Implement k-split basis for kBoundedSubspace of symmetric functions

[zabrocki]

The k-split basis of the subspace of symmetric functions is defined using the Hall-Littlewood creation operators. If lambda = mu + nu where lambda, mu and nu are lists representing k bounded partitions and mu has largest hook equal to k, then ksplit(nu).hl_creation_operator(mu) == ksplit(lambda).

This ticket implements the ksplit functions as a basis of the kBoundedSubspace. It is accessed from the symmetric functions either by

sage: SymmetricFunctions(QQ['t'].fraction_field()).kBoundedSubspace(3).ksplit()
3-bounded Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 3-split basis

or by the shortcut

sage: SymmetricFunctions(QQ['t'].fraction_field()).ksplit(3)

In either case, the value of t is specified as an optional argument whenever k is specified and the default value of this argument is 't' which needs to be an element of the ring if it is not specified.

A few minor changes are made to the other kBoundedSubspace bases (addition of a shortcut to access k and t in the basis, a minor change in the full name of the k-Schur basis to make it more consistent).

CC: @anneschilling

Component: combinatorics

Keywords: combinat, sf

Author: Mike Zabrocki

Branch/Commit: 011ff3f

Reviewer: Anne Schilling

Issue created by migration from https://trac.sagemath.org/ticket/22105

[anneschilling]

Reviewer: Anne Schilling

[anneschilling]

Changed author from zabrocki to Mike Zabrocki

[anneschilling]

comment:2

Looks great! Thank you, Mike, for implementing this! I ran the tests and also tested the code on some examples. It seems to work as desired.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. New commits:

011ff3f

missing double colon in sf.py

[sagetrac-git]

Changed commit from 06edc02 to 011ff3f

[zabrocki]

comment:4

Sorry about that. I was reviewing the html doc strings and noticed this missing ::

[vbraun]

Changed branch from public/sf/ksplit to 011ff3f