff

* Introduction to Sage book

• Hugh, Rob, Nicolas, Jeff